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