index
int64 0
731k
| package
stringlengths 2
98
⌀ | name
stringlengths 1
76
| docstring
stringlengths 0
281k
⌀ | code
stringlengths 4
1.07M
⌀ | signature
stringlengths 2
42.8k
⌀ |
|---|---|---|---|---|---|
30,421 | networkx.algorithms.traversal.breadth_first_search | bfs_edges | Iterate over edges in a breadth-first-search starting at source.
Parameters
----------
G : NetworkX graph
source : node
Specify starting node for breadth-first search; this function
iterates over only those edges in the component reachable from
this node.
reverse : bool, optional
If True traverse a directed graph in the reverse direction
depth_limit : int, optional(default=len(G))
Specify the maximum search depth
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Yields
------
edge: 2-tuple of nodes
Yields edges resulting from the breadth-first search.
Examples
--------
To get the edges in a breadth-first search::
>>> G = nx.path_graph(3)
>>> list(nx.bfs_edges(G, 0))
[(0, 1), (1, 2)]
>>> list(nx.bfs_edges(G, source=0, depth_limit=1))
[(0, 1)]
To get the nodes in a breadth-first search order::
>>> G = nx.path_graph(3)
>>> root = 2
>>> edges = nx.bfs_edges(G, root)
>>> nodes = [root] + [v for u, v in edges]
>>> nodes
[2, 1, 0]
Notes
-----
The naming of this function is very similar to
:func:`~networkx.algorithms.traversal.edgebfs.edge_bfs`. The difference
is that ``edge_bfs`` yields edges even if they extend back to an already
explored node while this generator yields the edges of the tree that results
from a breadth-first-search (BFS) so no edges are reported if they extend
to already explored nodes. That means ``edge_bfs`` reports all edges while
``bfs_edges`` only reports those traversed by a node-based BFS. Yet another
description is that ``bfs_edges`` reports the edges traversed during BFS
while ``edge_bfs`` reports all edges in the order they are explored.
Based on the breadth-first search implementation in PADS [1]_
by D. Eppstein, July 2004; with modifications to allow depth limits
as described in [2]_.
References
----------
.. [1] http://www.ics.uci.edu/~eppstein/PADS/BFS.py.
.. [2] https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
bfs_tree
:func:`~networkx.algorithms.traversal.depth_first_search.dfs_edges`
:func:`~networkx.algorithms.traversal.edgebfs.edge_bfs`
| null | (G, source, reverse=False, depth_limit=None, sort_neighbors=None, *, backend=None, **backend_kwargs) |
30,422 | networkx.algorithms.traversal.breadth_first_search | bfs_labeled_edges | Iterate over edges in a breadth-first search (BFS) labeled by type.
We generate triple of the form (*u*, *v*, *d*), where (*u*, *v*) is the
edge being explored in the breadth-first search and *d* is one of the
strings 'tree', 'forward', 'level', or 'reverse'. A 'tree' edge is one in
which *v* is first discovered and placed into the layer below *u*. A
'forward' edge is one in which *u* is on the layer above *v* and *v* has
already been discovered. A 'level' edge is one in which both *u* and *v*
occur on the same layer. A 'reverse' edge is one in which *u* is on a layer
below *v*.
We emit each edge exactly once. In an undirected graph, 'reverse' edges do
not occur, because each is discovered either as a 'tree' or 'forward' edge.
Parameters
----------
G : NetworkX graph
A graph over which to find the layers using breadth-first search.
sources : node in `G` or list of nodes in `G`
Starting nodes for single source or multiple sources breadth-first search
Yields
------
edges: generator
A generator of triples (*u*, *v*, *d*) where (*u*, *v*) is the edge being
explored and *d* is described above.
Examples
--------
>>> G = nx.cycle_graph(4, create_using=nx.DiGraph)
>>> list(nx.bfs_labeled_edges(G, 0))
[(0, 1, 'tree'), (1, 2, 'tree'), (2, 3, 'tree'), (3, 0, 'reverse')]
>>> G = nx.complete_graph(3)
>>> list(nx.bfs_labeled_edges(G, 0))
[(0, 1, 'tree'), (0, 2, 'tree'), (1, 2, 'level')]
>>> list(nx.bfs_labeled_edges(G, [0, 1]))
[(0, 1, 'level'), (0, 2, 'tree'), (1, 2, 'forward')]
| null | (G, sources, *, backend=None, **backend_kwargs) |
30,423 | networkx.algorithms.traversal.breadth_first_search | bfs_layers | Returns an iterator of all the layers in breadth-first search traversal.
Parameters
----------
G : NetworkX graph
A graph over which to find the layers using breadth-first search.
sources : node in `G` or list of nodes in `G`
Specify starting nodes for single source or multiple sources breadth-first search
Yields
------
layer: list of nodes
Yields list of nodes at the same distance from sources
Examples
--------
>>> G = nx.path_graph(5)
>>> dict(enumerate(nx.bfs_layers(G, [0, 4])))
{0: [0, 4], 1: [1, 3], 2: [2]}
>>> H = nx.Graph()
>>> H.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
>>> dict(enumerate(nx.bfs_layers(H, [1])))
{0: [1], 1: [0, 3, 4], 2: [2], 3: [5, 6]}
>>> dict(enumerate(nx.bfs_layers(H, [1, 6])))
{0: [1, 6], 1: [0, 3, 4, 2], 2: [5]}
| null | (G, sources, *, backend=None, **backend_kwargs) |
30,424 | networkx.drawing.layout | bfs_layout | Position nodes according to breadth-first search algorithm.
Parameters
----------
G : NetworkX graph
A position will be assigned to every node in G.
start : node in `G`
Starting node for bfs
center : array-like or None
Coordinate pair around which to center the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.bfs_layout(G, 0)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
| def bfs_layout(G, start, *, align="vertical", scale=1, center=None):
"""Position nodes according to breadth-first search algorithm.
Parameters
----------
G : NetworkX graph
A position will be assigned to every node in G.
start : node in `G`
Starting node for bfs
center : array-like or None
Coordinate pair around which to center the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.bfs_layout(G, 0)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
"""
G, center = _process_params(G, center, 2)
# Compute layers with BFS
layers = dict(enumerate(nx.bfs_layers(G, start)))
if len(G) != sum(len(nodes) for nodes in layers.values()):
raise nx.NetworkXError(
"bfs_layout didn't include all nodes. Perhaps use input graph:\n"
" G.subgraph(nx.node_connected_component(G, start))"
)
# Compute node positions with multipartite_layout
return multipartite_layout(
G, subset_key=layers, align=align, scale=scale, center=center
)
| (G, start, *, align='vertical', scale=1, center=None) |
30,425 | networkx.algorithms.traversal.breadth_first_search | bfs_predecessors | Returns an iterator of predecessors in breadth-first-search from source.
Parameters
----------
G : NetworkX graph
source : node
Specify starting node for breadth-first search
depth_limit : int, optional(default=len(G))
Specify the maximum search depth
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
pred: iterator
(node, predecessor) iterator where `predecessor` is the predecessor of
`node` in a breadth first search starting from `source`.
Examples
--------
>>> G = nx.path_graph(3)
>>> dict(nx.bfs_predecessors(G, 0))
{1: 0, 2: 1}
>>> H = nx.Graph()
>>> H.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
>>> dict(nx.bfs_predecessors(H, 0))
{1: 0, 2: 0, 3: 1, 4: 1, 5: 2, 6: 2}
>>> M = nx.Graph()
>>> nx.add_path(M, [0, 1, 2, 3, 4, 5, 6])
>>> nx.add_path(M, [2, 7, 8, 9, 10])
>>> sorted(nx.bfs_predecessors(M, source=1, depth_limit=3))
[(0, 1), (2, 1), (3, 2), (4, 3), (7, 2), (8, 7)]
>>> N = nx.DiGraph()
>>> nx.add_path(N, [0, 1, 2, 3, 4, 7])
>>> nx.add_path(N, [3, 5, 6, 7])
>>> sorted(nx.bfs_predecessors(N, source=2))
[(3, 2), (4, 3), (5, 3), (6, 5), (7, 4)]
Notes
-----
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py
by D. Eppstein, July 2004. The modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited-search`_".
.. _Depth-limited-search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
bfs_tree
bfs_edges
edge_bfs
| null | (G, source, depth_limit=None, sort_neighbors=None, *, backend=None, **backend_kwargs) |
30,426 | networkx.algorithms.traversal.breadth_first_search | bfs_successors | Returns an iterator of successors in breadth-first-search from source.
Parameters
----------
G : NetworkX graph
source : node
Specify starting node for breadth-first search
depth_limit : int, optional(default=len(G))
Specify the maximum search depth
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
succ: iterator
(node, successors) iterator where `successors` is the non-empty list of
successors of `node` in a breadth first search from `source`.
To appear in the iterator, `node` must have successors.
Examples
--------
>>> G = nx.path_graph(3)
>>> dict(nx.bfs_successors(G, 0))
{0: [1], 1: [2]}
>>> H = nx.Graph()
>>> H.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
>>> dict(nx.bfs_successors(H, 0))
{0: [1, 2], 1: [3, 4], 2: [5, 6]}
>>> G = nx.Graph()
>>> nx.add_path(G, [0, 1, 2, 3, 4, 5, 6])
>>> nx.add_path(G, [2, 7, 8, 9, 10])
>>> dict(nx.bfs_successors(G, source=1, depth_limit=3))
{1: [0, 2], 2: [3, 7], 3: [4], 7: [8]}
>>> G = nx.DiGraph()
>>> nx.add_path(G, [0, 1, 2, 3, 4, 5])
>>> dict(nx.bfs_successors(G, source=3))
{3: [4], 4: [5]}
Notes
-----
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py
by D. Eppstein, July 2004.The modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited-search`_".
.. _Depth-limited-search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
bfs_tree
bfs_edges
edge_bfs
| null | (G, source, depth_limit=None, sort_neighbors=None, *, backend=None, **backend_kwargs) |
30,427 | networkx.algorithms.traversal.breadth_first_search | bfs_tree | Returns an oriented tree constructed from of a breadth-first-search
starting at source.
Parameters
----------
G : NetworkX graph
source : node
Specify starting node for breadth-first search
reverse : bool, optional
If True traverse a directed graph in the reverse direction
depth_limit : int, optional(default=len(G))
Specify the maximum search depth
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
T: NetworkX DiGraph
An oriented tree
Examples
--------
>>> G = nx.path_graph(3)
>>> list(nx.bfs_tree(G, 1).edges())
[(1, 0), (1, 2)]
>>> H = nx.Graph()
>>> nx.add_path(H, [0, 1, 2, 3, 4, 5, 6])
>>> nx.add_path(H, [2, 7, 8, 9, 10])
>>> sorted(list(nx.bfs_tree(H, source=3, depth_limit=3).edges()))
[(1, 0), (2, 1), (2, 7), (3, 2), (3, 4), (4, 5), (5, 6), (7, 8)]
Notes
-----
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py
by D. Eppstein, July 2004. The modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited-search`_".
.. _Depth-limited-search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
dfs_tree
bfs_edges
edge_bfs
| null | (G, source, reverse=False, depth_limit=None, sort_neighbors=None, *, backend=None, **backend_kwargs) |
30,429 | networkx.algorithms.components.biconnected | biconnected_component_edges | Returns a generator of lists of edges, one list for each biconnected
component of the input graph.
Biconnected components are maximal subgraphs such that the removal of a
node (and all edges incident on that node) will not disconnect the
subgraph. Note that nodes may be part of more than one biconnected
component. Those nodes are articulation points, or cut vertices.
However, each edge belongs to one, and only one, biconnected component.
Notice that by convention a dyad is considered a biconnected component.
Parameters
----------
G : NetworkX Graph
An undirected graph.
Returns
-------
edges : generator of lists
Generator of lists of edges, one list for each bicomponent.
Raises
------
NetworkXNotImplemented
If the input graph is not undirected.
Examples
--------
>>> G = nx.barbell_graph(4, 2)
>>> print(nx.is_biconnected(G))
False
>>> bicomponents_edges = list(nx.biconnected_component_edges(G))
>>> len(bicomponents_edges)
5
>>> G.add_edge(2, 8)
>>> print(nx.is_biconnected(G))
True
>>> bicomponents_edges = list(nx.biconnected_component_edges(G))
>>> len(bicomponents_edges)
1
See Also
--------
is_biconnected,
biconnected_components,
articulation_points,
Notes
-----
The algorithm to find articulation points and biconnected
components is implemented using a non-recursive depth-first-search
(DFS) that keeps track of the highest level that back edges reach
in the DFS tree. A node `n` is an articulation point if, and only
if, there exists a subtree rooted at `n` such that there is no
back edge from any successor of `n` that links to a predecessor of
`n` in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all
edges of a bicomponent will be traversed consecutively between
articulation points.
References
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
| null | (G, *, backend=None, **backend_kwargs) |
30,430 | networkx.algorithms.components.biconnected | biconnected_components | Returns a generator of sets of nodes, one set for each biconnected
component of the graph
Biconnected components are maximal subgraphs such that the removal of a
node (and all edges incident on that node) will not disconnect the
subgraph. Note that nodes may be part of more than one biconnected
component. Those nodes are articulation points, or cut vertices. The
removal of articulation points will increase the number of connected
components of the graph.
Notice that by convention a dyad is considered a biconnected component.
Parameters
----------
G : NetworkX Graph
An undirected graph.
Returns
-------
nodes : generator
Generator of sets of nodes, one set for each biconnected component.
Raises
------
NetworkXNotImplemented
If the input graph is not undirected.
Examples
--------
>>> G = nx.lollipop_graph(5, 1)
>>> print(nx.is_biconnected(G))
False
>>> bicomponents = list(nx.biconnected_components(G))
>>> len(bicomponents)
2
>>> G.add_edge(0, 5)
>>> print(nx.is_biconnected(G))
True
>>> bicomponents = list(nx.biconnected_components(G))
>>> len(bicomponents)
1
You can generate a sorted list of biconnected components, largest
first, using sort.
>>> G.remove_edge(0, 5)
>>> [len(c) for c in sorted(nx.biconnected_components(G), key=len, reverse=True)]
[5, 2]
If you only want the largest connected component, it's more
efficient to use max instead of sort.
>>> Gc = max(nx.biconnected_components(G), key=len)
To create the components as subgraphs use:
``(G.subgraph(c).copy() for c in biconnected_components(G))``
See Also
--------
is_biconnected
articulation_points
biconnected_component_edges
k_components : this function is a special case where k=2
bridge_components : similar to this function, but is defined using
2-edge-connectivity instead of 2-node-connectivity.
Notes
-----
The algorithm to find articulation points and biconnected
components is implemented using a non-recursive depth-first-search
(DFS) that keeps track of the highest level that back edges reach
in the DFS tree. A node `n` is an articulation point if, and only
if, there exists a subtree rooted at `n` such that there is no
back edge from any successor of `n` that links to a predecessor of
`n` in the DFS tree. By keeping track of all the edges traversed
by the DFS we can obtain the biconnected components because all
edges of a bicomponent will be traversed consecutively between
articulation points.
References
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
| null | (G, *, backend=None, **backend_kwargs) |
30,431 | networkx.algorithms.shortest_paths.weighted | bidirectional_dijkstra | Dijkstra's algorithm for shortest paths using bidirectional search.
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number or None to indicate a hidden edge.
Returns
-------
length, path : number and list
length is the distance from source to target.
path is a list of nodes on a path from source to target.
Raises
------
NodeNotFound
If either `source` or `target` is not in `G`.
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G = nx.path_graph(5)
>>> length, path = nx.bidirectional_dijkstra(G, 0, 4)
>>> print(length)
4
>>> print(path)
[0, 1, 2, 3, 4]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The weight function can be used to hide edges by returning None.
So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``
will find the shortest red path.
In practice bidirectional Dijkstra is much more than twice as fast as
ordinary Dijkstra.
Ordinary Dijkstra expands nodes in a sphere-like manner from the
source. The radius of this sphere will eventually be the length
of the shortest path. Bidirectional Dijkstra will expand nodes
from both the source and the target, making two spheres of half
this radius. Volume of the first sphere is `\pi*r*r` while the
others are `2*\pi*r/2*r/2`, making up half the volume.
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
shortest_path
shortest_path_length
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
| (G, source, target, weight='weight', *, backend=None, **backend_kwargs) |
30,432 | networkx.algorithms.shortest_paths.unweighted | bidirectional_shortest_path | Returns a list of nodes in a shortest path between source and target.
Parameters
----------
G : NetworkX graph
source : node label
starting node for path
target : node label
ending node for path
Returns
-------
path: list
List of nodes in a path from source to target.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Examples
--------
>>> G = nx.Graph()
>>> nx.add_path(G, [0, 1, 2, 3, 0, 4, 5, 6, 7, 4])
>>> nx.bidirectional_shortest_path(G, 2, 6)
[2, 1, 0, 4, 5, 6]
See Also
--------
shortest_path
Notes
-----
This algorithm is used by shortest_path(G, source, target).
| null | (G, source, target, *, backend=None, **backend_kwargs) |
30,434 | networkx.generators.random_graphs | gnp_random_graph | Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph
or a binomial graph.
The $G_{n,p}$ model chooses each of the possible edges with probability $p$.
Parameters
----------
n : int
The number of nodes.
p : float
Probability for edge creation.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
directed : bool, optional (default=False)
If True, this function returns a directed graph.
See Also
--------
fast_gnp_random_graph
Notes
-----
This algorithm [2]_ runs in $O(n^2)$ time. For sparse graphs (that is, for
small values of $p$), :func:`fast_gnp_random_graph` is a faster algorithm.
:func:`binomial_graph` and :func:`erdos_renyi_graph` are
aliases for :func:`gnp_random_graph`.
>>> nx.binomial_graph is nx.gnp_random_graph
True
>>> nx.erdos_renyi_graph is nx.gnp_random_graph
True
References
----------
.. [1] P. Erdős and A. Rényi, On Random Graphs, Publ. Math. 6, 290 (1959).
.. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
| def dual_barabasi_albert_graph(n, m1, m2, p, seed=None, initial_graph=None):
"""Returns a random graph using dual Barabási–Albert preferential attachment
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
are preferentially attached to existing nodes with high degree.
Parameters
----------
n : int
Number of nodes
m1 : int
Number of edges to link each new node to existing nodes with probability $p$
m2 : int
Number of edges to link each new node to existing nodes with probability $1-p$
p : float
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : Graph or None (default)
Initial network for Barabási–Albert algorithm.
A copy of `initial_graph` is used.
It should be connected for most use cases.
If None, starts from an star graph on max(m1, m2) + 1 nodes.
Returns
-------
G : Graph
Raises
------
NetworkXError
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
`p` does not satisfy ``0 <= p <= 1``, or
the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.
References
----------
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
"""
if m1 < 1 or m1 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
)
if m2 < 1 or m2 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
)
if p < 0 or p > 1:
raise nx.NetworkXError(
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
)
# For simplicity, if p == 0 or 1, just return BA
if p == 1:
return barabasi_albert_graph(n, m1, seed)
elif p == 0:
return barabasi_albert_graph(n, m2, seed)
if initial_graph is None:
# Default initial graph : empty graph on max(m1, m2) nodes
G = star_graph(max(m1, m2))
else:
if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
raise nx.NetworkXError(
f"Barabási–Albert initial graph must have between "
f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
)
G = initial_graph.copy()
# Target nodes for new edges
targets = list(G)
# List of existing nodes, with nodes repeated once for each adjacent edge
repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
# Start adding the remaining nodes.
source = len(G)
while source < n:
# Pick which m to use (m1 or m2)
if seed.random() < p:
m = m1
else:
m = m2
# Now choose m unique nodes from the existing nodes
# Pick uniformly from repeated_nodes (preferential attachment)
targets = _random_subset(repeated_nodes, m, seed)
# Add edges to m nodes from the source.
G.add_edges_from(zip([source] * m, targets))
# Add one node to the list for each new edge just created.
repeated_nodes.extend(targets)
# And the new node "source" has m edges to add to the list.
repeated_nodes.extend([source] * m)
source += 1
return G
| (n, p, seed=None, directed=False, *, backend=None, **backend_kwargs) |
30,435 | networkx.generators.classic | binomial_tree | Returns the Binomial Tree of order n.
The binomial tree of order 0 consists of a single node. A binomial tree of order k
is defined recursively by linking two binomial trees of order k-1: the root of one is
the leftmost child of the root of the other.
.. plot::
>>> nx.draw(nx.binomial_tree(3))
Parameters
----------
n : int
Order of the binomial tree.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : NetworkX graph
A binomial tree of $2^n$ nodes and $2^n - 1$ edges.
