def basic_stats(G, area=None, clean_intersects=False, tolerance=15,
                circuity_dist='gc'):
    """
    Calculate basic descriptive metric and topological stats for a graph.

    For an unprojected lat-lng graph, tolerance and graph units should be in
    degrees, and circuity_dist should be 'gc'. For a projected graph, tolerance
    and graph units should be in meters (or similar) and circuity_dist should be
    'euclidean'.

    Parameters
    ----------
    G : networkx multidigraph
    area : numeric
        the area covered by the street network, in square meters (typically land
        area); if none, will skip all density-based metrics
    clean_intersects : bool
        if True, calculate clean intersections count (and density, if area is
        provided)
    tolerance : numeric
        tolerance value passed along if clean_intersects=True, see
        clean_intersections() function documentation for details and usage
    circuity_dist : str
        'gc' or 'euclidean', how to calculate straight-line distances for
        circuity measurement; use former for lat-lng networks and latter for
        projected networks

    Returns
    -------
    stats : dict
        dictionary of network measures containing the following elements (some
        keys may not be present, based on the arguments passed into the function):

          - n = number of nodes in the graph
          - m = number of edges in the graph
          - k_avg = average node degree of the graph
          - intersection_count = number of intersections in graph, that is,
                nodes with >1 street emanating from them
          - streets_per_node_avg = how many streets (edges in the undirected
                representation of the graph) emanate from each node (ie,
                intersection or dead-end) on average (mean)
          - streets_per_node_counts = dict, with keys of number of streets
                emanating from the node, and values of number of nodes with this
                count
          - streets_per_node_proportion = dict, same as previous, but as a
                proportion of the total, rather than counts
          - edge_length_total = sum of all edge lengths in the graph, in meters
          - edge_length_avg = mean edge length in the graph, in meters
          - street_length_total = sum of all edges in the undirected
                representation of the graph
          - street_length_avg = mean edge length in the undirected
                representation of the graph, in meters
          - street_segments_count = number of edges in the undirected
                representation of the graph
          - node_density_km = n divided by area in square kilometers
          - intersection_density_km = intersection_count divided by area in
                square kilometers
          - edge_density_km = edge_length_total divided by area in square
                kilometers
          - street_density_km = street_length_total divided by area in square
                kilometers
          - circuity_avg = edge_length_total divided by the sum of the great
                circle distances between the nodes of each edge
          - self_loop_proportion = proportion of edges that have a single node
                as its two endpoints (ie, the edge links nodes u and v, and u==v)
          - clean_intersection_count = number of intersections in street
                network, merging complex ones into single points
          - clean_intersection_density_km = clean_intersection_count divided by
                area in square kilometers
    """

    sq_m_in_sq_km = 1e6 #there are 1 million sq meters in 1 sq km
    G_undirected = None

    # calculate the number of nodes, n, and the number of edges, m, in the graph
    n = len(list(G.nodes()))
    m = len(list(G.edges()))

    # calculate the average degree of the graph
    k_avg = 2 * m / n

    if 'streets_per_node' in G.graph:
        # get the degrees saved as a graph attribute (from an undirected
        # representation of the graph). this is not the degree of the nodes in
        # the directed graph, but rather represents the number of streets
        # (unidirected edges) emanating from each node. see
        # count_streets_per_node function.
        streets_per_node = G.graph['streets_per_node']
    else:
        # count how many street segments emanate from each node in this graph
        streets_per_node = count_streets_per_node(G)

    # count number of intersections in graph, as nodes with >1 street emanating
    # from them
    node_ids = set(G.nodes())
    intersection_count = len([True for node, count in streets_per_node.items() if (count > 1) and (node in node_ids)])

    # calculate the average number of streets (unidirected edges) incident to
    # each node
    streets_per_node_avg = sum(streets_per_node.values()) / n

    # create a dict where key = number of streets (unidirected edges) incident
    # to each node, and value = how many nodes are of this number in the graph
    streets_per_node_counts = {num:list(streets_per_node.values()).count(num) for num in range(max(streets_per_node.values()) + 1)}

    # degree proportions: dict where key = each degree and value = what
    # proportion of nodes are of this degree in the graph
    streets_per_node_proportion = {num:count/n for num, count in streets_per_node_counts.items()}

    # calculate the total and average edge lengths
    edge_length_total = sum([d['length'] for u, v, d in G.edges(data=True)])
    edge_length_avg = edge_length_total / m

    # calculate the total and average street segment lengths (so, edges without
    # double-counting two-way streets)
    if G_undirected is None:
        G_undirected = G.to_undirected(reciprocal=False)
    street_length_total = sum([d['length'] for u, v, d in G_undirected.edges(data=True)])
    street_segments_count = len(list(G_undirected.edges(keys=True)))
    street_length_avg = street_length_total / street_segments_count

    # calculate clean intersection counts
    if clean_intersects:
        clean_intersection_points = clean_intersections(G, tolerance=tolerance, dead_ends=False )
        clean_intersection_count = len(clean_intersection_points)
    else:
        clean_intersection_count = None

    # we can calculate density metrics only if area is not null
    if area is not None:
        area_km = area / sq_m_in_sq_km

        # calculate node density as nodes per sq km
        node_density_km = n / area_km

        # calculate intersection density as nodes with >1 street emanating from
        # them, per sq km
        intersection_density_km = intersection_count / area_km

