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Source code for networkx.algorithms.centrality.current_flow_betweenness

"""
Current-flow betweenness centrality measures.
"""
#    Copyright (C) 2010-2012 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.
import random
import networkx as nx
from networkx.algorithms.centrality.flow_matrix import *
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""

__all__ = ['current_flow_betweenness_centrality',
           'approximate_current_flow_betweenness_centrality',
           'edge_current_flow_betweenness_centrality']


[docs]def approximate_current_flow_betweenness_centrality(G, normalized=True, weight='weight', dtype=float, solver='full', epsilon=0.5, kmax=10000): r"""Compute the approximate current-flow betweenness centrality for nodes. Approximates the current-flow betweenness centrality within absolute error of epsilon with high probability [1]_. 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='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). epsilon: float Absolute error tolerance. kmax: int Maximum number of sample node pairs to use for approximation. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- current_flow_betweenness_centrality Notes ----- The running time is `O((1/\epsilon^2)m{\sqrt k} \log n)` and the space required is `O(m)` for n nodes and m edges. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. 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. http://www.inf.uni-konstanz.de/algo/publications/bf-cmbcf-05.pdf """ 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: from scipy import sparse from scipy.sparse import linalg 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.") solvername={"full" :FullInverseLaplacian, "lu": SuperLUInverseLaplacian, "cg": CGInverseLaplacian} 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 H = nx.relabel_nodes(G,dict(zip(ordering,range(n)))) L = laplacian_sparse_matrix(H, nodelist=range(n), weight=weight, dtype=dtype, format='csc') C = solvername[solver](L, dtype=dtype) # initialize solver betweenness = dict.fromkeys(H,0.0) nb = (n-1.0)*(n-2.0) # normalization factor cstar = n*(n-1)/nb l = 1 # parameter in approximation, adjustable k = l*int(np.ceil((cstar/epsilon)**2*np.log(n))) if k > kmax: raise nx.NetworkXError('Number random pairs k>kmax (%d>%d) '%(k,kmax), 'Increase kmax or epsilon') cstar2k = cstar/(2*k) for i in range(k): s,t = random.sample(range(n),2) b = np.zeros(n, dtype=dtype) b[s] = 1 b[t] = -1 p = C.solve(b) for v in H: if v==s or v==t: continue for nbr in H[v]: w = H[v][nbr].get(weight,1.0) betweenness[v] += w*np.abs(p[v]-p[nbr])*cstar2k if normalized: factor = 1.0 else: factor = nb/2.0 # remap to original node names and "unnormalize" if required return dict((ordering[k],float(v*factor)) for k,v in betweenness.items())
[docs]def current_flow_betweenness_centrality(G, normalized=True, weight='weight', dtype=float, solver='full'): r"""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='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 H = nx.relabel_nodes(G,dict(zip(ordering,range(n)))) 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): pos = dict(zip(row.argsort()[::-1],range(n))) for i in range(n): betweenness[s] += (i-pos[i])*row[i] betweenness[t] += (n-i-1-pos[i])*row[i] if normalized: nb = (n-1.0)*(n-2.0) # normalization factor else: nb = 2.0 for i,v in enumerate(H): # map integers to nodes betweenness[v] = float((betweenness[v]-i)*2.0/nb) return dict((ordering[k],v) for k,v in betweenness.items())
[docs]def edge_current_flow_betweenness_centrality(G, normalized=True, weight='weight', dtype=float, solver='full'): """Compute current-flow betweenness centrality for edges. 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='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 H = nx.relabel_nodes(G,dict(zip(ordering,range(n)))) 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): pos=dict(zip(row.argsort()[::-1],range(1,n+1))) for i in range(n): betweenness[e]+=(i+1-pos[i])*row[i] betweenness[e]+=(n-i-pos[i])*row[i] betweenness[e]/=nb return dict(((ordering[s],ordering[t]),float(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")