""" Current-flow betweenness centrality measures for subsets of nodes. """ # Copyright (C) 2010-2011 by # Aric Hagberg # Dan Schult # Pieter Swart # All rights reserved. # BSD license. __author__ = """Aric Hagberg (hagberg@lanl.gov)""" __all__ = ['current_flow_betweenness_centrality_subset', 'edge_current_flow_betweenness_centrality_subset'] import itertools import networkx as nx from networkx.algorithms.centrality.flow_matrix import * def current_flow_betweenness_centrality_subset(G,sources,targets, normalized=True, weight='weight', 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='weight') Key for edge data used as the edge weight. If None, then use 1 as each edge weight. 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. http://www.inf.uni-konstanz.de/algo/publications/bf-cmbcf-05.pdf .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). """ from networkx.utils import reverse_cuthill_mckee_ordering try: import numpy as np except ImportError: raise ImportError('current_flow_betweenness_centrality requires NumPy ', 'http://scipy.org/') try: import scipy except ImportError: raise ImportError('current_flow_betweenness_centrality requires SciPy ', 'http://scipy.org/') if G.is_directed(): raise nx.NetworkXError('current_flow_betweenness_centrality() ', 'not defined for digraphs.') 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[v]=0 for v 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*np.abs(row[i]-row[j]) betweenness[t]+=0.5*np.abs(row[i]-row[j]) if normalized: nb=(n-1.0)*(n-2.0) # normalization factor else: nb=2.0 for v in H: betweenness[v]=betweenness[v]/nb+1.0/(2-n) return dict((ordering[k],v) for k,v in betweenness.items()) def edge_current_flow_betweenness_centrality_subset(G, sources, targets, normalized=True, weight='weight', dtype=float, solver='lu'): """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='weight') Key for edge data used as the edge weight. If None, then use 1 as each edge weight. 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 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. http://www.inf.uni-konstanz.de/algo/publications/bf-cmbcf-05.pdf .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). """ from networkx.utils import reverse_cuthill_mckee_ordering try: import numpy as np except ImportError: raise ImportError('current_flow_betweenness_centrality requires NumPy ', 'http://scipy.org/') try: import scipy except ImportError: raise ImportError('current_flow_betweenness_centrality requires SciPy ', 'http://scipy.org/') if G.is_directed(): raise nx.NetworkXError('edge_current_flow_betweenness_centrality ', 'not defined for digraphs.') 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.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*np.abs(row[i]-row[j]) betweenness[e]/=nb return dict(((ordering[s],ordering[t]),v) for (s,t),v in betweenness.items()) # fixture for nose tests def setup_module(module): from nose import SkipTest try: import numpy import scipy except: raise SkipTest("NumPy not available")