code: def network_simplex(
      G, demand="demand", capacity="capacity", weight="weight"
  ):
      r"""Find a minimum cost flow satisfying all demands in digraph G.
  
      This is a primal network simplex algorithm that uses the leaving
      arc rule to prevent cycling.
  
      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 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'.
  
      Returns
      -------
      flowCost : integer, float
          Cost of a minimum cost flow satisfying all demands.
  
      flowDict : dictionary
          Dictionary of dictionaries keyed by nodes such that
          flowDict[u][v] is the flow edge (u, v).
  
      Raises
      ------
      NetworkXError
          This exception is raised if the input graph is not directed or
          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 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).
  
      See also
      --------
      cost_of_flow, max_flow_min_cost, min_cost_flow, min_cost_flow_cost
  
      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.network_simplex(G)
      >>> flowCost
      24
      >>> flowDict
      {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
  
      The mincost flow algorithm can also be used to solve shortest path
      problems. To find the shortest path between two nodes u and v,
      give all edges an infinite capacity, give node u a demand of -1 and
      node v a demand a 1. Then run the network simplex. The value of a
      min cost flow will be the distance between u and v and edges
      carrying positive flow will indicate the path.
  
      >>> G = nx.DiGraph()
      >>> G.add_weighted_edges_from(
      ...     [
      ...         ("s", "u", 10),
      ...         ("s", "x", 5),
      ...         ("u", "v", 1),
      ...         ("u", "x", 2),
      ...         ("v", "y", 1),
      ...         ("x", "u", 3),
      ...         ("x", "v", 5),
      ...         ("x", "y", 2),
      ...         ("y", "s", 7),
      ...         ("y", "v", 6),
      ...     ]
      ... )
      >>> G.add_node("s", demand=-1)
      >>> G.add_node("v", demand=1)
      >>> flowCost, flowDict = nx.network_simplex(G)
      >>> flowCost == nx.shortest_path_length(G, "s", "v", weight="weight")
      True
      >>> sorted([(u, v) for u in flowDict for v in flowDict[u] if flowDict[u][v] > 0])
      [('s', 'x'), ('u', 'v'), ('x', 'u')]
      >>> nx.shortest_path(G, "s", "v", weight="weight")
      ['s', 'x', 'u', 'v']
  
      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.network_simplex(
      ...     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': {}}
  
      References
      ----------
      .. [1] Z. Kiraly, P. Kovacs.
             Efficient implementation of minimum-cost flow algorithms.
             Acta Universitatis Sapientiae, Informatica 4(1):67--118. 2012.
      .. [2] R. Barr, F. Glover, D. Klingman.
             Enhancement of spanning tree labeling procedures for network
             optimization.
             INFOR 17(1):16--34. 1979.
      """
      ###########################################################################
      # Problem essentials extraction and sanity check
      ###########################################################################
  
      if len(G) == 0:
          raise nx.NetworkXError("graph has no nodes")
  
      multigraph = G.is_multigraph()
  
      # extracting data essential to problem
      DEAF = _DataEssentialsAndFunctions(
          G, multigraph, demand=demand, capacity=capacity, weight=weight
      )
  
      ###########################################################################
      # Quick Error Detection
      ###########################################################################
  
      inf = float("inf")
      for u, d in zip(DEAF.node_list, DEAF.node_demands):
          if abs(d) == inf:
              raise nx.NetworkXError(f"node {u!r} has infinite demand")
      for e, w in zip(DEAF.edge_indices, DEAF.edge_weights):
          if abs(w) == inf:
              raise nx.NetworkXError(f"edge {e!r} has infinite weight")
      if not multigraph:
          edges = nx.selfloop_edges(G, data=True)
      else:
          edges = nx.selfloop_edges(G, data=True, keys=True)
      for e in edges:
          if abs(e[-1].get(weight, 0)) == inf:
              raise nx.NetworkXError(
                  f"edge {e[:-1]!r} has infinite weight"
              )
  
      ###########################################################################
      # Quick Infeasibility Detection
      ###########################################################################
  
      if sum(DEAF.node_demands) != 0:
          raise nx.NetworkXUnfeasible("total node demand is not zero")
      for e, c in zip(DEAF.edge_indices, DEAF.edge_capacities):
          if c < 0:
              raise nx.NetworkXUnfeasible(
                  f"edge {e!r} has negative capacity"
              )
      if not multigraph:
          edges = nx.selfloop_edges(G, data=True)
      else:
          edges = nx.selfloop_edges(G, data=True, keys=True)
      for e in edges:
          if e[-1].get(capacity, inf) < 0:
              raise nx.NetworkXUnfeasible(
                  f"edge {e[:-1]!r} has negative capacity"
              )
  
      ###########################################################################
      # Initialization
      ###########################################################################
  