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, create_using=None, *, backend=None, **backend_kwargs) |
30,437 | networkx.drawing.layout | bipartite_layout | Position nodes in two straight lines.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
nodes : list or container
Nodes in one node set of the bipartite graph.
This set will be placed on left or top.
align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
aspect_ratio : number (default=4/3):
The ratio of the width to the height of the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.bipartite.gnmk_random_graph(3, 5, 10, seed=123)
>>> top = nx.bipartite.sets(G)[0]
>>> pos = nx.bipartite_layout(G, top)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
| def bipartite_layout(
G, nodes, align="vertical", scale=1, center=None, aspect_ratio=4 / 3
):
"""Position nodes in two straight lines.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
nodes : list or container
Nodes in one node set of the bipartite graph.
This set will be placed on left or top.
align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
aspect_ratio : number (default=4/3):
The ratio of the width to the height of the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.bipartite.gnmk_random_graph(3, 5, 10, seed=123)
>>> top = nx.bipartite.sets(G)[0]
>>> pos = nx.bipartite_layout(G, top)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
"""
import numpy as np
if align not in ("vertical", "horizontal"):
msg = "align must be either vertical or horizontal."
raise ValueError(msg)
G, center = _process_params(G, center=center, dim=2)
if len(G) == 0:
return {}
height = 1
width = aspect_ratio * height
offset = (width / 2, height / 2)
top = dict.fromkeys(nodes)
bottom = [v for v in G if v not in top]
nodes = list(top) + bottom
left_xs = np.repeat(0, len(top))
right_xs = np.repeat(width, len(bottom))
left_ys = np.linspace(0, height, len(top))
right_ys = np.linspace(0, height, len(bottom))
top_pos = np.column_stack([left_xs, left_ys]) - offset
bottom_pos = np.column_stack([right_xs, right_ys]) - offset
pos = np.concatenate([top_pos, bottom_pos])
pos = rescale_layout(pos, scale=scale) + center
if align == "horizontal":
pos = pos[:, ::-1] # swap x and y coords
pos = dict(zip(nodes, pos))
return pos
| (G, nodes, align='vertical', scale=1, center=None, aspect_ratio=1.3333333333333333) |
30,439 | networkx.algorithms.cuts | boundary_expansion | Returns the boundary expansion of the set `S`.
The *boundary expansion* is the quotient of the size
of the node boundary and the cardinality of *S*. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
Returns
-------
number
The boundary expansion of the set `S`.
See also
--------
edge_expansion
mixing_expansion
node_expansion
References
----------
.. [1] Vadhan, Salil P.
"Pseudorandomness."
*Foundations and Trends in Theoretical Computer Science*
7.1–3 (2011): 1–336.
<https://doi.org/10.1561/0400000010>
| null | (G, S, *, backend=None, **backend_kwargs) |
30,441 | networkx.algorithms.bridges | bridges | Generate all bridges in a graph.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase. Equivalently, a bridge is an
edge that does not belong to any cycle. Bridges are also known as cut-edges,
isthmuses, or cut arcs.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be returned.
Yields
------
e : edge
An edge in the graph whose removal disconnects the graph (or
causes the number of connected components to increase).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
NetworkXNotImplemented
If `G` is a directed graph.
Examples
--------
The barbell graph with parameter zero has a single bridge:
>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]
Notes
-----
This is an implementation of the algorithm described in [1]_. An edge is a
bridge if and only if it is not contained in any chain. Chains are found
using the :func:`networkx.chain_decomposition` function.
The algorithm described in [1]_ requires a simple graph. If the provided
graph is a multigraph, we convert it to a simple graph and verify that any
bridges discovered by the chain decomposition algorithm are not multi-edges.
Ignoring polylogarithmic factors, the worst-case time complexity is the
same as the :func:`networkx.chain_decomposition` function,
$O(m + n)$, where $n$ is the number of nodes in the graph and $m$ is
the number of edges.
References
----------
.. [1] https://en.wikipedia.org/wiki/Bridge_%28graph_theory%29#Bridge-Finding_with_Chain_Decompositions
| null | (G, root=None, *, backend=None, **backend_kwargs) |
30,443 | networkx.generators.small | bull_graph |
Returns the Bull Graph
The Bull Graph has 5 nodes and 5 edges. It is a planar undirected
graph in the form of a triangle with two disjoint pendant edges [1]_
The name comes from the triangle and pendant edges representing
respectively the body and legs of a bull.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
A bull graph with 5 nodes
References
----------
.. [1] https://en.wikipedia.org/wiki/Bull_graph.
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) |
30,444 | networkx.algorithms.flow.capacityscaling | capacity_scaling | Find a minimum cost flow satisfying all demands in digraph G.
This is a capacity scaling successive shortest augmenting path algorithm.
G is a digraph with edge costs and capacities and in which nodes
have demand, i.e., they want to send or receive some amount of
flow. A negative demand means that the node wants to send flow, a
positive demand means that the node want to receive flow. A flow on
the digraph G satisfies all demand if the net flow into each node
is equal to the demand of that node.
Parameters
----------
G : NetworkX graph
DiGraph or MultiDiGraph on which a minimum cost flow satisfying all
demands is to be found.
demand : string
Nodes of the graph G are expected to have an attribute demand
that indicates how much flow a node wants to send (negative
demand) or receive (positive demand). Note that the sum of the
demands should be 0 otherwise the problem in not feasible. If
this attribute is not present, a node is considered to have 0
demand. Default value: 'demand'.
capacity : string
Edges of the graph G are expected to have an attribute capacity
that indicates how much flow the edge can support. If this
attribute is not present, the edge is considered to have
infinite capacity. Default value: 'capacity'.
weight : string
Edges of the graph G are expected to have an attribute weight
that indicates the cost incurred by sending one unit of flow on
that edge. If not present, the weight is considered to be 0.
Default value: 'weight'.
heap : class
Type of heap to be used in the algorithm. It should be a subclass of
:class:`MinHeap` or implement a compatible interface.
If a stock heap implementation is to be used, :class:`BinaryHeap` is
recommended over :class:`PairingHeap` for Python implementations without
optimized attribute accesses (e.g., CPython) despite a slower
asymptotic running time. For Python implementations with optimized
attribute accesses (e.g., PyPy), :class:`PairingHeap` provides better
performance. Default value: :class:`BinaryHeap`.
Returns
-------
flowCost : integer
Cost of a minimum cost flow satisfying all demands.
flowDict : dictionary
If G is a digraph, a dict-of-dicts keyed by nodes such that
flowDict[u][v] is the flow on edge (u, v).
If G is a MultiDiGraph, a dict-of-dicts-of-dicts keyed by nodes
so that flowDict[u][v][key] is the flow on edge (u, v, key).
Raises
------
NetworkXError
This exception is raised if the input graph is not directed,
not connected.
NetworkXUnfeasible
This exception is raised in the following situations:
* The sum of the demands is not zero. Then, there is no
flow satisfying all demands.
* There is no flow satisfying all demand.
NetworkXUnbounded
This exception is raised if the digraph G has a cycle of
negative cost and infinite capacity. Then, the cost of a flow
satisfying all demands is unbounded below.
Notes
-----
This algorithm does not work if edge weights are floating-point numbers.
See also
--------
:meth:`network_simplex`
Examples
--------
A simple example of a min cost flow problem.
>>> G = nx.DiGraph()
>>> G.add_node("a", demand=-5)
>>> G.add_node("d", demand=5)
>>> G.add_edge("a", "b", weight=3, capacity=4)
>>> G.add_edge("a", "c", weight=6, capacity=10)
>>> G.add_edge("b", "d", weight=1, capacity=9)
>>> G.add_edge("c", "d", weight=2, capacity=5)
>>> flowCost, flowDict = nx.capacity_scaling(G)
>>> flowCost
24
>>> flowDict
{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
It is possible to change the name of the attributes used for the
algorithm.
>>> G = nx.DiGraph()
>>> G.add_node("p", spam=-4)
>>> G.add_node("q", spam=2)
>>> G.add_node("a", spam=-2)
>>> G.add_node("d", spam=-1)
>>> G.add_node("t", spam=2)
>>> G.add_node("w", spam=3)
>>> G.add_edge("p", "q", cost=7, vacancies=5)
>>> G.add_edge("p", "a", cost=1, vacancies=4)
>>> G.add_edge("q", "d", cost=2, vacancies=3)
>>> G.add_edge("t", "q", cost=1, vacancies=2)
>>> G.add_edge("a", "t", cost=2, vacancies=4)
>>> G.add_edge("d", "w", cost=3, vacancies=4)
>>> G.add_edge("t", "w", cost=4, vacancies=1)
>>> flowCost, flowDict = nx.capacity_scaling(
... G, demand="spam", capacity="vacancies", weight="cost"
... )
>>> flowCost
37
>>> flowDict
{'p': {'q': 2, 'a': 2}, 'q': {'d': 1}, 'a': {'t': 4}, 'd': {'w': 2}, 't': {'q': 1, 'w': 1}, 'w': {}}
| null | (G, demand='demand', capacity='capacity', weight='weight', heap=<class 'networkx.utils.heaps.BinaryHeap'>, *, backend=None, **backend_kwargs) |
30,445 | networkx.algorithms.operators.product | cartesian_product | Returns the Cartesian product of G and H.
The Cartesian product $P$ of the graphs $G$ and $H$ has a node set that
is the Cartesian product of the node sets, $V(P)=V(G) \times V(H)$.
$P$ has an edge $((u,v),(x,y))$ if and only if either $u$ is equal to $x$
and both $v$ and $y$ are adjacent in $H$ or if $v$ is equal to $y$ and
both $u$ and $x$ are adjacent in $G$.
Parameters
----------
G, H: graphs
Networkx graphs.
Returns
-------
P: NetworkX graph
The Cartesian product of G and H. P will be a multi-graph if either G
or H is a multi-graph. Will be a directed if G and H are directed,
and undirected if G and H are undirected.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Notes
-----
Node attributes in P are two-tuple of the G and H node attributes.
Missing attributes are assigned None.
Examples
--------
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0, a1=True)
>>> H.add_node("a", a2="Spam")
>>> P = nx.cartesian_product(G, H)
>>> list(P)
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the
new product graph
| null | (G, H, *, backend=None, **backend_kwargs) |
30,446 | networkx.generators.community | caveman_graph | Returns a caveman graph of `l` cliques of size `k`.
Parameters
----------
l : int
Number of cliques
k : int
Size of cliques
Returns
-------
G : NetworkX Graph
caveman graph
Notes
-----
This returns an undirected graph, it can be converted to a directed
graph using :func:`nx.to_directed`, or a multigraph using
``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is
described in [1]_ and it is unclear which of the directed
generalizations is most useful.
Examples
--------
>>> G = nx.caveman_graph(3, 3)
See also
--------
connected_caveman_graph
References
----------
.. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
Amer. J. Soc. 105, 493-527, 1999.
| def _generate_communities(degree_seq, community_sizes, mu, max_iters, seed):
"""Returns a list of sets, each of which represents a community.
``degree_seq`` is the degree sequence that must be met by the
graph.
``community_sizes`` is the community size distribution that must be
met by the generated list of sets.
``mu`` is a float in the interval [0, 1] indicating the fraction of
intra-community edges incident to each node.
``max_iters`` is the number of times to try to add a node to a
community. This must be greater than the length of
``degree_seq``, otherwise this function will always fail. If
the number of iterations exceeds this value,
:exc:`~networkx.exception.ExceededMaxIterations` is raised.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
The communities returned by this are sets of integers in the set {0,
..., *n* - 1}, where *n* is the length of ``degree_seq``.
"""
# This assumes the nodes in the graph will be natural numbers.
result = [set() for _ in community_sizes]
n = len(degree_seq)
free = list(range(n))
for i in range(max_iters):
v = free.pop()
c = seed.choice(range(len(community_sizes)))
# s = int(degree_seq[v] * (1 - mu) + 0.5)
s = round(degree_seq[v] * (1 - mu))
# If the community is large enough, add the node to the chosen
# community. Otherwise, return it to the list of unaffiliated
# nodes.
if s < community_sizes[c]:
result[c].add(v)
else:
free.append(v)
# If the community is too big, remove a node from it.
if len(result[c]) > community_sizes[c]:
free.append(result[c].pop())
if not free:
return result
msg = "Could not assign communities; try increasing min_community"
raise nx.ExceededMaxIterations(msg)
| (l, k, *, backend=None, **backend_kwargs) |
30,447 | networkx.algorithms.time_dependent | cd_index | Compute the CD index for `node` within the graph `G`.
Calculates the CD index for the given node of the graph,
considering only its predecessors who have the `time` attribute
smaller than or equal to the `time` attribute of the `node`
plus `time_delta`.
Parameters
----------
G : graph
A directed networkx graph whose nodes have `time` attributes and optionally
`weight` attributes (if a weight is not given, it is considered 1).
node : node
The node for which the CD index is calculated.
time_delta : numeric or timedelta
Amount of time after the `time` attribute of the `node`. The value of
`time_delta` must support comparison with the `time` node attribute. For
example, if the `time` attribute of the nodes are `datetime.datetime`
objects, then `time_delta` should be a `datetime.timedelta` object.
time : string (Optional, default is "time")
The name of the node attribute that will be used for the calculations.
weight : string (Optional, default is None)
The name of the node attribute used as weight.
Returns
-------
float
The CD index calculated for the node `node` within the graph `G`.
Raises
------
NetworkXError
If not all nodes have a `time` attribute or
`time_delta` and `time` attribute types are not compatible or
`n` equals 0.
NetworkXNotImplemented
If `G` is a non-directed graph or a multigraph.
Examples
--------
>>> from datetime import datetime, timedelta
>>> G = nx.DiGraph()
>>> nodes = {
... 1: {"time": datetime(2015, 1, 1)},
... 2: {"time": datetime(2012, 1, 1), "weight": 4},
... 3: {"time": datetime(2010, 1, 1)},
... 4: {"time": datetime(2008, 1, 1)},
... 5: {"time": datetime(2014, 1, 1)},
... }
>>> G.add_nodes_from([(n, nodes[n]) for n in nodes])
>>> edges = [(1, 3), (1, 4), (2, 3), (3, 4), (3, 5)]
>>> G.add_edges_from(edges)
>>> delta = timedelta(days=5 * 365)
>>> nx.cd_index(G, 3, time_delta=delta, time="time")
0.5
>>> nx.cd_index(G, 3, time_delta=delta, time="time", weight="weight")
0.12
Integers can also be used for the time values:
>>> node_times = {1: 2015, 2: 2012, 3: 2010, 4: 2008, 5: 2014}
>>> nx.set_node_attributes(G, node_times, "new_time")
>>> nx.cd_index(G, 3, time_delta=4, time="new_time")
0.5
>>> nx.cd_index(G, 3, time_delta=4, time="new_time", weight="weight")
0.12
Notes
-----
This method implements the algorithm for calculating the CD index,
as described in the paper by Funk and Owen-Smith [1]_. The CD index
is used in order to check how consolidating or destabilizing a patent
is, hence the nodes of the graph represent patents and the edges show
the citations between these patents. The mathematical model is given
below:
.. math::
CD_{t}=\frac{1}{n_{t}}\sum_{i=1}^{n}\frac{-2f_{it}b_{it}+f_{it}}{w_{it}},
where `f_{it}` equals 1 if `i` cites the focal patent else 0, `b_{it}` equals
1 if `i` cites any of the focal patents successors else 0, `n_{t}` is the number
of forward citations in `i` and `w_{it}` is a matrix of weight for patent `i`
at time `t`.
The `datetime.timedelta` package can lead to off-by-one issues when converting
from years to days. In the example above `timedelta(days=5 * 365)` looks like
5 years, but it isn't because of leap year days. So it gives the same result
as `timedelta(days=4 * 365)`. But using `timedelta(days=5 * 365 + 1)` gives
a 5 year delta **for this choice of years** but may not if the 5 year gap has
more than 1 leap year. To avoid these issues, use integers to represent years,
or be very careful when you convert units of time.
References
----------
.. [1] Funk, Russell J., and Jason Owen-Smith.
"A dynamic network measure of technological change."
Management science 63, no. 3 (2017): 791-817.
http://russellfunk.org/cdindex/static/papers/funk_ms_2017.pdf
| null | (G, node, time_delta, *, time='time', weight=None, backend=None, **backend_kwargs) |
30,448 | networkx.algorithms.distance_measures | center | Returns the center of the graph G.
The center is the set of nodes with eccentricity equal to radius.
Parameters
----------
G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
weight : string, function, or None
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
If this is None, every edge has weight/distance/cost 1.
Weights stored as floating point values can lead to small round-off
errors in distances. Use integer weights to avoid this.
Weights should be positive, since they are distances.
Returns
-------
c : list
List of nodes in center
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> list(nx.center(G))
[1, 3, 4]
See Also
--------
barycenter
periphery
| def effective_graph_resistance(G, weight=None, invert_weight=True):
"""Returns the Effective graph resistance of G.
Also known as the Kirchhoff index.
The effective graph resistance is defined as the sum
of the resistance distance of every node pair in G [1]_.
If weight is not provided, then a weight of 1 is used for all edges.
The effective graph resistance of a disconnected graph is infinite.
Parameters
----------
G : NetworkX graph
A graph
weight : string or None, optional (default=None)
The edge data key used to compute the effective graph resistance.
If None, then each edge has weight 1.
invert_weight : boolean (default=True)
Proper calculation of resistance distance requires building the
Laplacian matrix with the reciprocal of the weight. Not required
if the weight is already inverted. Weight cannot be zero.
Returns
-------
RG : float
The effective graph resistance of `G`.
Raises
------
NetworkXNotImplemented
If `G` is a directed graph.
NetworkXError
If `G` does not contain any nodes.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> round(nx.effective_graph_resistance(G), 10)
10.25
Notes
-----
The implementation is based on Theorem 2.2 in [2]_. Self-loops are ignored.
Multi-edges are contracted in one edge with weight equal to the harmonic sum of the weights.
References
----------
.. [1] Wolfram
"Kirchhoff Index."
https://mathworld.wolfram.com/KirchhoffIndex.html
.. [2] W. Ellens, F. M. Spieksma, P. Van Mieghem, A. Jamakovic, R. E. Kooij.
Effective graph resistance.
Lin. Alg. Appl. 435:2491-2506, 2011.
"""
import numpy as np
if len(G) == 0:
raise nx.NetworkXError("Graph G must contain at least one node.")
# Disconnected graphs have infinite Effective graph resistance
if not nx.is_connected(G):
return float("inf")
# Invert weights
G = G.copy()
if invert_weight and weight is not None:
if G.is_multigraph():
for u, v, k, d in G.edges(keys=True, data=True):
d[weight] = 1 / d[weight]
else:
for u, v, d in G.edges(data=True):
d[weight] = 1 / d[weight]
# Get Laplacian eigenvalues
mu = np.sort(nx.laplacian_spectrum(G, weight=weight))
# Compute Effective graph resistance based on spectrum of the Laplacian
# Self-loops are ignored
return float(np.sum(1 / mu[1:]) * G.number_of_nodes())
| (G, e=None, usebounds=False, weight=None, *, backend=None, **backend_kwargs) |
30,450 | networkx.algorithms.chains | chain_decomposition | Returns the chain decomposition of a graph.
The *chain decomposition* of a graph with respect a depth-first
search tree is a set of cycles or paths derived from the set of
fundamental cycles of the tree in the following manner. Consider
each fundamental cycle with respect to the given tree, represented
as a list of edges beginning with the nontree edge oriented away
from the root of the tree. For each fundamental cycle, if it
overlaps with any previous fundamental cycle, just take the initial
non-overlapping segment, which is a path instead of a cycle. Each
cycle or path is called a *chain*. For more information, see [1]_.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the chain
decomposition for the connected component containing this node
will be returned. This node indicates the root of the depth-first
search tree.
Yields
------
chain : list
A list of edges representing a chain. There is no guarantee on
the orientation of the edges in each chain (for example, if a
chain includes the edge joining nodes 1 and 2, the chain may
include either (1, 2) or (2, 1)).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> list(nx.chain_decomposition(G))
[[(4, 5), (5, 3), (3, 4)]]
Notes
-----
The worst-case running time of this implementation is linear in the
number of nodes and number of edges [1]_.
References
----------
.. [1] Jens M. Schmidt (2013). "A simple test on 2-vertex-
and 2-edge-connectivity." *Information Processing Letters*,
113, 241–244. Elsevier. <https://doi.org/10.1016/j.ipl.2013.01.016>
| null | (G, root=None, *, backend=None, **backend_kwargs) |
30,452 | networkx.algorithms.planarity | check_planarity | Check if a graph is planar and return a counterexample or an embedding.
