"""Current-flow betweenness centrality measures for subsets of nodes.""" import networkx as nx from networkx.algorithms.centrality.flow_matrix import flow_matrix_row from networkx.utils import not_implemented_for, reverse_cuthill_mckee_ordering __all__ = [ "current_flow_betweenness_centrality_subset", "edge_current_flow_betweenness_centrality_subset", ] @not_implemented_for("directed") @nx._dispatchable(edge_attrs="weight") def current_flow_betweenness_centrality_subset( G, sources, targets, normalized=True, weight=None, dtype=float, solver="lu" ): r"""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). """ import numpy as np from networkx.utils import reverse_cuthill_mckee_ordering if not nx.is_connected(G): raise nx.NetworkXError("Graph not connected.") N = G.number_of_nodes() ordering = list(reverse_cuthill_mckee_ordering(G)) # make a copy with integer labels according to rcm ordering # this could be done without a copy if we really wanted to mapping = dict(zip(ordering, range(N))) H = nx.relabel_nodes(G, mapping) betweenness = dict.fromkeys(H, 0.0) # b[n]=0 for n in H for row, (s, t) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): for ss in sources: i = mapping[ss] for tt in targets: j = mapping[tt] betweenness[s] += 0.5 * abs(row.item(i) - row.item(j)) betweenness[t] += 0.5 * abs(row.item(i) - row.item(j)) if normalized: nb = (N - 1.0) * (N - 2.0) # normalization factor else: nb = 2.0 for node in H: betweenness[node] = betweenness[node] / nb + 1.0 / (2 - N) return {ordering[node]: value for node, value in betweenness.items()} @not_implemented_for("directed") @nx._dispatchable(edge_attrs="weight") def edge_current_flow_betweenness_centrality_subset( G, sources, targets, normalized=True, weight=None, dtype=float, solver="lu" ): r"""Compute current-flow betweenness centrality for edges using 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 : dict Dictionary of edge tuples with betweenness centrality as the value. See Also -------- betweenness_centrality edge_betweenness_centrality 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). """ import numpy as np if not nx.is_connected(G): raise nx.NetworkXError("Graph not connected.") N = G.number_of_nodes() ordering = list(reverse_cuthill_mckee_ordering(G)) # make a copy with integer labels according to rcm ordering # this could be done without a copy if we really wanted to mapping = dict(zip(ordering, range(N))) H = nx.relabel_nodes(G, mapping) edges = (tuple(sorted((u, v))) for u, v in H.edges()) betweenness = dict.fromkeys(edges, 0.0) if normalized: nb = (N - 1.0) * (N - 2.0) # normalization factor else: nb = 2.0 for row, (e) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): for ss in sources: i = mapping[ss] for tt in targets: j = mapping[tt] betweenness[e] += 0.5 * abs(row.item(i) - row.item(j)) betweenness[e] /= nb return {(ordering[s], ordering[t]): value for (s, t), value in betweenness.items()}