325 lines
9.3 KiB
Python
325 lines
9.3 KiB
Python
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# -*- coding: utf-8 -*-
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"""
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Subraph centrality and communicability betweenness.
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"""
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# Copyright (C) 2011 by
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# Aric Hagberg <hagberg@lanl.gov>
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# Dan Schult <dschult@colgate.edu>
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# Pieter Swart <swart@lanl.gov>
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# All rights reserved.
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# BSD license.
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import networkx as nx
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from networkx.utils import *
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__author__ = "\n".join(['Aric Hagberg (hagberg@lanl.gov)',
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'Franck Kalala (franckkalala@yahoo.fr'])
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__all__ = ['subgraph_centrality_exp',
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'subgraph_centrality',
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'communicability_betweenness_centrality',
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'estrada_index'
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]
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@not_implemented_for('directed')
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@not_implemented_for('multigraph')
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def subgraph_centrality_exp(G):
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r"""Returns the subgraph centrality for each node of G.
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Subgraph centrality of a node `n` is the sum of weighted closed
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walks of all lengths starting and ending at node `n`. The weights
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decrease with path length. Each closed walk is associated with a
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connected subgraph ([1]_).
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Parameters
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----------
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G: graph
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Returns
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-------
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nodes:dictionary
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Dictionary of nodes with subgraph centrality as the value.
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Raises
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------
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NetworkXError
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If the graph is not undirected and simple.
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See Also
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--------
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subgraph_centrality:
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Alternative algorithm of the subgraph centrality for each node of G.
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Notes
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-----
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This version of the algorithm exponentiates the adjacency matrix.
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The subgraph centrality of a node `u` in G can be found using
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the matrix exponential of the adjacency matrix of G [1]_,
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.. math::
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SC(u)=(e^A)_{uu} .
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References
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----------
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.. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
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"Subgraph centrality in complex networks",
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Physical Review E 71, 056103 (2005).
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https://arxiv.org/abs/cond-mat/0504730
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Examples
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--------
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(Example from [1]_)
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>>> G = nx.Graph([(1,2),(1,5),(1,8),(2,3),(2,8),(3,4),(3,6),(4,5),(4,7),(5,6),(6,7),(7,8)])
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>>> sc = nx.subgraph_centrality_exp(G)
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>>> print(['%s %0.2f'%(node,sc[node]) for node in sorted(sc)])
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['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
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"""
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# alternative implementation that calculates the matrix exponential
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import scipy.linalg
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nodelist = list(G) # ordering of nodes in matrix
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A = nx.to_numpy_array(G, nodelist)
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# convert to 0-1 matrix
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A[A != 0.0] = 1
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expA = scipy.linalg.expm(A)
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# convert diagonal to dictionary keyed by node
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sc = dict(zip(nodelist, map(float, expA.diagonal())))
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return sc
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@not_implemented_for('directed')
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@not_implemented_for('multigraph')
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def subgraph_centrality(G):
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r"""Returns subgraph centrality for each node in G.
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Subgraph centrality of a node `n` is the sum of weighted closed
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walks of all lengths starting and ending at node `n`. The weights
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decrease with path length. Each closed walk is associated with a
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connected subgraph ([1]_).
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Parameters
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----------
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G: graph
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with subgraph centrality as the value.
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Raises
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------
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NetworkXError
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If the graph is not undirected and simple.
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See Also
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--------
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subgraph_centrality_exp:
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Alternative algorithm of the subgraph centrality for each node of G.
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Notes
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-----
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This version of the algorithm computes eigenvalues and eigenvectors
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of the adjacency matrix.
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Subgraph centrality of a node `u` in G can be found using
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a spectral decomposition of the adjacency matrix [1]_,
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.. math::
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SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},
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where `v_j` is an eigenvector of the adjacency matrix `A` of G
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corresponding corresponding to the eigenvalue `\lambda_j`.
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Examples
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--------
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(Example from [1]_)
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>>> G = nx.Graph([(1,2),(1,5),(1,8),(2,3),(2,8),(3,4),(3,6),(4,5),(4,7),(5,6),(6,7),(7,8)])
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>>> sc = nx.subgraph_centrality(G)
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>>> print(['%s %0.2f'%(node,sc[node]) for node in sorted(sc)])
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['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
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References
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----------
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.. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
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"Subgraph centrality in complex networks",
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Physical Review E 71, 056103 (2005).