A graph is planar iff it can be drawn in a plane without
any edge intersections.
Parameters
----------
G : NetworkX graph
counterexample : bool
A Kuratowski subgraph (to proof non planarity) is only returned if set
to true.
Returns
-------
(is_planar, certificate) : (bool, NetworkX graph) tuple
is_planar is true if the graph is planar.
If the graph is planar `certificate` is a PlanarEmbedding
otherwise it is a Kuratowski subgraph.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2)])
>>> is_planar, P = nx.check_planarity(G)
>>> print(is_planar)
True
When `G` is planar, a `PlanarEmbedding` instance is returned:
>>> P.get_data()
{0: [1, 2], 1: [0], 2: [0]}
Notes
-----
A (combinatorial) embedding consists of cyclic orderings of the incident
edges at each vertex. Given such an embedding there are multiple approaches
discussed in literature to drawing the graph (subject to various
constraints, e.g. integer coordinates), see e.g. [2].
The planarity check algorithm and extraction of the combinatorial embedding
is based on the Left-Right Planarity Test [1].
A counterexample is only generated if the corresponding parameter is set,
because the complexity of the counterexample generation is higher.
See also
--------
is_planar :
Check for planarity without creating a `PlanarEmbedding` or counterexample.
References
----------
.. [1] Ulrik Brandes:
The Left-Right Planarity Test
2009
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.9208
.. [2] Takao Nishizeki, Md Saidur Rahman:
Planar graph drawing
Lecture Notes Series on Computing: Volume 12
2004
| def sign_recursive(self, e):
"""Recursive version of :meth:`sign`."""
if self.ref[e] is not None:
self.side[e] = self.side[e] * self.sign_recursive(self.ref[e])
self.ref[e] = None
return self.side[e]
| (G, counterexample=False, *, backend=None, **backend_kwargs) |
30,454 | networkx.generators.expanders | chordal_cycle_graph | Returns the chordal cycle graph on `p` nodes.
The returned graph is a cycle graph on `p` nodes with chords joining each
vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit)
3-regular expander [1]_.
`p` *must* be a prime number.
Parameters
----------
p : a prime number
The number of vertices in the graph. This also indicates where the
chordal edges in the cycle will be created.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : graph
The constructed undirected multigraph.
Raises
------
NetworkXError
If `create_using` indicates directed or not a multigraph.
References
----------
.. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and
invariant measures", volume 125 of Progress in Mathematics.
Birkhäuser Verlag, Basel, 1994.
| null | (p, create_using=None, *, backend=None, **backend_kwargs) |
30,455 | networkx.algorithms.chordal | chordal_graph_cliques | Returns all maximal cliques of a chordal graph.
The algorithm breaks the graph in connected components and performs a
maximum cardinality search in each component to get the cliques.
Parameters
----------
G : graph
A NetworkX graph
Yields
------
frozenset of nodes
Maximal cliques, each of which is a frozenset of
nodes in `G`. The order of cliques is arbitrary.
Raises
------
NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
The algorithm can only be applied to chordal graphs. If the input
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
Examples
--------
>>> e = [
... (1, 2),
... (1, 3),
... (2, 3),
... (2, 4),
... (3, 4),
... (3, 5),
... (3, 6),
... (4, 5),
... (4, 6),
... (5, 6),
... (7, 8),
... ]
>>> G = nx.Graph(e)
>>> G.add_node(9)
>>> cliques = [c for c in chordal_graph_cliques(G)]
>>> cliques[0]
frozenset({1, 2, 3})
| null | (G, *, backend=None, **backend_kwargs) |
30,456 | networkx.algorithms.chordal | chordal_graph_treewidth | Returns the treewidth of the chordal graph G.
Parameters
----------
G : graph
A NetworkX graph
Returns
-------
treewidth : int
The size of the largest clique in the graph minus one.
Raises
------
NetworkXError
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
The algorithm can only be applied to chordal graphs. If the input
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
Examples
--------
>>> e = [
... (1, 2),
... (1, 3),
... (2, 3),
... (2, 4),
... (3, 4),
... (3, 5),
... (3, 6),
... (4, 5),
... (4, 6),
... (5, 6),
... (7, 8),
... ]
>>> G = nx.Graph(e)
>>> G.add_node(9)
>>> nx.chordal_graph_treewidth(G)
3
References
----------
.. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
| null | (G, *, backend=None, **backend_kwargs) |
30,457 | networkx.algorithms.cycles | chordless_cycles | Find simple chordless cycles of a graph.
A `simple cycle` is a closed path where no node appears twice. In a simple
cycle, a `chord` is an additional edge between two nodes in the cycle. A
`chordless cycle` is a simple cycle without chords. Said differently, a
chordless cycle is a cycle C in a graph G where the number of edges in the
induced graph G[C] is equal to the length of `C`.
Note that some care must be taken in the case that G is not a simple graph
nor a simple digraph. Some authors limit the definition of chordless cycles
to have a prescribed minimum length; we do not.
1. We interpret self-loops to be chordless cycles, except in multigraphs
with multiple loops in parallel. Likewise, in a chordless cycle of
length greater than 1, there can be no nodes with self-loops.
2. We interpret directed two-cycles to be chordless cycles, except in
multi-digraphs when any edge in a two-cycle has a parallel copy.
3. We interpret parallel pairs of undirected edges as two-cycles, except
when a third (or more) parallel edge exists between the two nodes.
4. Generalizing the above, edges with parallel clones may not occur in
chordless cycles.
In a directed graph, two chordless cycles are distinct if they are not
cyclic permutations of each other. In an undirected graph, two chordless
cycles are distinct if they are not cyclic permutations of each other nor of
the other's reversal.
Optionally, the cycles are bounded in length.
We use an algorithm strongly inspired by that of Dias et al [1]_. It has
been modified in the following ways:
1. Recursion is avoided, per Python's limitations
2. The labeling function is not necessary, because the starting paths
are chosen (and deleted from the host graph) to prevent multiple
occurrences of the same path
3. The search is optionally bounded at a specified length
4. Support for directed graphs is provided by extending cycles along
forward edges, and blocking nodes along forward and reverse edges
5. Support for multigraphs is provided by omitting digons from the set
of forward edges
Parameters
----------
G : NetworkX DiGraph
A directed graph
length_bound : int or None, optional (default=None)
If length_bound is an int, generate all simple cycles of G with length at
most length_bound. Otherwise, generate all simple cycles of G.
Yields
------
list of nodes
Each cycle is represented by a list of nodes along the cycle.
Examples
--------
>>> sorted(list(nx.chordless_cycles(nx.complete_graph(4))))
[[1, 0, 2], [1, 0, 3], [2, 0, 3], [2, 1, 3]]
Notes
-----
When length_bound is None, and the graph is simple, the time complexity is
$O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$ chordless cycles.
Raises
------
ValueError
when length_bound < 0.
References
----------
.. [1] Efficient enumeration of chordless cycles
E. Dias and D. Castonguay and H. Longo and W.A.R. Jradi
https://arxiv.org/abs/1309.1051
See Also
--------
simple_cycles
| def recursive_simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
A `simple cycle`, or `elementary circuit`, is a closed path where
no node appears twice. Two elementary circuits are distinct if they
are not cyclic permutations of each other.
This version uses a recursive algorithm to build a list of cycles.
You should probably use the iterator version called simple_cycles().
Warning: This recursive version uses lots of RAM!
It appears in NetworkX for pedagogical value.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of cycles, where each cycle is represented by a list of nodes
along the cycle.
Example:
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> nx.recursive_simple_cycles(G)
[[0], [2], [0, 1, 2], [0, 2], [1, 2]]
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
https://doi.org/10.1137/0204007
See Also
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""
if blocked[thisnode]:
blocked[thisnode] = False
while B[thisnode]:
_unblock(B[thisnode].pop())
def circuit(thisnode, startnode, component):
closed = False # set to True if elementary path is closed
path.append(thisnode)
blocked[thisnode] = True
for nextnode in component[thisnode]: # direct successors of thisnode
if nextnode == startnode:
result.append(path[:])
closed = True
elif not blocked[nextnode]:
if circuit(nextnode, startnode, component):
closed = True
if closed:
_unblock(thisnode)
else:
for nextnode in component[thisnode]:
if thisnode not in B[nextnode]: # TODO: use set for speedup?
B[nextnode].append(thisnode)
path.pop() # remove thisnode from path
return closed
path = [] # stack of nodes in current path
blocked = defaultdict(bool) # vertex: blocked from search?
B = defaultdict(list) # graph portions that yield no elementary circuit
result = [] # list to accumulate the circuits found
# Johnson's algorithm exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in G:
if G.has_edge(v, v):
result.append([v])
G.remove_edge(v, v)
# Johnson's algorithm requires some ordering of the nodes.
# They might not be sortable so we assign an arbitrary ordering.
ordering = dict(zip(G, range(len(G))))
for s in ordering:
# Build the subgraph induced by s and following nodes in the ordering
subgraph = G.subgraph(node for node in G if ordering[node] >= ordering[s])
# Find the strongly connected component in the subgraph
# that contains the least node according to the ordering
strongcomp = nx.strongly_connected_components(subgraph)
mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
component = G.subgraph(mincomp)
if len(component) > 1:
# smallest node in the component according to the ordering
startnode = min(component, key=ordering.__getitem__)
for node in component:
blocked[node] = False
B[node][:] = []
dummy = circuit(startnode, startnode, component)
return result
| (G, length_bound=None, *, backend=None, **backend_kwargs) |
30,458 | networkx.algorithms.polynomials | chromatic_polynomial | Returns the chromatic polynomial of `G`
This function computes the chromatic polynomial via an iterative version of
the deletion-contraction algorithm.
The chromatic polynomial `X_G(x)` is a fundamental graph polynomial
invariant in one variable. Evaluating `X_G(k)` for an natural number `k`
enumerates the proper k-colorings of `G`.
There are several equivalent definitions; here are three:
Def 1 (explicit formula):
For `G` an undirected graph, `c(G)` the number of connected components of
`G`, `E` the edge set of `G`, and `G(S)` the spanning subgraph of `G` with
edge set `S` [1]_:
.. math::
X_G(x) = \sum_{S \subseteq E} (-1)^{|S|} x^{c(G(S))}
Def 2 (interpolating polynomial):
For `G` an undirected graph, `n(G)` the number of vertices of `G`, `k_0 = 0`,
and `k_i` the number of distinct ways to color the vertices of `G` with `i`
unique colors (for `i` a natural number at most `n(G)`), `X_G(x)` is the
unique Lagrange interpolating polynomial of degree `n(G)` through the points
`(0, k_0), (1, k_1), \dots, (n(G), k_{n(G)})` [2]_.
Def 3 (chromatic recurrence):
For `G` an undirected graph, `G-e` the graph obtained from `G` by deleting
edge `e`, `G/e` the graph obtained from `G` by contracting edge `e`, `n(G)`
the number of vertices of `G`, and `e(G)` the number of edges of `G` [3]_:
.. math::
X_G(x) = \begin{cases}
x^{n(G)}, & \text{if $e(G)=0$} \\
X_{G-e}(x) - X_{G/e}(x), & \text{otherwise, for an arbitrary edge $e$}
\end{cases}
This formulation is also known as the Fundamental Reduction Theorem [4]_.
Parameters
----------
G : NetworkX graph
Returns
-------
instance of `sympy.core.add.Add`
A Sympy expression representing the chromatic polynomial for `G`.
Examples
--------
>>> C = nx.cycle_graph(5)
>>> nx.chromatic_polynomial(C)
x**5 - 5*x**4 + 10*x**3 - 10*x**2 + 4*x
>>> G = nx.complete_graph(4)
>>> nx.chromatic_polynomial(G)
x**4 - 6*x**3 + 11*x**2 - 6*x
Notes
-----
Interpretation of the coefficients is discussed in [5]_. Several special
cases are listed in [2]_.
The chromatic polynomial is a specialization of the Tutte polynomial; in
particular, ``X_G(x) = T_G(x, 0)`` [6]_.
The chromatic polynomial may take negative arguments, though evaluations
may not have chromatic interpretations. For instance, ``X_G(-1)`` enumerates
the acyclic orientations of `G` [7]_.
References
----------
.. [1] D. B. West,
"Introduction to Graph Theory," p. 222
.. [2] E. W. Weisstein
"Chromatic Polynomial"
MathWorld--A Wolfram Web Resource
https://mathworld.wolfram.com/ChromaticPolynomial.html
.. [3] D. B. West,
"Introduction to Graph Theory," p. 221
.. [4] J. Zhang, J. Goodall,
"An Introduction to Chromatic Polynomials"
https://math.mit.edu/~apost/courses/18.204_2018/Julie_Zhang_paper.pdf
.. [5] R. C. Read,
"An Introduction to Chromatic Polynomials"
Journal of Combinatorial Theory, 1968
https://math.berkeley.edu/~mrklug/ReadChromatic.pdf
.. [6] W. T. Tutte,
"Graph-polynomials"
Advances in Applied Mathematics, 2004
https://www.sciencedirect.com/science/article/pii/S0196885803000411
.. [7] R. P. Stanley,
"Acyclic orientations of graphs"
Discrete Mathematics, 2006
https://math.mit.edu/~rstan/pubs/pubfiles/18.pdf
| null | (G, *, backend=None, **backend_kwargs) |
30,459 | networkx.generators.small | chvatal_graph |
Returns the Chvátal Graph
The Chvátal Graph is an undirected graph with 12 nodes and 24 edges [1]_.
It has 370 distinct (directed) Hamiltonian cycles, giving a unique generalized
LCF notation of order 4, two of order 6 , and 43 of order 1 [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
The Chvátal graph with 12 nodes and 24 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Chv%C3%A1tal_graph
.. [2] https://mathworld.wolfram.com/ChvatalGraph.html
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) |
30,460 | networkx.generators.classic | circulant_graph | Returns the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ nodes.
The circulant graph $Ci_n(x_1, ..., x_m)$ consists of $n$ nodes $0, ..., n-1$
such that node $i$ is connected to nodes $(i + x) \mod n$ and $(i - x) \mod n$
for all $x$ in $x_1, ..., x_m$. Thus $Ci_n(1)$ is a cycle graph.
.. plot::
>>> nx.draw(nx.circulant_graph(10, [1]))
Parameters
----------
n : integer
The number of nodes in the graph.
offsets : list of integers
A list of node offsets, $x_1$ up to $x_m$, as described above.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
NetworkX Graph of type create_using
Examples
--------
Many well-known graph families are subfamilies of the circulant graphs;
for example, to create the cycle graph on n points, we connect every
node to nodes on either side (with offset plus or minus one). For n = 10,
>>> G = nx.circulant_graph(10, [1])
>>> edges = [
... (0, 9),
... (0, 1),
... (1, 2),
... (2, 3),
... (3, 4),
... (4, 5),
... (5, 6),
... (6, 7),
... (7, 8),
... (8, 9),
... ]
>>> sorted(edges) == sorted(G.edges())
True
Similarly, we can create the complete graph
on 5 points with the set of offsets [1, 2]:
>>> G = nx.circulant_graph(5, [1, 2])
>>> edges = [
... (0, 1),
... (0, 2),
... (0, 3),
... (0, 4),
... (1, 2),
... (1, 3),
... (1, 4),
... (2, 3),
... (2, 4),
... (3, 4),
... ]
>>> sorted(edges) == sorted(G.edges())
True
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, offsets, create_using=None, *, backend=None, **backend_kwargs) |
30,461 | networkx.generators.classic | circular_ladder_graph | Returns the circular ladder graph $CL_n$ of length n.
$CL_n$ consists of two concentric n-cycles in which
each of the n pairs of concentric nodes are joined by an edge.
Node labels are the integers 0 to n-1
.. plot::
>>> nx.draw(nx.circular_ladder_graph(5))
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, create_using=None, *, backend=None, **backend_kwargs) |
30,462 | networkx.drawing.layout | circular_layout | Position nodes on a circle.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
If dim>2, the remaining dimensions are set to zero
in the returned positions.
If dim<2, a ValueError is raised.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Raises
------
ValueError
If dim < 2
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.circular_layout(G)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
| def circular_layout(G, scale=1, center=None, dim=2):
# dim=2 only
"""Position nodes on a circle.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
If dim>2, the remaining dimensions are set to zero
in the returned positions.
If dim<2, a ValueError is raised.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Raises
------
ValueError
If dim < 2
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.circular_layout(G)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
"""
import numpy as np
if dim < 2:
raise ValueError("cannot handle dimensions < 2")
G, center = _process_params(G, center, dim)
paddims = max(0, (dim - 2))
if len(G) == 0:
pos = {}
elif len(G) == 1:
pos = {nx.utils.arbitrary_element(G): center}
else:
# Discard the extra angle since it matches 0 radians.
theta = np.linspace(0, 1, len(G) + 1)[:-1] * 2 * np.pi
theta = theta.astype(np.float32)
pos = np.column_stack(
[np.cos(theta), np.sin(theta), np.zeros((len(G), paddims))]
)
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos
| (G, scale=1, center=None, dim=2) |
30,467 | networkx.algorithms.centrality.closeness | closeness_centrality | Compute closeness centrality for nodes.
Closeness centrality [1]_ of a node `u` is the reciprocal of the
average shortest path distance to `u` over all `n-1` reachable nodes.
.. math::
C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},
where `d(v, u)` is the shortest-path distance between `v` and `u`,
and `n-1` is the number of nodes reachable from `u`. Notice that the
closeness distance function computes the incoming distance to `u`
for directed graphs. To use outward distance, act on `G.reverse()`.
Notice that higher values of closeness indicate higher centrality.
Wasserman and Faust propose an improved formula for graphs with
more than one connected component. The result is "a ratio of the
fraction of actors in the group who are reachable, to the average
distance" from the reachable actors [2]_. You might think this
scale factor is inverted but it is not. As is, nodes from small
components receive a smaller closeness value. Letting `N` denote
the number of nodes in the graph,
.. math::
C_{WF}(u) = \frac{n-1}{N-1} \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},
Parameters
----------
G : graph
A NetworkX graph
u : node, optional
Return only the value for node u
distance : edge attribute key, optional (default=None)
Use the specified edge attribute as the edge distance in shortest
path calculations. If `None` (the default) all edges have a distance of 1.
Absent edge attributes are assigned a distance of 1. Note that no check
is performed to ensure that edges have the provided attribute.
wf_improved : bool, optional (default=True)
If True, scale by the fraction of nodes reachable. This gives the
Wasserman and Faust improved formula. For single component graphs
it is the same as the original formula.
Returns
-------
nodes : dictionary
Dictionary of nodes with closeness centrality as the value.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
>>> nx.closeness_centrality(G)
{0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}
See Also
--------
betweenness_centrality, load_centrality, eigenvector_centrality,
degree_centrality, incremental_closeness_centrality
Notes
-----
The closeness centrality is normalized to `(n-1)/(|G|-1)` where
`n` is the number of nodes in the connected part of graph
containing the node. If the graph is not completely connected,
this algorithm computes the closeness centrality for each
connected part separately scaled by that parts size.
If the 'distance' keyword is set to an edge attribute key then the
shortest-path length will be computed using Dijkstra's algorithm with
that edge attribute as the edge weight.
The closeness centrality uses *inward* distance to a node, not outward.
If you want to use outword distances apply the function to `G.reverse()`
In NetworkX 2.2 and earlier a bug caused Dijkstra's algorithm to use the
outward distance rather than the inward distance. If you use a 'distance'
keyword and a DiGraph, your results will change between v2.2 and v2.3.
References
----------
.. [1] Linton C. Freeman: Centrality in networks: I.
Conceptual clarification. Social Networks 1:215-239, 1979.
https://doi.org/10.1016/0378-8733(78)90021-7
.. [2] pg. 201 of Wasserman, S. and Faust, K.,
Social Network Analysis: Methods and Applications, 1994,
Cambridge University Press.
| null | (G, u=None, distance=None, wf_improved=True, *, backend=None, **backend_kwargs) |
30,468 | networkx.algorithms.vitality | closeness_vitality | Returns the closeness vitality for nodes in the graph.
The *closeness vitality* of a node, defined in Section 3.6.2 of [1],
is the change in the sum of distances between all node pairs when
excluding that node.
Parameters
----------
G : NetworkX graph
A strongly-connected graph.
weight : string
The name of the edge attribute used as weight. This is passed
directly to the :func:`~networkx.wiener_index` function.
node : object
If specified, only the closeness vitality for this node will be
returned. Otherwise, a dictionary mapping each node to its
closeness vitality will be returned.
Other parameters
----------------
wiener_index : number
If you have already computed the Wiener index of the graph
`G`, you can provide that value here. Otherwise, it will be
computed for you.