      # Add a dummy node -1 and connect all existing nodes to it with infinite-
      # capacity dummy edges. Node -1 will serve as the root of the
      # spanning tree of the network simplex method. The new edges will used to
      # trivially satisfy the node demands and create an initial strongly
      # feasible spanning tree.
      for i, d in enumerate(DEAF.node_demands):
          # Must be greater-than here. Zero-demand nodes must have
          # edges pointing towards the root to ensure strong feasibility.
          if d > 0:
              DEAF.edge_sources.append(-1)
              DEAF.edge_targets.append(i)
          else:
              DEAF.edge_sources.append(i)
              DEAF.edge_targets.append(-1)
      faux_inf = (
          3
          * max(
              chain(
                  [
                      sum(c for c in DEAF.edge_capacities if c < inf),
                      sum(abs(w) for w in DEAF.edge_weights),
                  ],
                  (abs(d) for d in DEAF.node_demands),
              )
          )
          or 1
      )
  
      n = len(DEAF.node_list)  # number of nodes
      DEAF.edge_weights.extend(repeat(faux_inf, n))
      DEAF.edge_capacities.extend(repeat(faux_inf, n))
  
      # Construct the initial spanning tree.
      DEAF.initialize_spanning_tree(n, faux_inf)
  
      ###########################################################################
      # Pivot loop
      ###########################################################################
  
      for i, p, q in DEAF.find_entering_edges():
          Wn, We = DEAF.find_cycle(i, p, q)
          j, s, t = DEAF.find_leaving_edge(Wn, We)
          DEAF.augment_flow(Wn, We, DEAF.residual_capacity(j, s))
          # Do nothing more if the entering edge is the same as the leaving edge.
          if i != j:
              if DEAF.parent[t] != s:
                  # Ensure that s is the parent of t.
                  s, t = t, s
              if We.index(i) > We.index(j):
                  # Ensure that q is in the subtree rooted at t.
                  p, q = q, p
              DEAF.remove_edge(s, t)
              DEAF.make_root(q)
              DEAF.add_edge(i, p, q)
              DEAF.update_potentials(i, p, q)
  
      ###########################################################################
      # Infeasibility and unboundedness detection
      ###########################################################################
  
      if any(DEAF.edge_flow[i] != 0 for i in range(-n, 0)):
          raise nx.NetworkXUnfeasible(
              "no flow satisfies all node demands"
          )
  
      if any(
          DEAF.edge_flow[i] * 2 >= faux_inf
          for i in range(DEAF.edge_count)
      ) or any(
          e[-1].get(capacity, inf) == inf and e[-1].get(weight, 0) < 0
          for e in nx.selfloop_edges(G, data=True)
      ):
          raise nx.NetworkXUnbounded(
              "negative cycle with infinite capacity found"
          )
  
      ###########################################################################
      # Flow cost calculation and flow dict construction
      ###########################################################################
  
      del DEAF.edge_flow[DEAF.edge_count :]
      flow_cost = sum(
          w * x for w, x in zip(DEAF.edge_weights, DEAF.edge_flow)
      )
      flow_dict = {n: {} for n in DEAF.node_list}
  
      def add_entry(e):
          """Add a flow dict entry."""
          d = flow_dict[e[0]]
          for k in e[1:-2]:
              try:
                  d = d[k]
              except KeyError:
                  t = {}
                  d[k] = t
                  d = t
          d[e[-2]] = e[-1]
  
      DEAF.edge_sources = (
          DEAF.node_list[s] for s in DEAF.edge_sources
      )  # Use original nodes.
      DEAF.edge_targets = (
          DEAF.node_list[t] for t in DEAF.edge_targets
      )  # Use original nodes.
      if not multigraph:
          for e in zip(
              DEAF.edge_sources, DEAF.edge_targets, DEAF.edge_flow
          ):
              add_entry(e)
          edges = G.edges(data=True)
      else:
          for e in zip(
              DEAF.edge_sources,
              DEAF.edge_targets,
              DEAF.edge_keys,
              DEAF.edge_flow,
          ):
              add_entry(e)
          edges = G.edges(data=True, keys=True)
      for e in edges:
          if e[0] != e[1]:
              if e[-1].get(capacity, inf) == 0:
                  add_entry(e[:-1] + (0,))
          else:
              w = e[-1].get(weight, 0)
              if w >= 0:
                  add_entry(e[:-1] + (0,))
              else:
                  c = e[-1][capacity]
                  flow_cost += w * c
                  add_entry(e[:-1] + (c,))
  
      return flow_cost, flow_dict
  
imports: from itertools import chain, islice, repeat
  from math import ceil, sqrt
  import networkx
  from networkx.utils import not_implemented_for
label_desc: Use image node_id 1 for calling a global function with example usage: network_simplex(G, demand, capacity, weight) and returns: flow_cost,  flow_dict