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https://arxiv.org/abs/cond-mat/0504730
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"""
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import numpy
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import numpy.linalg
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nodelist = list(G) # ordering of nodes in matrix
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A = nx.to_numpy_matrix(G, nodelist)
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# convert to 0-1 matrix
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A[A != 0.0] = 1
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w, v = numpy.linalg.eigh(A.A)
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vsquare = numpy.array(v)**2
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expw = numpy.exp(w)
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xg = numpy.dot(vsquare, expw)
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# convert vector dictionary keyed by node
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sc = dict(zip(nodelist, map(float, xg)))
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return sc
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@not_implemented_for('directed')
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@not_implemented_for('multigraph')
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def communicability_betweenness_centrality(G, normalized=True):
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r"""Returns subgraph communicability for all pairs of nodes in G.
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Communicability betweenness measure makes use of the number of walks
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connecting every pair of nodes as the basis of a betweenness centrality
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measure.
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Parameters
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----------
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G: graph
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with communicability betweenness as the value.
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Raises
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------
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NetworkXError
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If the graph is not undirected and simple.
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Notes
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-----
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Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges,
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and `A` denote the adjacency matrix of `G`.
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Let `G(r)=(V,E(r))` be the graph resulting from
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removing all edges connected to node `r` but not the node itself.
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The adjacency matrix for `G(r)` is `A+E(r)`, where `E(r)` has nonzeros
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only in row and column `r`.
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The subraph betweenness of a node `r` is [1]_
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.. math::
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\omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}},
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p\neq q, q\neq r,
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where
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`G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}` is the number of walks
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involving node r,
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`G_{pq}=(e^{A})_{pq}` is the number of closed walks starting
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at node `p` and ending at node `q`,
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and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the
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number of terms in the sum.
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The resulting `\omega_{r}` takes values between zero and one.
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The lower bound cannot be attained for a connected
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graph, and the upper bound is attained in the star graph.
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References
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----------
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.. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano,
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"Communicability Betweenness in Complex Networks"
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Physica A 388 (2009) 764-774.
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https://arxiv.org/abs/0905.4102
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Examples
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--------
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>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
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>>> cbc = nx.communicability_betweenness_centrality(G)
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"""
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import scipy
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import scipy.linalg
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nodelist = list(G) # ordering of nodes in matrix
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n = len(nodelist)
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A = nx.to_numpy_matrix(G, nodelist)
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# convert to 0-1 matrix
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A[A != 0.0] = 1
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expA = scipy.linalg.expm(A.A)
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mapping = dict(zip(nodelist, range(n)))
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cbc = {}
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for v in G:
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# remove row and col of node v
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i = mapping[v]
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row = A[i, :].copy()
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col = A[:, i].copy()
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A[i, :] = 0
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A[:, i] = 0
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B = (expA - scipy.linalg.expm(A.A)) / expA
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# sum with row/col of node v and diag set to zero
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B[i, :] = 0
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B[:, i] = 0
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B -= scipy.diag(scipy.diag(B))
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cbc[v] = float(B.sum())
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# put row and col back
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A[i, :] = row
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A[:, i] = col
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# rescaling
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cbc = _rescale(cbc, normalized=normalized)
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return cbc
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def _rescale(cbc, normalized):
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# helper to rescale betweenness centrality
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if normalized is True:
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order = len(cbc)
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if order <= 2:
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scale = None
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else:
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scale = 1.0 / ((order - 1.0)**2 - (order - 1.0))
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if scale is not None:
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for v in cbc:
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cbc[v] *= scale
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return cbc
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def estrada_index(G):
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r"""Returns the Estrada index of a the graph G.
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The Estrada Index is a topological index of folding or 3D "compactness" ([1]_).
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Parameters
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----------
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G: graph
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Returns
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-------
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estrada index: float
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Raises
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------
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NetworkXError
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If the graph is not undirected and simple.
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Notes
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-----
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Let `G=(V,E)` be a simple undirected graph with `n` nodes and let
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`\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{n}`
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be a non-increasing ordering of the eigenvalues of its adjacency
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matrix `A`. The Estrada index is ([1]_, [2]_)
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.. math::
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EE(G)=\sum_{j=1}^n e^{\lambda _j}.
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References
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----------
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.. [1] E. Estrada, "Characterization of 3D molecular structure",
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Chem. Phys. Lett. 319, 713 (2000).
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https://doi.org/10.1016/S0009-2614(00)00158-5
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.. [2] José Antonio de la Peñaa, Ivan Gutman, Juan Rada,
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"Estimating the Estrada index",
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Linear Algebra and its Applications. 427, 1 (2007).
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https://doi.org/10.1016/j.laa.2007.06.020
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Examples
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--------
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>>> G=nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
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>>> ei=nx.estrada_index(G)
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"""
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return sum(subgraph_centrality(G).values())
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# fixture for pytest
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def setup_module(module):
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import pytest
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numpy = pytest.importorskip('numpy')
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scipy = pytest.importorskip('scipy')
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