Returns
-------
dictionary or float
If `node` is None, this function returns a dictionary
with nodes as keys and closeness vitality as the
value. Otherwise, it returns only the closeness vitality for the
specified `node`.
The closeness vitality of a node may be negative infinity if
removing that node would disconnect the graph.
Examples
--------
>>> G = nx.cycle_graph(3)
>>> nx.closeness_vitality(G)
{0: 2.0, 1: 2.0, 2: 2.0}
See Also
--------
closeness_centrality
References
----------
.. [1] Ulrik Brandes, Thomas Erlebach (eds.).
*Network Analysis: Methodological Foundations*.
Springer, 2005.
<http://books.google.com/books?id=TTNhSm7HYrIC>
| null | (G, node=None, weight=None, wiener_index=None, *, backend=None, **backend_kwargs) |
30,470 | networkx.algorithms.cluster | clustering | Compute the clustering coefficient for nodes.
For unweighted graphs, the clustering of a node :math:`u`
is the fraction of possible triangles through that node that exist,
.. math::
c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)},
where :math:`T(u)` is the number of triangles through node :math:`u` and
:math:`deg(u)` is the degree of :math:`u`.
For weighted graphs, there are several ways to define clustering [1]_.
the one used here is defined
as the geometric average of the subgraph edge weights [2]_,
.. math::
c_u = \frac{1}{deg(u)(deg(u)-1))}
\sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}.
The edge weights :math:`\hat{w}_{uv}` are normalized by the maximum weight
in the network :math:`\hat{w}_{uv} = w_{uv}/\max(w)`.
The value of :math:`c_u` is assigned to 0 if :math:`deg(u) < 2`.
Additionally, this weighted definition has been generalized to support negative edge weights [3]_.
For directed graphs, the clustering is similarly defined as the fraction
of all possible directed triangles or geometric average of the subgraph
edge weights for unweighted and weighted directed graph respectively [4]_.
.. math::
c_u = \frac{T(u)}{2(deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u))},
where :math:`T(u)` is the number of directed triangles through node
:math:`u`, :math:`deg^{tot}(u)` is the sum of in degree and out degree of
:math:`u` and :math:`deg^{\leftrightarrow}(u)` is the reciprocal degree of
:math:`u`.
Parameters
----------
G : graph
nodes : node, iterable of nodes, or None (default=None)
If a singleton node, return the number of triangles for that node.
If an iterable, compute the number of triangles for each of those nodes.
If `None` (the default) compute the number of triangles for all nodes in `G`.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight.
If None, then each edge has weight 1.
Returns
-------
out : float, or dictionary
Clustering coefficient at specified nodes
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.clustering(G, 0))
1.0
>>> print(nx.clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
Notes
-----
Self loops are ignored.
References
----------
.. [1] Generalizations of the clustering coefficient to weighted
complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
http://jponnela.com/web_documents/a9.pdf
.. [2] Intensity and coherence of motifs in weighted complex
networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski,
Physical Review E, 71(6), 065103 (2005).
.. [3] Generalization of Clustering Coefficients to Signed Correlation Networks
by G. Costantini and M. Perugini, PloS one, 9(2), e88669 (2014).
.. [4] Clustering in complex directed networks by G. Fagiolo,
Physical Review E, 76(2), 026107 (2007).
| null | (G, nodes=None, weight=None, *, backend=None, **backend_kwargs) |
30,471 | networkx.algorithms.link_prediction | cn_soundarajan_hopcroft | Count the number of common neighbors of all node pairs in ebunch
using community information.
For two nodes $u$ and $v$, this function computes the number of
common neighbors and bonus one for each common neighbor belonging to
the same community as $u$ and $v$. Mathematically,
.. math::
|\Gamma(u) \cap \Gamma(v)| + \sum_{w \in \Gamma(u) \cap \Gamma(v)} f(w)
where $f(w)$ equals 1 if $w$ belongs to the same community as $u$
and $v$ or 0 otherwise and $\Gamma(u)$ denotes the set of
neighbors of $u$.
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The score will be computed for each pair of nodes given in the
iterable. The pairs must be given as 2-tuples (u, v) where u
and v are nodes in the graph. If ebunch is None then all
nonexistent edges in the graph will be used.
Default value: None.
community : string, optional (default = 'community')
Nodes attribute name containing the community information.
G[u][community] identifies which community u belongs to. Each
node belongs to at most one community. Default value: 'community'.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their score.
Raises
------
NetworkXNotImplemented
If `G` is a `DiGraph`, a `Multigraph` or a `MultiDiGraph`.
NetworkXAlgorithmError
If no community information is available for a node in `ebunch` or in `G` (if `ebunch` is `None`).
NodeNotFound
If `ebunch` has a node that is not in `G`.
Examples
--------
>>> G = nx.path_graph(3)
>>> G.nodes[0]["community"] = 0
>>> G.nodes[1]["community"] = 0
>>> G.nodes[2]["community"] = 0
>>> preds = nx.cn_soundarajan_hopcroft(G, [(0, 2)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p}")
(0, 2) -> 2
References
----------
.. [1] Sucheta Soundarajan and John Hopcroft.
Using community information to improve the precision of link
prediction methods.
In Proceedings of the 21st international conference companion on
World Wide Web (WWW '12 Companion). ACM, New York, NY, USA, 607-608.
http://doi.acm.org/10.1145/2187980.2188150
| null | (G, ebunch=None, community='community', *, backend=None, **backend_kwargs) |
30,474 | networkx.algorithms.planar_drawing | combinatorial_embedding_to_pos | Assigns every node a (x, y) position based on the given embedding
The algorithm iteratively inserts nodes of the input graph in a certain
order and rearranges previously inserted nodes so that the planar drawing
stays valid. This is done efficiently by only maintaining relative
positions during the node placements and calculating the absolute positions
at the end. For more information see [1]_.
Parameters
----------
embedding : nx.PlanarEmbedding
This defines the order of the edges
fully_triangulate : bool
If set to True the algorithm adds edges to a copy of the input
embedding and makes it chordal.
Returns
-------
pos : dict
Maps each node to a tuple that defines the (x, y) position
References
----------
.. [1] M. Chrobak and T.H. Payne:
A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677
| def combinatorial_embedding_to_pos(embedding, fully_triangulate=False):
"""Assigns every node a (x, y) position based on the given embedding
The algorithm iteratively inserts nodes of the input graph in a certain
order and rearranges previously inserted nodes so that the planar drawing
stays valid. This is done efficiently by only maintaining relative
positions during the node placements and calculating the absolute positions
at the end. For more information see [1]_.
Parameters
----------
embedding : nx.PlanarEmbedding
This defines the order of the edges
fully_triangulate : bool
If set to True the algorithm adds edges to a copy of the input
embedding and makes it chordal.
Returns
-------
pos : dict
Maps each node to a tuple that defines the (x, y) position
References
----------
.. [1] M. Chrobak and T.H. Payne:
A Linear-time Algorithm for Drawing a Planar Graph on a Grid 1989
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.6677
"""
if len(embedding.nodes()) < 4:
# Position the node in any triangle
default_positions = [(0, 0), (2, 0), (1, 1)]
pos = {}
for i, v in enumerate(embedding.nodes()):
pos[v] = default_positions[i]
return pos
embedding, outer_face = triangulate_embedding(embedding, fully_triangulate)
# The following dicts map a node to another node
# If a node is not in the key set it means that the node is not yet in G_k
# If a node maps to None then the corresponding subtree does not exist
left_t_child = {}
right_t_child = {}
# The following dicts map a node to an integer
delta_x = {}
y_coordinate = {}
node_list = get_canonical_ordering(embedding, outer_face)
# 1. Phase: Compute relative positions
# Initialization
v1, v2, v3 = node_list[0][0], node_list[1][0], node_list[2][0]
delta_x[v1] = 0
y_coordinate[v1] = 0
right_t_child[v1] = v3
left_t_child[v1] = None
delta_x[v2] = 1
y_coordinate[v2] = 0
right_t_child[v2] = None
left_t_child[v2] = None
delta_x[v3] = 1
y_coordinate[v3] = 1
right_t_child[v3] = v2
left_t_child[v3] = None
for k in range(3, len(node_list)):
vk, contour_nbrs = node_list[k]
wp = contour_nbrs[0]
wp1 = contour_nbrs[1]
wq = contour_nbrs[-1]
wq1 = contour_nbrs[-2]
adds_mult_tri = len(contour_nbrs) > 2
# Stretch gaps:
delta_x[wp1] += 1
delta_x[wq] += 1
delta_x_wp_wq = sum(delta_x[x] for x in contour_nbrs[1:])
# Adjust offsets
delta_x[vk] = (-y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2
y_coordinate[vk] = (y_coordinate[wp] + delta_x_wp_wq + y_coordinate[wq]) // 2
delta_x[wq] = delta_x_wp_wq - delta_x[vk]
if adds_mult_tri:
delta_x[wp1] -= delta_x[vk]
# Install v_k:
right_t_child[wp] = vk
right_t_child[vk] = wq
if adds_mult_tri:
left_t_child[vk] = wp1
right_t_child[wq1] = None
else:
left_t_child[vk] = None
# 2. Phase: Set absolute positions
pos = {}
pos[v1] = (0, y_coordinate[v1])
remaining_nodes = [v1]
while remaining_nodes:
parent_node = remaining_nodes.pop()
# Calculate position for left child
set_position(
parent_node, left_t_child, remaining_nodes, delta_x, y_coordinate, pos
)
# Calculate position for right child
set_position(
parent_node, right_t_child, remaining_nodes, delta_x, y_coordinate, pos
)
return pos
| (embedding, fully_triangulate=False) |
30,475 | networkx.algorithms.link_prediction | common_neighbor_centrality | Return the CCPA score for each pair of nodes.
Compute the Common Neighbor and Centrality based Parameterized Algorithm(CCPA)
score of all node pairs in ebunch.
CCPA score of `u` and `v` is defined as
.. math::
\alpha \cdot (|\Gamma (u){\cap }^{}\Gamma (v)|)+(1-\alpha )\cdot \frac{N}{{d}_{uv}}
where $\Gamma(u)$ denotes the set of neighbors of $u$, $\Gamma(v)$ denotes the
set of neighbors of $v$, $\alpha$ is parameter varies between [0,1], $N$ denotes
total number of nodes in the Graph and ${d}_{uv}$ denotes shortest distance
between $u$ and $v$.
This algorithm is based on two vital properties of nodes, namely the number
of common neighbors and their centrality. Common neighbor refers to the common
nodes between two nodes. Centrality refers to the prestige that a node enjoys
in a network.
.. seealso::
:func:`common_neighbors`
Parameters
----------
G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Preferential attachment score will be computed for each pair of
nodes given in the iterable. The pairs must be given as
2-tuples (u, v) where u and v are nodes in the graph. If ebunch
is None then all nonexistent edges in the graph will be used.
Default value: None.
alpha : Parameter defined for participation of Common Neighbor
and Centrality Algorithm share. Values for alpha should
normally be between 0 and 1. Default value set to 0.8
because author found better performance at 0.8 for all the
dataset.
Default value: 0.8
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their Common Neighbor and Centrality based
Parameterized Algorithm(CCPA) score.
Raises
------
NetworkXNotImplemented
If `G` is a `DiGraph`, a `Multigraph` or a `MultiDiGraph`.
NetworkXAlgorithmError
If self loops exsists in `ebunch` or in `G` (if `ebunch` is `None`).
NodeNotFound
If `ebunch` has a node that is not in `G`.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.common_neighbor_centrality(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p}")
(0, 1) -> 3.4000000000000004
(2, 3) -> 3.4000000000000004
References
----------
.. [1] Ahmad, I., Akhtar, M.U., Noor, S. et al.
Missing Link Prediction using Common Neighbor and Centrality based Parameterized Algorithm.
Sci Rep 10, 364 (2020).
https://doi.org/10.1038/s41598-019-57304-y
| null | (G, ebunch=None, alpha=0.8, *, backend=None, **backend_kwargs) |
30,476 | networkx.classes.function | common_neighbors | Returns the common neighbors of two nodes in a graph.
Parameters
----------
G : graph
A NetworkX undirected graph.
u, v : nodes
Nodes in the graph.
Returns
-------
cnbors : set
Set of common neighbors of u and v in the graph.
Raises
------
NetworkXError
If u or v is not a node in the graph.
Examples
--------
>>> G = nx.complete_graph(5)
>>> sorted(nx.common_neighbors(G, 0, 1))
[2, 3, 4]
| def set_edge_attributes(G, values, name=None):
"""Sets edge attributes from a given value or dictionary of values.
.. Warning:: The call order of arguments `values` and `name`
switched between v1.x & v2.x.
Parameters
----------
G : NetworkX Graph
values : scalar value, dict-like
What the edge attribute should be set to. If `values` is
not a dictionary, then it is treated as a single attribute value
that is then applied to every edge in `G`. This means that if
you provide a mutable object, like a list, updates to that object
will be reflected in the edge attribute for each edge. The attribute
name will be `name`.
If `values` is a dict or a dict of dict, it should be keyed
by edge tuple to either an attribute value or a dict of attribute
key/value pairs used to update the edge's attributes.
For multigraphs, the edge tuples must be of the form ``(u, v, key)``,
where `u` and `v` are nodes and `key` is the edge key.
For non-multigraphs, the keys must be tuples of the form ``(u, v)``.
name : string (optional, default=None)
Name of the edge attribute to set if values is a scalar.
Examples
--------
After computing some property of the edges of a graph, you may want
to assign a edge attribute to store the value of that property for
each edge::
>>> G = nx.path_graph(3)
>>> bb = nx.edge_betweenness_centrality(G, normalized=False)
>>> nx.set_edge_attributes(G, bb, "betweenness")
>>> G.edges[1, 2]["betweenness"]
2.0
If you provide a list as the second argument, updates to the list
will be reflected in the edge attribute for each edge::
>>> labels = []
>>> nx.set_edge_attributes(G, labels, "labels")
>>> labels.append("foo")
>>> G.edges[0, 1]["labels"]
['foo']
>>> G.edges[1, 2]["labels"]
['foo']
If you provide a dictionary of dictionaries as the second argument,
the entire dictionary will be used to update edge attributes::
>>> G = nx.path_graph(3)
>>> attrs = {(0, 1): {"attr1": 20, "attr2": "nothing"}, (1, 2): {"attr2": 3}}
>>> nx.set_edge_attributes(G, attrs)
>>> G[0][1]["attr1"]
20
>>> G[0][1]["attr2"]
'nothing'
>>> G[1][2]["attr2"]
3
The attributes of one Graph can be used to set those of another.
>>> H = nx.path_graph(3)
>>> nx.set_edge_attributes(H, G.edges)
Note that if the dict contains edges that are not in `G`, they are
silently ignored::
>>> G = nx.Graph([(0, 1)])
>>> nx.set_edge_attributes(G, {(1, 2): {"weight": 2.0}})
>>> (1, 2) in G.edges()
False
For multigraphs, the `values` dict is expected to be keyed by 3-tuples
including the edge key::
>>> MG = nx.MultiGraph()
>>> edges = [(0, 1), (0, 1)]
>>> MG.add_edges_from(edges) # Returns list of edge keys
[0, 1]
>>> attributes = {(0, 1, 0): {"cost": 21}, (0, 1, 1): {"cost": 7}}
>>> nx.set_edge_attributes(MG, attributes)
>>> MG[0][1][0]["cost"]
21
>>> MG[0][1][1]["cost"]
7
If MultiGraph attributes are desired for a Graph, you must convert the 3-tuple
multiedge to a 2-tuple edge and the last multiedge's attribute value will
overwrite the previous values. Continuing from the previous case we get::
>>> H = nx.path_graph([0, 1, 2])
>>> nx.set_edge_attributes(H, {(u, v): ed for u, v, ed in MG.edges.data()})
>>> nx.get_edge_attributes(H, "cost")
{(0, 1): 7}
"""
if name is not None:
# `values` does not contain attribute names
try:
# if `values` is a dict using `.items()` => {edge: value}
if G.is_multigraph():
for (u, v, key), value in values.items():
try:
G._adj[u][v][key][name] = value
except KeyError:
pass
else:
for (u, v), value in values.items():
try:
G._adj[u][v][name] = value
except KeyError:
pass
except AttributeError:
# treat `values` as a constant
for u, v, data in G.edges(data=True):
data[name] = values
else:
# `values` consists of doct-of-dict {edge: {attr: value}} shape
if G.is_multigraph():
for (u, v, key), d in values.items():
try:
G._adj[u][v][key].update(d)
except KeyError:
pass
else:
for (u, v), d in values.items():
try:
G._adj[u][v].update(d)
except KeyError:
pass
nx._clear_cache(G)
| (G, u, v) |
30,477 | networkx.algorithms.communicability_alg | communicability | Returns communicability between all pairs of nodes in G.
The communicability between pairs of nodes in G is the sum of
walks of different lengths starting at node u and ending at node v.
Parameters
----------
G: graph
Returns
-------
comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability
as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability_exp:
Communicability between all pairs of nodes in G using spectral
decomposition.
communicability_betweenness_centrality:
Communicability betweenness centrality for each node in G.
Notes
-----
This algorithm uses a spectral decomposition of the adjacency matrix.
Let G=(V,E) be a simple undirected graph. Using the connection between
the powers of the adjacency matrix and the number of walks in the graph,
the communicability between nodes `u` and `v` based on the graph spectrum
is [1]_
.. math::
C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},
where `\phi_{j}(u)` is the `u\rm{th}` element of the `j\rm{th}` orthonormal
eigenvector of the adjacency matrix associated with the eigenvalue
`\lambda_{j}`.
References
----------
.. [1] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
https://arxiv.org/abs/0707.0756
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
>>> c = nx.communicability(G)
| null | (G, *, backend=None, **backend_kwargs) |
30,479 | networkx.algorithms.centrality.subgraph_alg | communicability_betweenness_centrality | Returns subgraph communicability for all pairs of nodes in G.
Communicability betweenness measure makes use of the number of walks
connecting every pair of nodes as the basis of a betweenness centrality
measure.
Parameters
----------
G: graph
Returns
-------
nodes : dictionary
Dictionary of nodes with communicability betweenness as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
Notes
-----
Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges,
and `A` denote the adjacency matrix of `G`.
Let `G(r)=(V,E(r))` be the graph resulting from
removing all edges connected to node `r` but not the node itself.
The adjacency matrix for `G(r)` is `A+E(r)`, where `E(r)` has nonzeros
only in row and column `r`.
The subraph betweenness of a node `r` is [1]_
.. math::
\omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}},
p\neq q, q\neq r,
where
`G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}` is the number of walks
involving node r,
`G_{pq}=(e^{A})_{pq}` is the number of closed walks starting
at node `p` and ending at node `q`,
and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the
number of terms in the sum.
The resulting `\omega_{r}` takes values between zero and one.
The lower bound cannot be attained for a connected
graph, and the upper bound is attained in the star graph.
References
----------
.. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano,
"Communicability Betweenness in Complex Networks"
Physica A 388 (2009) 764-774.
https://arxiv.org/abs/0905.4102
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
>>> cbc = nx.communicability_betweenness_centrality(G)
>>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)])
['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03']
| null | (G, *, backend=None, **backend_kwargs) |
30,480 | networkx.algorithms.communicability_alg | communicability_exp | Returns communicability between all pairs of nodes in G.
Communicability between pair of node (u,v) of node in G is the sum of
walks of different lengths starting at node u and ending at node v.
Parameters
----------
G: graph
Returns
-------
comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability
as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability:
Communicability between pairs of nodes in G.
communicability_betweenness_centrality:
Communicability betweenness centrality for each node in G.
Notes
-----
This algorithm uses matrix exponentiation of the adjacency matrix.
Let G=(V,E) be a simple undirected graph. Using the connection between
the powers of the adjacency matrix and the number of walks in the graph,
the communicability between nodes u and v is [1]_,
.. math::
C(u,v) = (e^A)_{uv},
where `A` is the adjacency matrix of G.
References
----------
.. [1] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
https://arxiv.org/abs/0707.0756
Examples
--------
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
>>> c = nx.communicability_exp(G)
| null | (G, *, backend=None, **backend_kwargs) |
30,482 | networkx.algorithms.operators.unary | complement | Returns the graph complement of G.
Parameters
----------
G : graph
A NetworkX graph
Returns
-------
GC : A new graph.
Notes
-----
Note that `complement` does not create self-loops and also
does not produce parallel edges for MultiGraphs.
Graph, node, and edge data are not propagated to the new graph.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (3, 5)])
>>> G_complement = nx.complement(G)
>>> G_complement.edges() # This shows the edges of the complemented graph
EdgeView([(1, 4), (1, 5), (2, 4), (2, 5), (4, 5)])
| null | (G, *, backend=None, **backend_kwargs) |
30,483 | networkx.algorithms.bipartite.generators | complete_bipartite_graph | Returns the complete bipartite graph `K_{n_1,n_2}`.