        # calculate edge density as linear meters per sq km
        edge_density_km = edge_length_total / area_km

        # calculate street density as linear meters per sq km
        street_density_km = street_length_total / area_km

        if clean_intersects:
            clean_intersection_density_km = clean_intersection_count / area_km
        else:
            clean_intersection_density_km = None
    else:
        # if area is None, then we cannot calculate density
        node_density_km = None
        intersection_density_km = None
        edge_density_km = None
        street_density_km = None
        clean_intersection_density_km = None

    # average circuity: sum of edge lengths divided by sum of straight-line
    # distance between edge endpoints. first load all the edges origin and
    # destination coordinates as a dataframe, then calculate the straight-line
    # distance
    coords = np.array([[G.nodes[u]['y'], G.nodes[u]['x'], G.nodes[v]['y'], G.nodes[v]['x']] for u, v, k in G.edges(keys=True)])
    df_coords = pd.DataFrame(coords, columns=['u_y', 'u_x', 'v_y', 'v_x'])
    if circuity_dist == 'gc':
        gc_distances = great_circle_vec(lat1=df_coords['u_y'],
                                        lng1=df_coords['u_x'],
                                        lat2=df_coords['v_y'],
                                        lng2=df_coords['v_x'])
    elif circuity_dist == 'euclidean':
        gc_distances = euclidean_dist_vec(y1=df_coords['u_y'],
                                          x1=df_coords['u_x'],
                                          y2=df_coords['v_y'],
                                          x2=df_coords['v_x'])
    else:
        raise ValueError('circuity_dist must be "gc" or "euclidean"')

    gc_distances = gc_distances.fillna(value=0)
    try:
        circuity_avg = edge_length_total / gc_distances.sum()
    except ZeroDivisionError:
        circuity_avg = np.nan

    # percent of edges that are self-loops, ie both endpoints are the same node
    self_loops = [True for u, v, k in G.edges(keys=True) if u == v]
    self_loops_count = len(self_loops)
    self_loop_proportion = self_loops_count / m

    # assemble the results
    stats = {'n':n,
             'm':m,
             'k_avg':k_avg,
             'intersection_count':intersection_count,
             'streets_per_node_avg':streets_per_node_avg,
             'streets_per_node_counts':streets_per_node_counts,
             'streets_per_node_proportion':streets_per_node_proportion,
             'edge_length_total':edge_length_total,
             'edge_length_avg':edge_length_avg,
             'street_length_total':street_length_total,
             'street_length_avg':street_length_avg,
             'street_segments_count':street_segments_count,
             'node_density_km':node_density_km,
             'intersection_density_km':intersection_density_km,
             'edge_density_km':edge_density_km,
             'street_density_km':street_density_km,
             'circuity_avg':circuity_avg,
             'self_loop_proportion':self_loop_proportion,
             'clean_intersection_count':clean_intersection_count,
             'clean_intersection_density_km':clean_intersection_density_km}

    # return the results
    return stats
Calculate basic descriptive metric and topological stats for a graph.

    For an unprojected lat-lng graph, tolerance and graph units should be in
    degrees, and circuity_dist should be 'gc'. For a projected graph, tolerance
    and graph units should be in meters (or similar) and circuity_dist should be
    'euclidean'.

    Parameters
    ----------
    G : networkx multidigraph
    area : numeric
        the area covered by the street network, in square meters (typically land
        area); if none, will skip all density-based metrics
    clean_intersects : bool
        if True, calculate clean intersections count (and density, if area is
        provided)
    tolerance : numeric
        tolerance value passed along if clean_intersects=True, see
        clean_intersections() function documentation for details and usage
    circuity_dist : str
        'gc' or 'euclidean', how to calculate straight-line distances for
        circuity measurement; use former for lat-lng networks and latter for
        projected networks

    Returns
    -------
    stats : dict
        dictionary of network measures containing the following elements (some
        keys may not be present, based on the arguments passed into the function):

          - n = number of nodes in the graph
          - m = number of edges in the graph
          - k_avg = average node degree of the graph
          - intersection_count = number of intersections in graph, that is,
                nodes with >1 street emanating from them
          - streets_per_node_avg = how many streets (edges in the undirected
                representation of the graph) emanate from each node (ie,
                intersection or dead-end) on average (mean)
          - streets_per_node_counts = dict, with keys of number of streets
                emanating from the node, and values of number of nodes with this
                count
          - streets_per_node_proportion = dict, same as previous, but as a
                proportion of the total, rather than counts
          - edge_length_total = sum of all edge lengths in the graph, in meters
          - edge_length_avg = mean edge length in the graph, in meters
          - street_length_total = sum of all edges in the undirected
                representation of the graph
          - street_length_avg = mean edge length in the undirected
                representation of the graph, in meters
          - street_segments_count = number of edges in the undirected
                representation of the graph
          - node_density_km = n divided by area in square kilometers
          - intersection_density_km = intersection_count divided by area in
                square kilometers
          - edge_density_km = edge_length_total divided by area in square
                kilometers
          - street_density_km = street_length_total divided by area in square
                kilometers
          - circuity_avg = edge_length_total divided by the sum of the great
                circle distances between the nodes of each edge
          - self_loop_proportion = proportion of edges that have a single node
                as its two endpoints (ie, the edge links nodes u and v, and u==v)
          - clean_intersection_count = number of intersections in street
                network, merging complex ones into single points
          - clean_intersection_density_km = clean_intersection_count divided by
                area in square kilometers