The graph is composed of two partitions with nodes 0 to (n1 - 1)
in the first and nodes n1 to (n1 + n2 - 1) in the second.
Each node in the first is connected to each node in the second.
Parameters
----------
n1, n2 : integer or iterable container of nodes
If integers, nodes are from `range(n1)` and `range(n1, n1 + n2)`.
If a container, the elements are the nodes.
create_using : NetworkX graph instance, (default: nx.Graph)
Return graph of this type.
Notes
-----
Nodes are the integers 0 to `n1 + n2 - 1` unless either n1 or n2 are
containers of nodes. If only one of n1 or n2 are integers, that
integer is replaced by `range` of that integer.
The nodes are assigned the attribute 'bipartite' with the value 0 or 1
to indicate which bipartite set the node belongs to.
This function is not imported in the main namespace.
To use it use nx.bipartite.complete_bipartite_graph
| null | (n1, n2, create_using=None, *, backend=None, **backend_kwargs) |
30,484 | networkx.generators.classic | complete_graph | Return the complete graph `K_n` with n nodes.
A complete graph on `n` nodes means that all pairs
of distinct nodes have an edge connecting them.
.. plot::
>>> nx.draw(nx.complete_graph(5))
Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from range(n).
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Examples
--------
>>> G = nx.complete_graph(9)
>>> len(G)
9
>>> G.size()
36
>>> G = nx.complete_graph(range(11, 14))
>>> list(G.nodes())
[11, 12, 13]
>>> G = nx.complete_graph(4, nx.DiGraph())
>>> G.is_directed()
True
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, create_using=None, *, backend=None, **backend_kwargs) |
30,485 | networkx.generators.classic | complete_multipartite_graph | Returns the complete multipartite graph with the specified subset sizes.
.. plot::
>>> nx.draw(nx.complete_multipartite_graph(1, 2, 3))
Parameters
----------
subset_sizes : tuple of integers or tuple of node iterables
The arguments can either all be integer number of nodes or they
can all be iterables of nodes. If integers, they represent the
number of nodes in each subset of the multipartite graph.
If iterables, each is used to create the nodes for that subset.
The length of subset_sizes is the number of subsets.
Returns
-------
G : NetworkX Graph
Returns the complete multipartite graph with the specified subsets.
For each node, the node attribute 'subset' is an integer
indicating which subset contains the node.
Examples
--------
Creating a complete tripartite graph, with subsets of one, two, and three
nodes, respectively.
>>> G = nx.complete_multipartite_graph(1, 2, 3)
>>> [G.nodes[u]["subset"] for u in G]
[0, 1, 1, 2, 2, 2]
>>> list(G.edges(0))
[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)]
>>> list(G.edges(2))
[(2, 0), (2, 3), (2, 4), (2, 5)]
>>> list(G.edges(4))
[(4, 0), (4, 1), (4, 2)]
>>> G = nx.complete_multipartite_graph("a", "bc", "def")
>>> [G.nodes[u]["subset"] for u in sorted(G)]
[0, 1, 1, 2, 2, 2]
Notes
-----
This function generalizes several other graph builder functions.
- If no subset sizes are given, this returns the null graph.
- If a single subset size `n` is given, this returns the empty graph on
`n` nodes.
- If two subset sizes `m` and `n` are given, this returns the complete
bipartite graph on `m + n` nodes.
- If subset sizes `1` and `n` are given, this returns the star graph on
`n + 1` nodes.
See also
--------
complete_bipartite_graph
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (*subset_sizes, backend=None, **backend_kwargs) |
30,486 | networkx.algorithms.chordal | complete_to_chordal_graph | Return a copy of G completed to a chordal graph
Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is
called chordal if for each cycle with length bigger than 3, there exist
two non-adjacent nodes connected by an edge (called a chord).
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
H : NetworkX graph
The chordal enhancement of G
alpha : Dictionary
The elimination ordering of nodes of G
Notes
-----
There are different approaches to calculate the chordal
enhancement of a graph. The algorithm used here is called
MCS-M and gives at least minimal (local) triangulation of graph. Note
that this triangulation is not necessarily a global minimum.
https://en.wikipedia.org/wiki/Chordal_graph
References
----------
.. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004)
Maximum Cardinality Search for Computing Minimal Triangulations of
Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3.
Examples
--------
>>> from networkx.algorithms.chordal import complete_to_chordal_graph
>>> G = nx.wheel_graph(10)
>>> H, alpha = complete_to_chordal_graph(G)
| null | (G, *, backend=None, **backend_kwargs) |
30,488 | networkx.algorithms.operators.binary | compose | Compose graph G with H by combining nodes and edges into a single graph.
The node sets and edges sets do not need to be disjoint.
Composing preserves the attributes of nodes and edges.
Attribute values from H take precedent over attribute values from G.
Parameters
----------
G, H : graph
A NetworkX graph
Returns
-------
C: A new graph with the same type as G
See Also
--------
:func:`~networkx.Graph.update`
union
disjoint_union
Notes
-----
It is recommended that G and H be either both directed or both undirected.
For MultiGraphs, the edges are identified by incident nodes AND edge-key.
This can cause surprises (i.e., edge `(1, 2)` may or may not be the same
in two graphs) if you use MultiGraph without keeping track of edge keys.
If combining the attributes of common nodes is not desired, consider union(),
which raises an exception for name collisions.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2)])
>>> H = nx.Graph([(0, 1), (1, 2)])
>>> R = nx.compose(G, H)
>>> R.nodes
NodeView((0, 1, 2))
>>> R.edges
EdgeView([(0, 1), (0, 2), (1, 2)])
By default, the attributes from `H` take precedent over attributes from `G`.
If you prefer another way of combining attributes, you can update them after the compose operation:
>>> G = nx.Graph([(0, 1, {"weight": 2.0}), (3, 0, {"weight": 100.0})])
>>> H = nx.Graph([(0, 1, {"weight": 10.0}), (1, 2, {"weight": -1.0})])
>>> nx.set_node_attributes(G, {0: "dark", 1: "light", 3: "black"}, name="color")
>>> nx.set_node_attributes(H, {0: "green", 1: "orange", 2: "yellow"}, name="color")
>>> GcomposeH = nx.compose(G, H)
Normally, color attribute values of nodes of GcomposeH come from H. We can workaround this as follows:
>>> node_data = {
... n: G.nodes[n]["color"] + " " + H.nodes[n]["color"] for n in G.nodes & H.nodes
... }
>>> nx.set_node_attributes(GcomposeH, node_data, "color")
>>> print(GcomposeH.nodes[0]["color"])
dark green
>>> print(GcomposeH.nodes[3]["color"])
black
Similarly, we can update edge attributes after the compose operation in a way we prefer:
>>> edge_data = {
... e: G.edges[e]["weight"] * H.edges[e]["weight"] for e in G.edges & H.edges
... }
>>> nx.set_edge_attributes(GcomposeH, edge_data, "weight")
>>> print(GcomposeH.edges[(0, 1)]["weight"])
20.0
>>> print(GcomposeH.edges[(3, 0)]["weight"])
100.0
| null | (G, H, *, backend=None, **backend_kwargs) |
30,489 | networkx.algorithms.operators.all | compose_all | Returns the composition of all graphs.
Composition is the simple union of the node sets and edge sets.
The node sets of the supplied graphs need not be disjoint.
Parameters
----------
graphs : iterable
Iterable of NetworkX graphs
Returns
-------
C : A graph with the same type as the first graph in list
Raises
------
ValueError
If `graphs` is an empty list.
NetworkXError
In case of mixed type graphs, like MultiGraph and Graph, or directed and undirected graphs.
Examples
--------
>>> G1 = nx.Graph([(1, 2), (2, 3)])
>>> G2 = nx.Graph([(3, 4), (5, 6)])
>>> C = nx.compose_all([G1, G2])
>>> list(C.nodes())
[1, 2, 3, 4, 5, 6]
>>> list(C.edges())
[(1, 2), (2, 3), (3, 4), (5, 6)]
Notes
-----
For operating on mixed type graphs, they should be converted to the same type.
Graph, edge, and node attributes are propagated to the union graph.
If a graph attribute is present in multiple graphs, then the value
from the last graph in the list with that attribute is used.
| null | (graphs, *, backend=None, **backend_kwargs) |
30,490 | networkx.algorithms.dag | compute_v_structures | Iterate through the graph to compute all v-structures.
V-structures are triples in the directed graph where
two parent nodes point to the same child and the two parent nodes
are not adjacent.
Parameters
----------
G : graph
A networkx DiGraph.
Returns
-------
vstructs : iterator of tuples
The v structures within the graph. Each v structure is a 3-tuple with the
parent, collider, and other parent.
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_edges_from([(1, 2), (0, 5), (3, 1), (2, 4), (3, 1), (4, 5), (1, 5)])
>>> sorted(nx.compute_v_structures(G))
[(0, 5, 1), (0, 5, 4), (1, 5, 4)]
Notes
-----
`Wikipedia: Collider in causal graphs <https://en.wikipedia.org/wiki/Collider_(statistics)>`_
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, *, backend=None, **backend_kwargs) |
30,491 | networkx.algorithms.components.strongly_connected | condensation | Returns the condensation of G.
The condensation of G is the graph with each of the strongly connected
components contracted into a single node.
Parameters
----------
G : NetworkX DiGraph
A directed graph.
scc: list or generator (optional, default=None)
Strongly connected components. If provided, the elements in
`scc` must partition the nodes in `G`. If not provided, it will be
calculated as scc=nx.strongly_connected_components(G).
Returns
-------
C : NetworkX DiGraph
The condensation graph C of G. The node labels are integers
corresponding to the index of the component in the list of
strongly connected components of G. C has a graph attribute named
'mapping' with a dictionary mapping the original nodes to the
nodes in C to which they belong. Each node in C also has a node
attribute 'members' with the set of original nodes in G that
form the SCC that the node in C represents.
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
Contracting two sets of strongly connected nodes into two distinct SCC
using the barbell graph.
>>> G = nx.barbell_graph(4, 0)
>>> G.remove_edge(3, 4)
>>> G = nx.DiGraph(G)
>>> H = nx.condensation(G)
>>> H.nodes.data()
NodeDataView({0: {'members': {0, 1, 2, 3}}, 1: {'members': {4, 5, 6, 7}}})
>>> H.graph["mapping"]
{0: 0, 1: 0, 2: 0, 3: 0, 4: 1, 5: 1, 6: 1, 7: 1}
Contracting a complete graph into one single SCC.
>>> G = nx.complete_graph(7, create_using=nx.DiGraph)
>>> H = nx.condensation(G)
>>> H.nodes
NodeView((0,))
>>> H.nodes.data()
NodeDataView({0: {'members': {0, 1, 2, 3, 4, 5, 6}}})
Notes
-----
After contracting all strongly connected components to a single node,
the resulting graph is a directed acyclic graph.
| null | (G, scc=None, *, backend=None, **backend_kwargs) |
30,492 | networkx.algorithms.cuts | conductance | Returns the conductance of two sets of nodes.
The *conductance* is the quotient of the cut size and the smaller of
the volumes of the two sets. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The conductance between the two sets `S` and `T`.
See also
--------
cut_size
edge_expansion
normalized_cut_size
volume
References
----------
.. [1] David Gleich.
*Hierarchical Directed Spectral Graph Partitioning*.
<https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
| null | (G, S, T=None, weight=None, *, backend=None, **backend_kwargs) |
30,493 | networkx.generators.degree_seq | configuration_model | Returns a random graph with the given degree sequence.
The configuration model generates a random pseudograph (graph with
parallel edges and self loops) by randomly assigning edges to
match the given degree sequence.
Parameters
----------
deg_sequence : list of nonnegative integers
Each list entry corresponds to the degree of a node.
create_using : NetworkX graph constructor, optional (default MultiGraph)
Graph type to create. If graph instance, then cleared before populated.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
G : MultiGraph
A graph with the specified degree sequence.
Nodes are labeled starting at 0 with an index
corresponding to the position in deg_sequence.
Raises
------
NetworkXError
If the degree sequence does not have an even sum.
See Also
--------
is_graphical
Notes
-----
As described by Newman [1]_.
A non-graphical degree sequence (not realizable by some simple
graph) is allowed since this function returns graphs with self
loops and parallel edges. An exception is raised if the degree
sequence does not have an even sum.
This configuration model construction process can lead to
duplicate edges and loops. You can remove the self-loops and
parallel edges (see below) which will likely result in a graph
that doesn't have the exact degree sequence specified.
The density of self-loops and parallel edges tends to decrease as
the number of nodes increases. However, typically the number of
self-loops will approach a Poisson distribution with a nonzero mean,
and similarly for the number of parallel edges. Consider a node
with *k* stubs. The probability of being joined to another stub of
the same node is basically (*k* - *1*) / *N*, where *k* is the
degree and *N* is the number of nodes. So the probability of a
self-loop scales like *c* / *N* for some constant *c*. As *N* grows,
this means we expect *c* self-loops. Similarly for parallel edges.
References
----------
.. [1] M.E.J. Newman, "The structure and function of complex networks",
SIAM REVIEW 45-2, pp 167-256, 2003.
Examples
--------
You can create a degree sequence following a particular distribution
by using the one of the distribution functions in
:mod:`~networkx.utils.random_sequence` (or one of your own). For
example, to create an undirected multigraph on one hundred nodes
with degree sequence chosen from the power law distribution:
>>> sequence = nx.random_powerlaw_tree_sequence(100, tries=5000)
>>> G = nx.configuration_model(sequence)
>>> len(G)
100
>>> actual_degrees = [d for v, d in G.degree()]
>>> actual_degrees == sequence
True
The returned graph is a multigraph, which may have parallel
edges. To remove any parallel edges from the returned graph:
>>> G = nx.Graph(G)
Similarly, to remove self-loops:
>>> G.remove_edges_from(nx.selfloop_edges(G))
| def generate(self):
# remaining_degree is mapping from int->remaining degree
self.remaining_degree = dict(enumerate(self.degree))
# add all nodes to make sure we get isolated nodes
self.graph = nx.Graph()
self.graph.add_nodes_from(self.remaining_degree)
# remove zero degree nodes
for n, d in list(self.remaining_degree.items()):
if d == 0:
del self.remaining_degree[n]
if len(self.remaining_degree) > 0:
# build graph in three phases according to how many unmatched edges
self.phase1()
self.phase2()
self.phase3()
return self.graph
| (deg_sequence, create_using=None, seed=None, *, backend=None, **backend_kwargs) |
30,495 | networkx.generators.community | connected_caveman_graph | Returns a connected caveman graph of `l` cliques of size `k`.
The connected caveman graph is formed by creating `n` cliques of size
`k`, then a single edge in each clique is rewired to a node in an
adjacent clique.
Parameters
----------
l : int
number of cliques
k : int
size of cliques (k at least 2 or NetworkXError is raised)
Returns
-------
G : NetworkX Graph
connected caveman graph
Raises
------
NetworkXError
If the size of cliques `k` is smaller than 2.
Notes
-----
This returns an undirected graph, it can be converted to a directed
graph using :func:`nx.to_directed`, or a multigraph using
``nx.MultiGraph(nx.caveman_graph(l, k))``. Only the undirected version is
described in [1]_ and it is unclear which of the directed
generalizations is most useful.
Examples
--------
>>> G = nx.connected_caveman_graph(3, 3)
References
----------
.. [1] Watts, D. J. 'Networks, Dynamics, and the Small-World Phenomenon.'
Amer. J. Soc. 105, 493-527, 1999.
| def _generate_communities(degree_seq, community_sizes, mu, max_iters, seed):
"""Returns a list of sets, each of which represents a community.
``degree_seq`` is the degree sequence that must be met by the
graph.
``community_sizes`` is the community size distribution that must be
met by the generated list of sets.
``mu`` is a float in the interval [0, 1] indicating the fraction of
intra-community edges incident to each node.
``max_iters`` is the number of times to try to add a node to a
community. This must be greater than the length of
``degree_seq``, otherwise this function will always fail. If
the number of iterations exceeds this value,
:exc:`~networkx.exception.ExceededMaxIterations` is raised.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
The communities returned by this are sets of integers in the set {0,
..., *n* - 1}, where *n* is the length of ``degree_seq``.
"""
# This assumes the nodes in the graph will be natural numbers.
result = [set() for _ in community_sizes]
n = len(degree_seq)
free = list(range(n))
for i in range(max_iters):
v = free.pop()
c = seed.choice(range(len(community_sizes)))
# s = int(degree_seq[v] * (1 - mu) + 0.5)
s = round(degree_seq[v] * (1 - mu))
# If the community is large enough, add the node to the chosen
# community. Otherwise, return it to the list of unaffiliated
# nodes.
if s < community_sizes[c]:
result[c].add(v)
else:
free.append(v)
# If the community is too big, remove a node from it.
if len(result[c]) > community_sizes[c]:
free.append(result[c].pop())
if not free:
return result
msg = "Could not assign communities; try increasing min_community"
raise nx.ExceededMaxIterations(msg)
| (l, k, *, backend=None, **backend_kwargs) |
30,496 | networkx.algorithms.components.connected | connected_components | Generate connected components.
Parameters
----------
G : NetworkX graph
An undirected graph
Returns
-------
comp : generator of sets
A generator of sets of nodes, one for each component of G.
Raises
------
NetworkXNotImplemented
If G is directed.
Examples
--------
Generate a sorted list of connected components, largest first.
>>> G = nx.path_graph(4)
>>> nx.add_path(G, [10, 11, 12])
>>> [len(c) for c in sorted(nx.connected_components(G), key=len, reverse=True)]
[4, 3]
If you only want the largest connected component, it's more
efficient to use max instead of sort.
>>> largest_cc = max(nx.connected_components(G), key=len)
To create the induced subgraph of each component use:
>>> S = [G.subgraph(c).copy() for c in nx.connected_components(G)]
See Also
--------
strongly_connected_components
weakly_connected_components
Notes
-----
For undirected graphs only.
| null | (G, *, backend=None, **backend_kwargs) |
30,497 | networkx.algorithms.swap | connected_double_edge_swap | Attempts the specified number of double-edge swaps in the graph `G`.
A double-edge swap removes two randomly chosen edges `(u, v)` and `(x,
y)` and creates the new edges `(u, x)` and `(v, y)`::
u--v u v
becomes | |
x--y x y
If either `(u, x)` or `(v, y)` already exist, then no swap is performed
so the actual number of swapped edges is always *at most* `nswap`.
Parameters
----------
G : graph
An undirected graph
nswap : integer (optional, default=1)
Number of double-edge swaps to perform
_window_threshold : integer
The window size below which connectedness of the graph will be checked
after each swap.
The "window" in this function is a dynamically updated integer that
represents the number of swap attempts to make before checking if the
graph remains connected. It is an optimization used to decrease the
running time of the algorithm in exchange for increased complexity of
implementation.
If the window size is below this threshold, then the algorithm checks
after each swap if the graph remains connected by checking if there is a
path joining the two nodes whose edge was just removed. If the window
size is above this threshold, then the algorithm performs do all the
swaps in the window and only then check if the graph is still connected.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
int
The number of successful swaps
Raises
------
NetworkXError
If the input graph is not connected, or if the graph has fewer than four
nodes.
Notes
-----
The initial graph `G` must be connected, and the resulting graph is
connected. The graph `G` is modified in place.
References
----------
.. [1] C. Gkantsidis and M. Mihail and E. Zegura,
The Markov chain simulation method for generating connected
power law random graphs, 2003.
http://citeseer.ist.psu.edu/gkantsidis03markov.html
| null | (G, nswap=1, _window_threshold=3, seed=None, *, backend=None, **backend_kwargs) |
30,498 | networkx.generators.random_graphs | connected_watts_strogatz_graph | Returns a connected Watts–Strogatz small-world graph.
Attempts to generate a connected graph by repeated generation of
Watts–Strogatz small-world graphs. An exception is raised if the maximum
number of tries is exceeded.
Parameters
----------
n : int
The number of nodes
k : int
Each node is joined with its `k` nearest neighbors in a ring
topology.
p : float
The probability of rewiring each edge
tries : int
Number of attempts to generate a connected graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Notes
-----
First create a ring over $n$ nodes [1]_. Then each node in the ring is joined
to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd).
Then shortcuts are created by replacing some edges as follows: for each
edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors"
with probability $p$ replace it with a new edge $(u, w)$ with uniformly
random choice of existing node $w$.
The entire process is repeated until a connected graph results.
See Also
--------
newman_watts_strogatz_graph
watts_strogatz_graph
References
----------
.. [1] Duncan J. Watts and Steven H. Strogatz,
Collective dynamics of small-world networks,
Nature, 393, pp. 440--442, 1998.
| def dual_barabasi_albert_graph(n, m1, m2, p, seed=None, initial_graph=None):
"""Returns a random graph using dual Barabási–Albert preferential attachment
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
are preferentially attached to existing nodes with high degree.
Parameters
----------
n : int
Number of nodes
m1 : int
Number of edges to link each new node to existing nodes with probability $p$
m2 : int
Number of edges to link each new node to existing nodes with probability $1-p$
p : float
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : Graph or None (default)
Initial network for Barabási–Albert algorithm.
A copy of `initial_graph` is used.
It should be connected for most use cases.
If None, starts from an star graph on max(m1, m2) + 1 nodes.
Returns
-------
G : Graph
Raises
------
NetworkXError
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
`p` does not satisfy ``0 <= p <= 1``, or
the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.
References
----------
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
"""
if m1 < 1 or m1 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
)
if m2 < 1 or m2 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
)
if p < 0 or p > 1:
raise nx.NetworkXError(
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
)
# For simplicity, if p == 0 or 1, just return BA
if p == 1:
return barabasi_albert_graph(n, m1, seed)
elif p == 0:
return barabasi_albert_graph(n, m2, seed)
if initial_graph is None:
# Default initial graph : empty graph on max(m1, m2) nodes
G = star_graph(max(m1, m2))
else:
if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
raise nx.NetworkXError(
f"Barabási–Albert initial graph must have between "
f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
)
G = initial_graph.copy()
# Target nodes for new edges
targets = list(G)
# List of existing nodes, with nodes repeated once for each adjacent edge
repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
# Start adding the remaining nodes.
source = len(G)
while source < n:
# Pick which m to use (m1 or m2)
if seed.random() < p:
m = m1
else:
m = m2
# Now choose m unique nodes from the existing nodes
# Pick uniformly from repeated_nodes (preferential attachment)
targets = _random_subset(repeated_nodes, m, seed)
# Add edges to m nodes from the source.
G.add_edges_from(zip([source] * m, targets))
# Add one node to the list for each new edge just created.
repeated_nodes.extend(targets)
# And the new node "source" has m edges to add to the list.
repeated_nodes.extend([source] * m)
source += 1
return G
| (n, k, p, tries=100, seed=None, *, backend=None, **backend_kwargs) |
30,500 | networkx.algorithms.structuralholes | constraint | Returns the constraint on all nodes in the graph ``G``.
The *constraint* is a measure of the extent to which a node *v* is
invested in those nodes that are themselves invested in the
neighbors of *v*. Formally, the *constraint on v*, denoted `c(v)`,
is defined by
.. math::
c(v) = \sum_{w \in N(v) \setminus \{v\}} \ell(v, w)
where $N(v)$ is the subset of the neighbors of `v` that are either
predecessors or successors of `v` and $\ell(v, w)$ is the local
constraint on `v` with respect to `w` [1]_. For the definition of local
constraint, see :func:`local_constraint`.
Parameters
----------
G : NetworkX graph
The graph containing ``v``. This can be either directed or undirected.
nodes : container, optional
Container of nodes in the graph ``G`` to compute the constraint. If
None, the constraint of every node is computed.
weight : None or string, optional
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
Returns
-------
dict
Dictionary with nodes as keys and the constraint on the node as values.
See also
--------
local_constraint
References
----------
.. [1] Burt, Ronald S.
"Structural holes and good ideas".
American Journal of Sociology (110): 349–399.
| null | (G, nodes=None, weight=None, *, backend=None, **backend_kwargs) |
30,501 | networkx.algorithms.minors.contraction | contracted_edge | Returns the graph that results from contracting the specified edge.
Edge contraction identifies the two endpoints of the edge as a single node
incident to any edge that was incident to the original two nodes. A graph
that results from edge contraction is called a *minor* of the original
graph.
Parameters
----------
G : NetworkX graph
The graph whose edge will be contracted.
edge : tuple
Must be a pair of nodes in `G`.
self_loops : Boolean
If this is True, any edges (including `edge`) joining the
endpoints of `edge` in `G` become self-loops on the new node in the
returned graph.
copy : Boolean (default True)
If this is True, a the contraction will be performed on a copy of `G`,
otherwise the contraction will happen in place.
Returns
-------
Networkx graph
A new graph object of the same type as `G` (leaving `G` unmodified)
with endpoints of `edge` identified in a single node. The right node
of `edge` will be merged into the left one, so only the left one will
appear in the returned graph.
Raises
------
ValueError
If `edge` is not an edge in `G`.
Examples
--------
Attempting to contract two nonadjacent nodes yields an error:
>>> G = nx.cycle_graph(4)
>>> nx.contracted_edge(G, (1, 3))
Traceback (most recent call last):
...
ValueError: Edge (1, 3) does not exist in graph G; cannot contract it
Contracting two adjacent nodes in the cycle graph on *n* nodes yields the
cycle graph on *n - 1* nodes:
>>> C5 = nx.cycle_graph(5)
>>> C4 = nx.cycle_graph(4)
>>> M = nx.contracted_edge(C5, (0, 1), self_loops=False)
>>> nx.is_isomorphic(M, C4)
True
See also
--------
contracted_nodes
quotient_graph
| null | (G, edge, self_loops=True, copy=True, *, backend=None, **backend_kwargs) |
30,502 | networkx.algorithms.minors.contraction | contracted_nodes | Returns the graph that results from contracting `u` and `v`.
Node contraction identifies the two nodes as a single node incident to any
edge that was incident to the original two nodes.
Parameters
----------
G : NetworkX graph
The graph whose nodes will be contracted.
u, v : nodes
Must be nodes in `G`.
self_loops : Boolean
If this is True, any edges joining `u` and `v` in `G` become
self-loops on the new node in the returned graph.
copy : Boolean
If this is True (default True), make a copy of
`G` and return that instead of directly changing `G`.
Returns
-------
Networkx graph
If Copy is True,
A new graph object of the same type as `G` (leaving `G` unmodified)
with `u` and `v` identified in a single node. The right node `v`
will be merged into the node `u`, so only `u` will appear in the
returned graph.
If copy is False,
Modifies `G` with `u` and `v` identified in a single node.
The right node `v` will be merged into the node `u`, so
only `u` will appear in the returned graph.
Notes
-----
For multigraphs, the edge keys for the realigned edges may
not be the same as the edge keys for the old edges. This is
natural because edge keys are unique only within each pair of nodes.
For non-multigraphs where `u` and `v` are adjacent to a third node
`w`, the edge (`v`, `w`) will be contracted into the edge (`u`,
`w`) with its attributes stored into a "contraction" attribute.
This function is also available as `identified_nodes`.
Examples
--------
Contracting two nonadjacent nodes of the cycle graph on four nodes `C_4`
yields the path graph (ignoring parallel edges):
>>> G = nx.cycle_graph(4)
>>> M = nx.contracted_nodes(G, 1, 3)
>>> P3 = nx.path_graph(3)
>>> nx.is_isomorphic(M, P3)
True
>>> G = nx.MultiGraph(P3)
>>> M = nx.contracted_nodes(G, 0, 2)
>>> M.edges
MultiEdgeView([(0, 1, 0), (0, 1, 1)])
>>> G = nx.Graph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> list(H.nodes())
[1]
>>> list(H.edges())
[(1, 1)]
In a ``MultiDiGraph`` with a self loop, the in and out edges will
be treated separately as edges, so while contracting a node which
has a self loop the contraction will add multiple edges:
>>> G = nx.MultiDiGraph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2)
>>> list(H.edges()) # edge 1->2, 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1), (1, 1)]
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> list(H.edges()) # edge 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1)]
See Also
--------
contracted_edge
quotient_graph
| null | (G, u, v, self_loops=True, copy=True, *, backend=None, **backend_kwargs) |
30,505 | networkx.relabel | convert_node_labels_to_integers | Returns a copy of the graph G with the nodes relabeled using
consecutive integers.
Parameters
----------
G : graph
A NetworkX graph
first_label : int, optional (default=0)
An integer specifying the starting offset in numbering nodes.
The new integer labels are numbered first_label, ..., n-1+first_label.
ordering : string
"default" : inherit node ordering from G.nodes()
"sorted" : inherit node ordering from sorted(G.nodes())
"increasing degree" : nodes are sorted by increasing degree
"decreasing degree" : nodes are sorted by decreasing degree
label_attribute : string, optional (default=None)
Name of node attribute to store old label. If None no attribute
is created.
Notes
-----
Node and edge attribute data are copied to the new (relabeled) graph.
There is no guarantee that the relabeling of nodes to integers will
give the same two integers for two (even identical graphs).
Use the `ordering` argument to try to preserve the order.
See Also
--------
relabel_nodes
| null | (G, first_label=0, ordering='default', label_attribute=None, *, backend=None, **backend_kwargs) |
30,507 | networkx.algorithms.core | core_number | Returns the core number for each node.
A k-core is a maximal subgraph that contains nodes of degree k or more.
The core number of a node is the largest value k of a k-core containing
that node.
Parameters
----------
G : NetworkX graph
An undirected or directed graph
Returns
-------
core_number : dictionary
A dictionary keyed by node to the core number.
Raises
------
NetworkXNotImplemented
If `G` is a multigraph or contains self loops.
Notes
-----
For directed graphs the node degree is defined to be the
in-degree + out-degree.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> nx.core_number(H)
{0: 1, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 0}
>>> G = nx.DiGraph()
>>> G.add_edges_from([(1, 2), (2, 1), (2, 3), (2, 4), (3, 4), (4, 3)])
>>> nx.core_number(G)
{1: 2, 2: 2, 3: 2, 4: 2}
References
----------
.. [1] An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
https://arxiv.org/abs/cs.DS/0310049
| null | (G, *, backend=None, **backend_kwargs) |
30,509 | networkx.algorithms.operators.product | corona_product | Returns the Corona product of G and H.
The corona product of $G$ and $H$ is the graph $C = G \circ H$ obtained by
taking one copy of $G$, called the center graph, $|V(G)|$ copies of $H$,
called the outer graph, and making the $i$-th vertex of $G$ adjacent to
every vertex of the $i$-th copy of $H$, where $1 ≤ i ≤ |V(G)|$.
Parameters
----------
G, H: NetworkX graphs
The graphs to take the carona product of.
`G` is the center graph and `H` is the outer graph
Returns
-------
C: NetworkX graph
The Corona product of G and H.
Raises
------
NetworkXError
If G and H are not both directed or both undirected.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> H = nx.path_graph(2)
>>> C = nx.corona_product(G, H)
>>> list(C)
[0, 1, 2, 3, (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)]
>>> print(C)
Graph with 12 nodes and 16 edges
References
----------
[1] M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi,
"Studying the corona product of graphs under some graph invariants,"
Transactions on Combinatorics, vol. 3, no. 3, pp. 43–49, Sep. 2014,
doi: 10.22108/toc.2014.5542.
[2] A. Faraji, "Corona Product in Graph Theory," Ali Faraji, May 11, 2021.
https://blog.alifaraji.ir/math/graph-theory/corona-product.html (accessed Dec. 07, 2021).
| null | (G, H, *, backend=None, **backend_kwargs) |
30,511 | networkx.algorithms.flow.mincost | cost_of_flow | Compute the cost of the flow given by flowDict on graph G.
Note that this function does not check for the validity of the
flow flowDict. This function will fail if the graph G and the
flow don't have the same edge set.
Parameters
----------
G : NetworkX graph
DiGraph on which a minimum cost flow satisfying all demands is
to be found.
weight : string
Edges of the graph G are expected to have an attribute weight
that indicates the cost incurred by sending one unit of flow on
that edge. If not present, the weight is considered to be 0.
Default value: 'weight'.
flowDict : dictionary
Dictionary of dictionaries keyed by nodes such that
flowDict[u][v] is the flow edge (u, v).
Returns
-------
cost : Integer, float
The total cost of the flow. This is given by the sum over all
edges of the product of the edge's flow and the edge's weight.
See also
--------
max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex
Notes
-----
This algorithm is not guaranteed to work if edge weights or demands
are floating point numbers (overflows and roundoff errors can
cause problems). As a workaround you can use integer numbers by
multiplying the relevant edge attributes by a convenient
constant factor (eg 100).
Examples
--------
>>> G = nx.DiGraph()
>>> G.add_node("a", demand=-5)
>>> G.add_node("d", demand=5)
>>> G.add_edge("a", "b", weight=3, capacity=4)
>>> G.add_edge("a", "c", weight=6, capacity=10)
>>> G.add_edge("b", "d", weight=1, capacity=9)
>>> G.add_edge("c", "d", weight=2, capacity=5)
>>> flowDict = nx.min_cost_flow(G)
>>> flowDict
{'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
>>> nx.cost_of_flow(G, flowDict)
24
| null | (G, flowDict, weight='weight', *, backend=None, **backend_kwargs) |
30,512 | networkx.algorithms.isomorphism.isomorph | could_be_isomorphic | Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters
----------
G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
-----
Checks for matching degree, triangle, and number of cliques sequences.
The triangle sequence contains the number of triangles each node is part of.
The clique sequence contains for each node the number of maximal cliques
involving that node.
| null | (G1, G2, *, backend=None, **backend_kwargs) |
30,514 | networkx.classes.function | create_empty_copy | Returns a copy of the graph G with all of the edges removed.
Parameters
----------
G : graph
A NetworkX graph
with_data : bool (default=True)
Propagate Graph and Nodes data to the new graph.
See Also
--------
empty_graph
| def create_empty_copy(G, with_data=True):
"""Returns a copy of the graph G with all of the edges removed.
Parameters
----------
G : graph
A NetworkX graph
with_data : bool (default=True)
Propagate Graph and Nodes data to the new graph.
See Also
--------
empty_graph
"""
H = G.__class__()
H.add_nodes_from(G.nodes(data=with_data))
if with_data:
H.graph.update(G.graph)
return H
| (G, with_data=True) |
30,515 | networkx.generators.small | cubical_graph |
Returns the 3-regular Platonic Cubical Graph
The skeleton of the cube (the nodes and edges) form a graph, with 8
nodes, and 12 edges. It is a special case of the hypercube graph.
It is one of 5 Platonic graphs, each a skeleton of its
Platonic solid [1]_.
Such graphs arise in parallel processing in computers.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
A cubical graph with 8 nodes and 12 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Cube#Cubical_graph
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) |
30,517 | networkx.algorithms.centrality.current_flow_betweenness | current_flow_betweenness_centrality | Compute current-flow betweenness centrality for nodes.
Current-flow betweenness centrality uses an electrical current
model for information spreading in contrast to betweenness
centrality which uses shortest paths.
Current-flow betweenness centrality is also known as
random-walk betweenness centrality [2]_.
Parameters
----------
G : graph
A NetworkX graph
normalized : bool, optional (default=True)
If True the betweenness values are normalized by 2/[(n-1)(n-2)] where
n is the number of nodes in G.
weight : string or None, optional (default=None)
Key for edge data used as the edge weight.
If None, then use 1 as each edge weight.
The weight reflects the capacity or the strength of the
edge.
dtype : data type (float)
Default data type for internal matrices.
Set to np.float32 for lower memory consumption.
solver : string (default='full')
Type of linear solver to use for computing the flow matrix.
Options are "full" (uses most memory), "lu" (recommended), and
"cg" (uses least memory).
Returns
-------
nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See Also
--------
approximate_current_flow_betweenness_centrality
betweenness_centrality
edge_betweenness_centrality
edge_current_flow_betweenness_centrality
Notes
-----
Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$
time [1]_, where $I(n-1)$ is the time needed to compute the
inverse Laplacian. For a full matrix this is $O(n^3)$ but using
sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
Laplacian matrix condition number.
The space required is $O(nw)$ where $w$ is the width of the sparse
Laplacian matrix. Worse case is $w=n$ for $O(n^2)$.
If the edges have a 'weight' attribute they will be used as
weights in this algorithm. Unspecified weights are set to 1.
References
----------
.. [1] Centrality Measures Based on Current Flow.
Ulrik Brandes and Daniel Fleischer,
Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
https://doi.org/10.1007/978-3-540-31856-9_44
.. [2] A measure of betweenness centrality based on random walks,
M. E. J. Newman, Social Networks 27, 39-54 (2005).
| null | (G, normalized=True, weight=None, dtype=<class 'float'>, solver='full', *, backend=None, **backend_kwargs) |
30,518 | networkx.algorithms.centrality.current_flow_betweenness_subset | current_flow_betweenness_centrality_subset | Compute current-flow betweenness centrality for subsets of nodes.
Current-flow betweenness centrality uses an electrical current
model for information spreading in contrast to betweenness
centrality which uses shortest paths.
Current-flow betweenness centrality is also known as
random-walk betweenness centrality [2]_.
Parameters
----------
G : graph
A NetworkX graph
sources: list of nodes
Nodes to use as sources for current
targets: list of nodes
Nodes to use as sinks for current
normalized : bool, optional (default=True)
If True the betweenness values are normalized by b=b/(n-1)(n-2) where
n is the number of nodes in G.
weight : string or None, optional (default=None)
Key for edge data used as the edge weight.
If None, then use 1 as each edge weight.
The weight reflects the capacity or the strength of the
edge.
dtype: data type (float)
Default data type for internal matrices.
Set to np.float32 for lower memory consumption.
solver: string (default='lu')
Type of linear solver to use for computing the flow matrix.
Options are "full" (uses most memory), "lu" (recommended), and
"cg" (uses least memory).
Returns
-------
nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See Also
--------
approximate_current_flow_betweenness_centrality
betweenness_centrality
edge_betweenness_centrality
edge_current_flow_betweenness_centrality
Notes
-----
Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$
time [1]_, where $I(n-1)$ is the time needed to compute the
inverse Laplacian. For a full matrix this is $O(n^3)$ but using
sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
Laplacian matrix condition number.
The space required is $O(nw)$ where $w$ is the width of the sparse
Laplacian matrix. Worse case is $w=n$ for $O(n^2)$.
If the edges have a 'weight' attribute they will be used as
weights in this algorithm. Unspecified weights are set to 1.
References
----------
.. [1] Centrality Measures Based on Current Flow.
Ulrik Brandes and Daniel Fleischer,
Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
https://doi.org/10.1007/978-3-540-31856-9_44
.. [2] A measure of betweenness centrality based on random walks,
M. E. J. Newman, Social Networks 27, 39-54 (2005).
| null | (G, sources, targets, normalized=True, weight=None, dtype=<class 'float'>, solver='lu', *, backend=None, **backend_kwargs) |
30,521 | networkx.algorithms.centrality.current_flow_closeness | current_flow_closeness_centrality | Compute current-flow closeness centrality for nodes.
Current-flow closeness centrality is variant of closeness
centrality based on effective resistance between nodes in
a network. This metric is also known as information centrality.
Parameters
----------
G : graph
A NetworkX graph.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
The weight reflects the capacity or the strength of the
edge.
dtype: data type (default=float)
Default data type for internal matrices.
Set to np.float32 for lower memory consumption.
solver: string (default='lu')
Type of linear solver to use for computing the flow matrix.
Options are "full" (uses most memory), "lu" (recommended), and
"cg" (uses least memory).
Returns
-------
nodes : dictionary
Dictionary of nodes with current flow closeness centrality as the value.
See Also
--------
closeness_centrality
Notes
-----
The algorithm is from Brandes [1]_.
See also [2]_ for the original definition of information centrality.
References
----------
.. [1] Ulrik Brandes and Daniel Fleischer,
Centrality Measures Based on Current Flow.
Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
https://doi.org/10.1007/978-3-540-31856-9_44
.. [2] Karen Stephenson and Marvin Zelen:
Rethinking centrality: Methods and examples.
Social Networks 11(1):1-37, 1989.
https://doi.org/10.1016/0378-8733(89)90016-6
| null | (G, weight=None, dtype=<class 'float'>, solver='lu', *, backend=None, **backend_kwargs) |
30,522 | networkx.algorithms.cuts | cut_size | Returns the size of the cut between two sets of nodes.
A *cut* is a partition of the nodes of a graph into two sets. The
*cut size* is the sum of the weights of the edges "between" the two
sets of nodes.
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`. If not specified, this is taken to
be the set complement of `S`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
Total weight of all edges from nodes in set `S` to nodes in
set `T` (and, in the case of directed graphs, all edges from
nodes in `T` to nodes in `S`).
Examples
--------
In the graph with two cliques joined by a single edges, the natural
bipartition of the graph into two blocks, one for each clique,
yields a cut of weight one::
>>> G = nx.barbell_graph(3, 0)
>>> S = {0, 1, 2}
>>> T = {3, 4, 5}
>>> nx.cut_size(G, S, T)
1
Each parallel edge in a multigraph is counted when determining the
cut size::
>>> G = nx.MultiGraph(["ab", "ab"])
>>> S = {"a"}
>>> T = {"b"}
>>> nx.cut_size(G, S, T)
2
Notes
-----
In a multigraph, the cut size is the total weight of edges including
multiplicity.
| null | (G, S, T=None, weight=None, *, backend=None, **backend_kwargs) |
30,524 | networkx.algorithms.cycles | cycle_basis | Returns a list of cycles which form a basis for cycles of G.
A basis for cycles of a network is a minimal collection of
cycles such that any cycle in the network can be written
as a sum of cycles in the basis. Here summation of cycles
is defined as "exclusive or" of the edges. Cycle bases are
useful, e.g. when deriving equations for electric circuits
using Kirchhoff's Laws.
Parameters
----------
G : NetworkX Graph
root : node, optional
Specify starting node for basis.
Returns
-------
A list of cycle lists. Each cycle list is a list of nodes
which forms a cycle (loop) in G.
Examples
--------
>>> G = nx.Graph()
>>> nx.add_cycle(G, [0, 1, 2, 3])
>>> nx.add_cycle(G, [0, 3, 4, 5])
>>> nx.cycle_basis(G, 0)
[[3, 4, 5, 0], [1, 2, 3, 0]]
Notes
-----
This is adapted from algorithm CACM 491 [1]_.
References
----------
.. [1] Paton, K. An algorithm for finding a fundamental set of
cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.
See Also
--------
simple_cycles
minimum_cycle_basis
| def recursive_simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
A `simple cycle`, or `elementary circuit`, is a closed path where
no node appears twice. Two elementary circuits are distinct if they
are not cyclic permutations of each other.
This version uses a recursive algorithm to build a list of cycles.
You should probably use the iterator version called simple_cycles().
Warning: This recursive version uses lots of RAM!
It appears in NetworkX for pedagogical value.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of cycles, where each cycle is represented by a list of nodes
along the cycle.
Example:
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> nx.recursive_simple_cycles(G)
[[0], [2], [0, 1, 2], [0, 2], [1, 2]]
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
https://doi.org/10.1137/0204007
See Also
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""
if blocked[thisnode]:
blocked[thisnode] = False
while B[thisnode]:
_unblock(B[thisnode].pop())
def circuit(thisnode, startnode, component):
closed = False # set to True if elementary path is closed
path.append(thisnode)
blocked[thisnode] = True
for nextnode in component[thisnode]: # direct successors of thisnode
if nextnode == startnode:
result.append(path[:])
closed = True
elif not blocked[nextnode]:
if circuit(nextnode, startnode, component):
closed = True
if closed:
_unblock(thisnode)
else:
for nextnode in component[thisnode]:
if thisnode not in B[nextnode]: # TODO: use set for speedup?
B[nextnode].append(thisnode)
path.pop() # remove thisnode from path
return closed
path = [] # stack of nodes in current path
blocked = defaultdict(bool) # vertex: blocked from search?
B = defaultdict(list) # graph portions that yield no elementary circuit
result = [] # list to accumulate the circuits found
# Johnson's algorithm exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in G:
if G.has_edge(v, v):
result.append([v])
G.remove_edge(v, v)
# Johnson's algorithm requires some ordering of the nodes.
# They might not be sortable so we assign an arbitrary ordering.
ordering = dict(zip(G, range(len(G))))
for s in ordering:
# Build the subgraph induced by s and following nodes in the ordering
subgraph = G.subgraph(node for node in G if ordering[node] >= ordering[s])
# Find the strongly connected component in the subgraph
# that contains the least node according to the ordering
strongcomp = nx.strongly_connected_components(subgraph)
mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
component = G.subgraph(mincomp)
if len(component) > 1:
# smallest node in the component according to the ordering
startnode = min(component, key=ordering.__getitem__)
for node in component:
blocked[node] = False
B[node][:] = []
dummy = circuit(startnode, startnode, component)
return result
| (G, root=None, *, backend=None, **backend_kwargs) |
30,525 | networkx.generators.classic | cycle_graph | Returns the cycle graph $C_n$ of cyclically connected nodes.
$C_n$ is a path with its two end-nodes connected.
.. plot::
>>> nx.draw(nx.cycle_graph(5))
Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
If create_using is directed, the direction is in increasing order.
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, create_using=None, *, backend=None, **backend_kwargs) |
30,528 | networkx.readwrite.json_graph.cytoscape | cytoscape_data | Returns data in Cytoscape JSON format (cyjs).
Parameters
----------
G : NetworkX Graph
The graph to convert to cytoscape format
name : string
A string which is mapped to the 'name' node element in cyjs format.
Must not have the same value as `ident`.
ident : string
A string which is mapped to the 'id' node element in cyjs format.
Must not have the same value as `name`.
Returns
-------
data: dict
A dictionary with cyjs formatted data.
Raises
------
NetworkXError
If the values for `name` and `ident` are identical.
See Also
--------
cytoscape_graph: convert a dictionary in cyjs format to a graph
References
----------
.. [1] Cytoscape user's manual:
http://manual.cytoscape.org/en/stable/index.html
Examples
--------
>>> G = nx.path_graph(2)
>>> nx.cytoscape_data(G) # doctest: +SKIP
{'data': [],
'directed': False,
'multigraph': False,
'elements': {'nodes': [{'data': {'id': '0', 'value': 0, 'name': '0'}},
{'data': {'id': '1', 'value': 1, 'name': '1'}}],
'edges': [{'data': {'source': 0, 'target': 1}}]}}
| def cytoscape_data(G, name="name", ident="id"):
"""Returns data in Cytoscape JSON format (cyjs).
Parameters
----------
G : NetworkX Graph
The graph to convert to cytoscape format
name : string
A string which is mapped to the 'name' node element in cyjs format.
Must not have the same value as `ident`.
ident : string
A string which is mapped to the 'id' node element in cyjs format.
Must not have the same value as `name`.
Returns
-------
data: dict
A dictionary with cyjs formatted data.
Raises
------
NetworkXError
If the values for `name` and `ident` are identical.
See Also
--------
cytoscape_graph: convert a dictionary in cyjs format to a graph
References
----------
.. [1] Cytoscape user's manual:
http://manual.cytoscape.org/en/stable/index.html
Examples
--------
>>> G = nx.path_graph(2)
>>> nx.cytoscape_data(G) # doctest: +SKIP
{'data': [],
'directed': False,
'multigraph': False,
'elements': {'nodes': [{'data': {'id': '0', 'value': 0, 'name': '0'}},
{'data': {'id': '1', 'value': 1, 'name': '1'}}],
'edges': [{'data': {'source': 0, 'target': 1}}]}}
"""
if name == ident:
raise nx.NetworkXError("name and ident must be different.")
jsondata = {"data": list(G.graph.items())}
jsondata["directed"] = G.is_directed()
jsondata["multigraph"] = G.is_multigraph()
jsondata["elements"] = {"nodes": [], "edges": []}
nodes = jsondata["elements"]["nodes"]
edges = jsondata["elements"]["edges"]
for i, j in G.nodes.items():
n = {"data": j.copy()}
n["data"]["id"] = j.get(ident) or str(i)
n["data"]["value"] = i
n["data"]["name"] = j.get(name) or str(i)
nodes.append(n)
if G.is_multigraph():
for e in G.edges(keys=True):
n = {"data": G.adj[e[0]][e[1]][e[2]].copy()}
n["data"]["source"] = e[0]
n["data"]["target"] = e[1]
n["data"]["key"] = e[2]
edges.append(n)
else:
for e in G.edges():
n = {"data": G.adj[e[0]][e[1]].copy()}
n["data"]["source"] = e[0]
n["data"]["target"] = e[1]
edges.append(n)
return jsondata
| (G, name='name', ident='id') |
30,529 | networkx.readwrite.json_graph.cytoscape | cytoscape_graph |
Create a NetworkX graph from a dictionary in cytoscape JSON format.
Parameters
----------
data : dict
A dictionary of data conforming to cytoscape JSON format.
name : string
A string which is mapped to the 'name' node element in cyjs format.
Must not have the same value as `ident`.
ident : string
A string which is mapped to the 'id' node element in cyjs format.
Must not have the same value as `name`.
Returns
-------
graph : a NetworkX graph instance
The `graph` can be an instance of `Graph`, `DiGraph`, `MultiGraph`, or
`MultiDiGraph` depending on the input data.
Raises
------
NetworkXError
If the `name` and `ident` attributes are identical.
See Also
--------
cytoscape_data: convert a NetworkX graph to a dict in cyjs format
References
----------
.. [1] Cytoscape user's manual:
http://manual.cytoscape.org/en/stable/index.html
Examples
--------
>>> data_dict = {
... "data": [],
... "directed": False,
... "multigraph": False,
... "elements": {
... "nodes": [
... {"data": {"id": "0", "value": 0, "name": "0"}},
... {"data": {"id": "1", "value": 1, "name": "1"}},
... ],
... "edges": [{"data": {"source": 0, "target": 1}}],
... },
... }
>>> G = nx.cytoscape_graph(data_dict)
>>> G.name
''
>>> G.nodes()
NodeView((0, 1))
>>> G.nodes(data=True)[0]
{'id': '0', 'value': 0, 'name': '0'}
>>> G.edges(data=True)
EdgeDataView([(0, 1, {'source': 0, 'target': 1})])
| null | (data, name='name', ident='id', *, backend=None, **backend_kwargs) |
30,530 | networkx.algorithms.d_separation | d_separated | Return whether nodes sets ``x`` and ``y`` are d-separated by ``z``.
.. deprecated:: 3.3
This function is deprecated and will be removed in NetworkX v3.5.
Please use `is_d_separator(G, x, y, z)`.
| def d_separated(G, x, y, z):
"""Return whether nodes sets ``x`` and ``y`` are d-separated by ``z``.
.. deprecated:: 3.3
This function is deprecated and will be removed in NetworkX v3.5.
Please use `is_d_separator(G, x, y, z)`.
"""
import warnings
warnings.warn(
"d_separated is deprecated and will be removed in NetworkX v3.5."
"Please use `is_d_separator(G, x, y, z)`.",
category=DeprecationWarning,
stacklevel=2,
)
return nx.is_d_separator(G, x, y, z)
| (G, x, y, z) |
30,533 | networkx.algorithms.dag | dag_longest_path | Returns the longest path in a directed acyclic graph (DAG).
If `G` has edges with `weight` attribute the edge data are used as
weight values.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
weight : str, optional
Edge data key to use for weight
default_weight : int, optional
The weight of edges that do not have a weight attribute
topo_order: list or tuple, optional
A topological order for `G` (if None, the function will compute one)
Returns
-------
list
Longest path
Raises
------
NetworkXNotImplemented
If `G` is not directed
Examples
--------
>>> DG = nx.DiGraph([(0, 1, {"cost": 1}), (1, 2, {"cost": 1}), (0, 2, {"cost": 42})])
>>> list(nx.all_simple_paths(DG, 0, 2))
[[0, 1, 2], [0, 2]]
>>> nx.dag_longest_path(DG)
[0, 1, 2]
>>> nx.dag_longest_path(DG, weight="cost")
[0, 2]
In the case where multiple valid topological orderings exist, `topo_order`
can be used to specify a specific ordering:
>>> DG = nx.DiGraph([(0, 1), (0, 2)])
>>> sorted(nx.all_topological_sorts(DG)) # Valid topological orderings
[[0, 1, 2], [0, 2, 1]]
>>> nx.dag_longest_path(DG, topo_order=[0, 1, 2])
[0, 1]
>>> nx.dag_longest_path(DG, topo_order=[0, 2, 1])
[0, 2]
See also
--------
dag_longest_path_length
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, weight='weight', default_weight=1, topo_order=None, *, backend=None, **backend_kwargs) |
30,534 | networkx.algorithms.dag | dag_longest_path_length | Returns the longest path length in a DAG
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
weight : string, optional
Edge data key to use for weight
default_weight : int, optional
The weight of edges that do not have a weight attribute
Returns
-------
int
Longest path length
Raises
------
NetworkXNotImplemented
If `G` is not directed
Examples
--------
>>> DG = nx.DiGraph([(0, 1, {"cost": 1}), (1, 2, {"cost": 1}), (0, 2, {"cost": 42})])
>>> list(nx.all_simple_paths(DG, 0, 2))
[[0, 1, 2], [0, 2]]
>>> nx.dag_longest_path_length(DG)
2
>>> nx.dag_longest_path_length(DG, weight="cost")
42
See also
--------
dag_longest_path
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, weight='weight', default_weight=1, *, backend=None, **backend_kwargs) |
30,535 | networkx.algorithms.dag | dag_to_branching | Returns a branching representing all (overlapping) paths from
root nodes to leaf nodes in the given directed acyclic graph.
As described in :mod:`networkx.algorithms.tree.recognition`, a
*branching* is a directed forest in which each node has at most one
parent. In other words, a branching is a disjoint union of
*arborescences*. For this function, each node of in-degree zero in
`G` becomes a root of one of the arborescences, and there will be
one leaf node for each distinct path from that root to a leaf node
in `G`.
Each node `v` in `G` with *k* parents becomes *k* distinct nodes in
the returned branching, one for each parent, and the sub-DAG rooted
at `v` is duplicated for each copy. The algorithm then recurses on
the children of each copy of `v`.
Parameters
----------
G : NetworkX graph
A directed acyclic graph.
Returns
-------
DiGraph
The branching in which there is a bijection between root-to-leaf
paths in `G` (in which multiple paths may share the same leaf)
and root-to-leaf paths in the branching (in which there is a
unique path from a root to a leaf).
Each node has an attribute 'source' whose value is the original
node to which this node corresponds. No other graph, node, or
edge attributes are copied into this new graph.
Raises
------
NetworkXNotImplemented
If `G` is not directed, or if `G` is a multigraph.
HasACycle
If `G` is not acyclic.
Examples
--------
To examine which nodes in the returned branching were produced by
which original node in the directed acyclic graph, we can collect
the mapping from source node to new nodes into a dictionary. For
example, consider the directed diamond graph::
>>> from collections import defaultdict
>>> from operator import itemgetter
>>>
>>> G = nx.DiGraph(nx.utils.pairwise("abd"))
>>> G.add_edges_from(nx.utils.pairwise("acd"))
>>> B = nx.dag_to_branching(G)
>>>
>>> sources = defaultdict(set)
>>> for v, source in B.nodes(data="source"):
... sources[source].add(v)
>>> len(sources["a"])
1
>>> len(sources["d"])
2
To copy node attributes from the original graph to the new graph,
you can use a dictionary like the one constructed in the above
example::
>>> for source, nodes in sources.items():
... for v in nodes:
... B.nodes[v].update(G.nodes[source])
Notes
-----
This function is not idempotent in the sense that the node labels in
the returned branching may be uniquely generated each time the
function is invoked. In fact, the node labels may not be integers;
in order to relabel the nodes to be more readable, you can use the
:func:`networkx.convert_node_labels_to_integers` function.
The current implementation of this function uses
:func:`networkx.prefix_tree`, so it is subject to the limitations of
that function.
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, *, backend=None, **backend_kwargs) |
30,536 | networkx.generators.social | davis_southern_women_graph | Returns Davis Southern women social network.
This is a bipartite graph.
References
----------
.. [1] A. Davis, Gardner, B. B., Gardner, M. R., 1941. Deep South.
University of Chicago Press, Chicago, IL.
| null | (*, backend=None, **backend_kwargs) |
30,537 | networkx.algorithms.summarization | dedensify | Compresses neighborhoods around high-degree nodes
Reduces the number of edges to high-degree nodes by adding compressor nodes
that summarize multiple edges of the same type to high-degree nodes (nodes
with a degree greater than a given threshold). Dedensification also has
the added benefit of reducing the number of edges around high-degree nodes.
The implementation currently supports graphs with a single edge type.
Parameters
----------
G: graph
A networkx graph
threshold: int
Minimum degree threshold of a node to be considered a high degree node.
The threshold must be greater than or equal to 2.
prefix: str or None, optional (default: None)
An optional prefix for denoting compressor nodes
copy: bool, optional (default: True)
Indicates if dedensification should be done inplace
Returns
-------
dedensified networkx graph : (graph, set)
2-tuple of the dedensified graph and set of compressor nodes
Notes
-----
According to the algorithm in [1]_, removes edges in a graph by
compressing/decompressing the neighborhoods around high degree nodes by
adding compressor nodes that summarize multiple edges of the same type
to high-degree nodes. Dedensification will only add a compressor node when
doing so will reduce the total number of edges in the given graph. This
implementation currently supports graphs with a single edge type.
Examples
--------
Dedensification will only add compressor nodes when doing so would result
in fewer edges::
>>> original_graph = nx.DiGraph()
>>> original_graph.add_nodes_from(
... ["1", "2", "3", "4", "5", "6", "A", "B", "C"]
... )
>>> original_graph.add_edges_from(
... [
... ("1", "C"), ("1", "B"),
... ("2", "C"), ("2", "B"), ("2", "A"),
... ("3", "B"), ("3", "A"), ("3", "6"),
... ("4", "C"), ("4", "B"), ("4", "A"),
... ("5", "B"), ("5", "A"),
... ("6", "5"),
... ("A", "6")
... ]
... )
>>> c_graph, c_nodes = nx.dedensify(original_graph, threshold=2)
>>> original_graph.number_of_edges()
15
>>> c_graph.number_of_edges()
14
A dedensified, directed graph can be "densified" to reconstruct the
original graph::
>>> original_graph = nx.DiGraph()
>>> original_graph.add_nodes_from(
... ["1", "2", "3", "4", "5", "6", "A", "B", "C"]
... )
>>> original_graph.add_edges_from(
... [
... ("1", "C"), ("1", "B"),
... ("2", "C"), ("2", "B"), ("2", "A"),
... ("3", "B"), ("3", "A"), ("3", "6"),
... ("4", "C"), ("4", "B"), ("4", "A"),
... ("5", "B"), ("5", "A"),
... ("6", "5"),
... ("A", "6")
... ]
... )
>>> c_graph, c_nodes = nx.dedensify(original_graph, threshold=2)
>>> # re-densifies the compressed graph into the original graph
>>> for c_node in c_nodes:
... all_neighbors = set(nx.all_neighbors(c_graph, c_node))
... out_neighbors = set(c_graph.neighbors(c_node))
... for out_neighbor in out_neighbors:
... c_graph.remove_edge(c_node, out_neighbor)
... in_neighbors = all_neighbors - out_neighbors
... for in_neighbor in in_neighbors:
... c_graph.remove_edge(in_neighbor, c_node)
... for out_neighbor in out_neighbors:
... c_graph.add_edge(in_neighbor, out_neighbor)
... c_graph.remove_node(c_node)
...
>>> nx.is_isomorphic(original_graph, c_graph)
True
References
----------
.. [1] Maccioni, A., & Abadi, D. J. (2016, August).
Scalable pattern matching over compressed graphs via dedensification.
In Proceedings of the 22nd ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining (pp. 1755-1764).
http://www.cs.umd.edu/~abadi/papers/graph-dedense.pdf
| null | (G, threshold, prefix=None, copy=True, *, backend=None, **backend_kwargs) |
30,538 | networkx.classes.function | degree | Returns a degree view of single node or of nbunch of nodes.
If nbunch is omitted, then return degrees of *all* nodes.
This function wraps the :func:`G.degree <networkx.Graph.degree>` property.
| def degree(G, nbunch=None, weight=None):
"""Returns a degree view of single node or of nbunch of nodes.
If nbunch is omitted, then return degrees of *all* nodes.
This function wraps the :func:`G.degree <networkx.Graph.degree>` property.
"""
return G.degree(nbunch, weight)
| (G, nbunch=None, weight=None) |
30,540 | networkx.algorithms.assortativity.correlation | degree_assortativity_coefficient | Compute degree assortativity of graph.
Assortativity measures the similarity of connections
in the graph with respect to the node degree.
Parameters
----------
G : NetworkX graph
x: string ('in','out')
The degree type for source node (directed graphs only).
y: string ('in','out')
The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
nodes: list or iterable (optional)
Compute degree assortativity only for nodes in container.
The default is all nodes.
Returns
-------
r : float
Assortativity of graph by degree.
Examples
--------
>>> G = nx.path_graph(4)
>>> r = nx.degree_assortativity_coefficient(G)
>>> print(f"{r:3.1f}")
-0.5
See Also
--------
attribute_assortativity_coefficient
numeric_assortativity_coefficient
degree_mixing_dict
degree_mixing_matrix
Notes
-----
This computes Eq. (21) in Ref. [1]_ , where e is the joint
probability distribution (mixing matrix) of the degrees. If G is
directed than the matrix e is the joint probability of the
user-specified degree type for the source and target.
References
----------
.. [1] M. E. J. Newman, Mixing patterns in networks,
Physical Review E, 67 026126, 2003
.. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M.
Edge direction and the structure of networks, PNAS 107, 10815-20 (2010).
| null | (G, x='out', y='in', weight=None, nodes=None, *, backend=None, **backend_kwargs) |
30,541 | networkx.algorithms.centrality.degree_alg | degree_centrality | Compute the degree centrality for nodes.
The degree centrality for a node v is the fraction of nodes it
is connected to.
Parameters
----------
G : graph
A networkx graph
Returns
-------
nodes : dictionary
Dictionary of nodes with degree centrality as the value.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
>>> nx.degree_centrality(G)
{0: 1.0, 1: 1.0, 2: 0.6666666666666666, 3: 0.6666666666666666}
See Also
--------
betweenness_centrality, load_centrality, eigenvector_centrality
Notes
-----
The degree centrality values are normalized by dividing by the maximum
possible degree in a simple graph n-1 where n is the number of nodes in G.
For multigraphs or graphs with self loops the maximum degree might
be higher than n-1 and values of degree centrality greater than 1
are possible.
| null | (G, *, backend=None, **backend_kwargs) |
30,542 | networkx.classes.function | degree_histogram | Returns a list of the frequency of each degree value.
Parameters
----------
G : Networkx graph
A graph
Returns
-------
hist : list
A list of frequencies of degrees.
The degree values are the index in the list.
Notes
-----
Note: the bins are width one, hence len(list) can be large
(Order(number_of_edges))
| def degree_histogram(G):
"""Returns a list of the frequency of each degree value.
Parameters
----------
G : Networkx graph
A graph
Returns
-------
hist : list
A list of frequencies of degrees.
The degree values are the index in the list.
Notes
-----
Note: the bins are width one, hence len(list) can be large
(Order(number_of_edges))
"""
counts = Counter(d for n, d in G.degree())
return [counts.get(i, 0) for i in range(max(counts) + 1 if counts else 0)]
| (G) |
30,543 | networkx.algorithms.assortativity.mixing | degree_mixing_dict | Returns dictionary representation of mixing matrix for degree.
Parameters
----------
G : graph
NetworkX graph object.
x: string ('in','out')
The degree type for source node (directed graphs only).
y: string ('in','out')
The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns
-------
d: dictionary
Counts or joint probability of occurrence of degree pairs.
| null | (G, x='out', y='in', weight=None, nodes=None, normalized=False, *, backend=None, **backend_kwargs) |
30,544 | networkx.algorithms.assortativity.mixing | degree_mixing_matrix | Returns mixing matrix for attribute.
Parameters
----------
G : graph
NetworkX graph object.
x: string ('in','out')
The degree type for source node (directed graphs only).
y: string ('in','out')
The degree type for target node (directed graphs only).
nodes: list or iterable (optional)
Build the matrix using only nodes in container.
The default is all nodes.
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
normalized : bool (default=True)
Return counts if False or probabilities if True.
mapping : dictionary, optional
Mapping from node degree to integer index in matrix.
If not specified, an arbitrary ordering will be used.
Returns
-------
m: numpy array
Counts, or joint probability, of occurrence of node degree.
Notes
-----
Definitions of degree mixing matrix vary on whether the matrix
should include rows for degree values that don't arise. Here we
do not include such empty-rows. But you can force them to appear
by inputting a `mapping` that includes those values. See examples.
Examples
--------
>>> G = nx.star_graph(3)
>>> mix_mat = nx.degree_mixing_matrix(G)
>>> mix_mat
array([[0. , 0.5],
[0.5, 0. ]])
If you want every possible degree to appear as a row, even if no nodes
have that degree, use `mapping` as follows,
>>> max_degree = max(deg for n, deg in G.degree)
>>> mapping = {x: x for x in range(max_degree + 1)} # identity mapping
>>> mix_mat = nx.degree_mixing_matrix(G, mapping=mapping)
>>> mix_mat
array([[0. , 0. , 0. , 0. ],
[0. , 0. , 0. , 0.5],
[0. , 0. , 0. , 0. ],
[0. , 0.5, 0. , 0. ]])
| null | (G, x='out', y='in', weight=None, nodes=None, normalized=True, mapping=None, *, backend=None, **backend_kwargs) |
30,545 | networkx.algorithms.assortativity.correlation | degree_pearson_correlation_coefficient | Compute degree assortativity of graph.
Assortativity measures the similarity of connections
in the graph with respect to the node degree.
This is the same as degree_assortativity_coefficient but uses the
potentially faster scipy.stats.pearsonr function.
Parameters
----------
G : NetworkX graph
x: string ('in','out')
The degree type for source node (directed graphs only).
y: string ('in','out')
The degree type for target node (directed graphs only).
weight: string or None, optional (default=None)
The edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
nodes: list or iterable (optional)
Compute pearson correlation of degrees only for specified nodes.
The default is all nodes.
Returns
-------
r : float
Assortativity of graph by degree.
Examples
--------
>>> G = nx.path_graph(4)
>>> r = nx.degree_pearson_correlation_coefficient(G)
>>> print(f"{r:3.1f}")
-0.5
Notes
-----
This calls scipy.stats.pearsonr.
References
----------
.. [1] M. E. J. Newman, Mixing patterns in networks
Physical Review E, 67 026126, 2003
.. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M.
Edge direction and the structure of networks, PNAS 107, 10815-20 (2010).
| null | (G, x='out', y='in', weight=None, nodes=None, *, backend=None, **backend_kwargs) |
30,547 | networkx.generators.degree_seq | degree_sequence_tree | Make a tree for the given degree sequence.
A tree has #nodes-#edges=1 so
the degree sequence must have
len(deg_sequence)-sum(deg_sequence)/2=1
| def generate(self):
# remaining_degree is mapping from int->remaining degree
self.remaining_degree = dict(enumerate(self.degree))
# add all nodes to make sure we get isolated nodes
self.graph = nx.Graph()
self.graph.add_nodes_from(self.remaining_degree)
# remove zero degree nodes
for n, d in list(self.remaining_degree.items()):
if d == 0:
del self.remaining_degree[n]
if len(self.remaining_degree) > 0:
# build graph in three phases according to how many unmatched edges
self.phase1()
self.phase2()
self.phase3()
return self.graph
| (deg_sequence, create_using=None, *, backend=None, **backend_kwargs) |
30,549 | networkx.generators.random_graphs | dense_gnm_random_graph | Returns a $G_{n,m}$ random graph.
In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
of all graphs with $n$ nodes and $m$ edges.
This algorithm should be faster than :func:`gnm_random_graph` for dense
graphs.
Parameters
----------
n : int
The number of nodes.
m : int
The number of edges.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
See Also
--------
gnm_random_graph
Notes
-----
Algorithm by Keith M. Briggs Mar 31, 2006.
Inspired by Knuth's Algorithm S (Selection sampling technique),
in section 3.4.2 of [1]_.
References
----------
.. [1] Donald E. Knuth, The Art of Computer Programming,
Volume 2/Seminumerical algorithms, Third Edition, Addison-Wesley, 1997.
| def dual_barabasi_albert_graph(n, m1, m2, p, seed=None, initial_graph=None):
"""Returns a random graph using dual Barabási–Albert preferential attachment
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
are preferentially attached to existing nodes with high degree.
Parameters
----------
n : int
Number of nodes
m1 : int
Number of edges to link each new node to existing nodes with probability $p$
m2 : int
Number of edges to link each new node to existing nodes with probability $1-p$
p : float
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : Graph or None (default)
Initial network for Barabási–Albert algorithm.
A copy of `initial_graph` is used.
It should be connected for most use cases.
If None, starts from an star graph on max(m1, m2) + 1 nodes.
Returns
-------
G : Graph
Raises
------
NetworkXError
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
`p` does not satisfy ``0 <= p <= 1``, or
the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.
References
----------
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
"""
if m1 < 1 or m1 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
)
if m2 < 1 or m2 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
)
if p < 0 or p > 1:
raise nx.NetworkXError(
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
)
# For simplicity, if p == 0 or 1, just return BA
if p == 1:
return barabasi_albert_graph(n, m1, seed)
elif p == 0:
return barabasi_albert_graph(n, m2, seed)
if initial_graph is None:
# Default initial graph : empty graph on max(m1, m2) nodes
G = star_graph(max(m1, m2))
else:
if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
raise nx.NetworkXError(
f"Barabási–Albert initial graph must have between "
f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
)
G = initial_graph.copy()
# Target nodes for new edges
targets = list(G)
# List of existing nodes, with nodes repeated once for each adjacent edge
repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
# Start adding the remaining nodes.
source = len(G)
while source < n:
# Pick which m to use (m1 or m2)
if seed.random() < p:
m = m1
else:
m = m2
# Now choose m unique nodes from the existing nodes
# Pick uniformly from repeated_nodes (preferential attachment)
targets = _random_subset(repeated_nodes, m, seed)
# Add edges to m nodes from the source.
G.add_edges_from(zip([source] * m, targets))
# Add one node to the list for each new edge just created.
repeated_nodes.extend(targets)
# And the new node "source" has m edges to add to the list.
repeated_nodes.extend([source] * m)
source += 1
return G
| (n, m, seed=None, *, backend=None, **backend_kwargs) |
30,550 | networkx.classes.function | density | Returns the density of a graph.
The density for undirected graphs is
.. math::
d = \frac{2m}{n(n-1)},
and for directed graphs is
.. math::
d = \frac{m}{n(n-1)},
where `n` is the number of nodes and `m` is the number of edges in `G`.
Notes
-----
The density is 0 for a graph without edges and 1 for a complete graph.
The density of multigraphs can be higher than 1.
Self loops are counted in the total number of edges so graphs with self
loops can have density higher than 1.
| def density(G):
r"""Returns the density of a graph.
The density for undirected graphs is
.. math::
d = \frac{2m}{n(n-1)},
and for directed graphs is
.. math::
d = \frac{m}{n(n-1)},
where `n` is the number of nodes and `m` is the number of edges in `G`.
Notes
-----
The density is 0 for a graph without edges and 1 for a complete graph.
The density of multigraphs can be higher than 1.
Self loops are counted in the total number of edges so graphs with self
loops can have density higher than 1.
"""
n = number_of_nodes(G)
m = number_of_edges(G)
if m == 0 or n <= 1:
return 0
d = m / (n * (n - 1))
if not G.is_directed():
d *= 2
return d
| (G) |
30,552 | networkx.generators.small | desargues_graph |
Returns the Desargues Graph
The Desargues Graph is a non-planar, distance-transitive cubic graph
with 20 nodes and 30 edges [1]_.
It is a symmetric graph. It can be represented in LCF notation
as [5,-5,9,-9]^5 [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Desargues Graph with 20 nodes and 30 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Desargues_graph
.. [2] https://mathworld.wolfram.com/DesarguesGraph.html
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
| (create_using=None, *, backend=None, **backend_kwargs) |
30,553 | networkx.algorithms.dag | descendants | Returns all nodes reachable from `source` in `G`.
Parameters
----------
G : NetworkX Graph
source : node in `G`
Returns
-------
set()
The descendants of `source` in `G`
Raises
------
NetworkXError
If node `source` is not in `G`.
Examples
--------
>>> DG = nx.path_graph(5, create_using=nx.DiGraph)
>>> sorted(nx.descendants(DG, 2))
[3, 4]
The `source` node is not a descendant of itself, but can be included manually:
>>> sorted(nx.descendants(DG, 2) | {2})
[2, 3, 4]
See also
--------
ancestors
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, source, *, backend=None, **backend_kwargs) |
30,554 | networkx.algorithms.traversal.breadth_first_search | descendants_at_distance | Returns all nodes at a fixed `distance` from `source` in `G`.
Parameters
----------
G : NetworkX graph
A graph
source : node in `G`
distance : the distance of the wanted nodes from `source`
Returns
-------
set()
The descendants of `source` in `G` at the given `distance` from `source`
Examples
--------
>>> G = nx.path_graph(5)
>>> nx.descendants_at_distance(G, 2, 2)
{0, 4}
>>> H = nx.DiGraph()
>>> H.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
>>> nx.descendants_at_distance(H, 0, 2)
{3, 4, 5, 6}
>>> nx.descendants_at_distance(H, 5, 0)
{5}
>>> nx.descendants_at_distance(H, 5, 1)
set()
| null | (G, source, distance, *, backend=None, **backend_kwargs) |
30,555 | networkx.algorithms.traversal.depth_first_search | dfs_edges | Iterate over edges in a depth-first-search (DFS).
Perform a depth-first-search over the nodes of `G` and yield
the edges in order. This may not generate all edges in `G`
(see `~networkx.algorithms.traversal.edgedfs.edge_dfs`).
Parameters
----------
G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and yield edges in
the component reachable from source.
depth_limit : int, optional (default=len(G))
Specify the maximum search depth.
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Yields
------
edge: 2-tuple of nodes
Yields edges resulting from the depth-first-search.
Examples
--------
>>> G = nx.path_graph(5)
>>> list(nx.dfs_edges(G, source=0))
[(0, 1), (1, 2), (2, 3), (3, 4)]
>>> list(nx.dfs_edges(G, source=0, depth_limit=2))
[(0, 1), (1, 2)]
Notes
-----
If a source is not specified then a source is chosen arbitrarily and
repeatedly until all components in the graph are searched.
The implementation of this function is adapted from David Eppstein's
depth-first search function in PADS [1]_, with modifications
to allow depth limits based on the Wikipedia article
"Depth-limited search" [2]_.
See Also
--------
dfs_preorder_nodes
dfs_postorder_nodes
dfs_labeled_edges
:func:`~networkx.algorithms.traversal.edgedfs.edge_dfs`
:func:`~networkx.algorithms.traversal.breadth_first_search.bfs_edges`
References
----------
.. [1] http://www.ics.uci.edu/~eppstein/PADS
.. [2] https://en.wikipedia.org/wiki/Depth-limited_search
| null | (G, source=None, depth_limit=None, *, sort_neighbors=None, backend=None, **backend_kwargs) |
30,556 | networkx.algorithms.traversal.depth_first_search | dfs_labeled_edges | Iterate over edges in a depth-first-search (DFS) labeled by type.
Parameters
----------
G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return edges in
the component reachable from source.
depth_limit : int, optional (default=len(G))
Specify the maximum search depth.
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
edges: generator
A generator of triples of the form (*u*, *v*, *d*), where (*u*,
*v*) is the edge being explored in the depth-first search and *d*
is one of the strings 'forward', 'nontree', 'reverse', or 'reverse-depth_limit'.
A 'forward' edge is one in which *u* has been visited but *v* has
not. A 'nontree' edge is one in which both *u* and *v* have been
visited but the edge is not in the DFS tree. A 'reverse' edge is
one in which both *u* and *v* have been visited and the edge is in
the DFS tree. When the `depth_limit` is reached via a 'forward' edge,
a 'reverse' edge is immediately generated rather than the subtree
being explored. To indicate this flavor of 'reverse' edge, the string
yielded is 'reverse-depth_limit'.
Examples
--------
The labels reveal the complete transcript of the depth-first search
algorithm in more detail than, for example, :func:`dfs_edges`::
>>> from pprint import pprint
>>>
>>> G = nx.DiGraph([(0, 1), (1, 2), (2, 1)])
>>> pprint(list(nx.dfs_labeled_edges(G, source=0)))
[(0, 0, 'forward'),
(0, 1, 'forward'),
(1, 2, 'forward'),
(2, 1, 'nontree'),
(1, 2, 'reverse'),
(0, 1, 'reverse'),
(0, 0, 'reverse')]
Notes
-----
If a source is not specified then a source is chosen arbitrarily and
repeatedly until all components in the graph are searched.
The implementation of this function is adapted from David Eppstein's
depth-first search function in `PADS`_, with modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited search`_".
.. _PADS: http://www.ics.uci.edu/~eppstein/PADS
.. _Depth-limited search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
dfs_edges
dfs_preorder_nodes
dfs_postorder_nodes
| null | (G, source=None, depth_limit=None, *, sort_neighbors=None, backend=None, **backend_kwargs) |
30,557 | networkx.algorithms.traversal.depth_first_search | dfs_postorder_nodes | Generate nodes in a depth-first-search post-ordering starting at source.
Parameters
----------
G : NetworkX graph
source : node, optional
Specify starting node for depth-first search.
depth_limit : int, optional (default=len(G))
Specify the maximum search depth.
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
nodes: generator
A generator of nodes in a depth-first-search post-ordering.
Examples
--------
>>> G = nx.path_graph(5)
>>> list(nx.dfs_postorder_nodes(G, source=0))
[4, 3, 2, 1, 0]
>>> list(nx.dfs_postorder_nodes(G, source=0, depth_limit=2))
[1, 0]
Notes
-----
If a source is not specified then a source is chosen arbitrarily and
repeatedly until all components in the graph are searched.
The implementation of this function is adapted from David Eppstein's
depth-first search function in `PADS`_, with modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited search`_".
.. _PADS: http://www.ics.uci.edu/~eppstein/PADS
.. _Depth-limited search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
dfs_edges
dfs_preorder_nodes
dfs_labeled_edges
:func:`~networkx.algorithms.traversal.edgedfs.edge_dfs`
:func:`~networkx.algorithms.traversal.breadth_first_search.bfs_tree`
| null | (G, source=None, depth_limit=None, *, sort_neighbors=None, backend=None, **backend_kwargs) |
30,558 | networkx.algorithms.traversal.depth_first_search | dfs_predecessors | Returns dictionary of predecessors in depth-first-search from source.
Parameters
----------
G : NetworkX graph
source : node, optional
Specify starting node for depth-first search.
Note that you will get predecessors for all nodes in the
component containing `source`. This input only specifies
where the DFS starts.
depth_limit : int, optional (default=len(G))
Specify the maximum search depth.
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
pred: dict
A dictionary with nodes as keys and predecessor nodes as values.
Examples
--------
>>> G = nx.path_graph(4)
>>> nx.dfs_predecessors(G, source=0)
{1: 0, 2: 1, 3: 2}
>>> nx.dfs_predecessors(G, source=0, depth_limit=2)
{1: 0, 2: 1}
Notes
-----
If a source is not specified then a source is chosen arbitrarily and
repeatedly until all components in the graph are searched.
The implementation of this function is adapted from David Eppstein's
depth-first search function in `PADS`_, with modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited search`_".
.. _PADS: http://www.ics.uci.edu/~eppstein/PADS
.. _Depth-limited search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
dfs_preorder_nodes
dfs_postorder_nodes
dfs_labeled_edges
:func:`~networkx.algorithms.traversal.edgedfs.edge_dfs`
:func:`~networkx.algorithms.traversal.breadth_first_search.bfs_tree`
| null | (G, source=None, depth_limit=None, *, sort_neighbors=None, backend=None, **backend_kwargs) |
30,559 | networkx.algorithms.traversal.depth_first_search | dfs_preorder_nodes | Generate nodes in a depth-first-search pre-ordering starting at source.
Parameters
----------
G : NetworkX graph
source : node, optional
Specify starting node for depth-first search and return nodes in
the component reachable from source.
depth_limit : int, optional (default=len(G))
Specify the maximum search depth.
sort_neighbors : function (default=None)
A function that takes an iterator over nodes as the input, and
returns an iterable of the same nodes with a custom ordering.
For example, `sorted` will sort the nodes in increasing order.
Returns
-------
nodes: generator
A generator of nodes in a depth-first-search pre-ordering.
Examples
--------
>>> G = nx.path_graph(5)
>>> list(nx.dfs_preorder_nodes(G, source=0))
[0, 1, 2, 3, 4]
>>> list(nx.dfs_preorder_nodes(G, source=0, depth_limit=2))
[0, 1, 2]
Notes
-----
If a source is not specified then a source is chosen arbitrarily and
repeatedly until all components in the graph are searched.
The implementation of this function is adapted from David Eppstein's
depth-first search function in `PADS`_, with modifications
to allow depth limits based on the Wikipedia article
"`Depth-limited search`_".
.. _PADS: http://www.ics.uci.edu/~eppstein/PADS
.. _Depth-limited search: https://en.wikipedia.org/wiki/Depth-limited_search
See Also
--------
dfs_edges
dfs_postorder_nodes
dfs_labeled_edges
:func:`~networkx.algorithms.traversal.breadth_first_search.bfs_edges`
| null | (G, source=None, depth_limit=None, *, sort_neighbors=None, backend=None, **backend_kwargs) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.