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Source code for networkx.algorithms.centrality.communicability_alg
"""
Communicability and centrality measures.
"""
# 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__ = ['communicability_centrality_exp',
'communicability_centrality',
'communicability_betweenness_centrality',
'communicability',
'communicability_exp',
'estrada_index',
]
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def communicability_centrality_exp(G):
r"""Return the communicability centrality for each node of G
Communicability centrality, also called subgraph centrality, of a node `n`
is the sum of closed walks of all lengths starting and ending at node `n`.
Parameters
----------
G: graph
Returns
-------
nodes:dictionary
Dictionary of nodes with communicability centrality as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability:
Communicability between all pairs of nodes in G.
communicability_centrality:
Communicability centrality for each node of G.
Notes
-----
This version of the algorithm exponentiates the adjacency matrix.
The communicability centrality of a node `u` in G can be found using
the matrix exponential of the adjacency matrix of G [1]_ [2]_,
.. 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).
http://arxiv.org/abs/cond-mat/0504730
.. [2] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
http://arxiv.org/abs/0707.0756
Examples
--------
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> sc = nx.communicability_centrality_exp(G)
"""
# alternative implementation that calculates the matrix exponential
import scipy.linalg
nodelist = G.nodes() # 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)
# convert diagonal to dictionary keyed by node
sc = dict(zip(nodelist,map(float,expA.diagonal())))
return sc
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def communicability_centrality(G):
r"""Return communicability centrality for each node in G.
Communicability centrality, also called subgraph centrality, of a node `n`
is the sum of closed walks of all lengths starting and ending at node `n`.
Parameters
----------
G: graph
Returns
-------
nodes: dictionary
Dictionary of nodes with communicability centrality as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability:
Communicability between all pairs of nodes in G.
communicability_centrality:
Communicability centrality for each node of G.
Notes
-----
This version of the algorithm computes eigenvalues and eigenvectors
of the adjacency matrix.
Communicability centrality of a node `u` in G can be found using
a spectral decomposition of the adjacency matrix [1]_ [2]_,
.. 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
--------
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> sc = nx.communicability_centrality(G)
References
----------
.. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
"Subgraph centrality in complex networks",
Physical Review E 71, 056103 (2005).
http://arxiv.org/abs/cond-mat/0504730
.. [2] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
http://arxiv.org/abs/0707.0756
"""
import numpy
import numpy.linalg
nodelist = G.nodes() # 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)
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
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def communicability_betweenness_centrality(G, normalized=True):
r"""Return communicability betweenness 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.
See Also
--------
communicability:
Communicability between all pairs of nodes in G.
communicability_centrality:
Communicability centrality for each node of G using matrix exponential.
communicability_centrality_exp:
Communicability centrality for each node in G using
spectral decomposition.
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 communicability 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.
http://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 = G.nodes() # 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)
mapping = dict(zip(nodelist,range(n)))
sc = {}
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)) / 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))
sc[v] = float(B.sum())
# put row and col back
A[i,:] = row
A[:,i] = col
# rescaling
sc = _rescale(sc,normalized=normalized)
return sc
def _rescale(sc,normalized):
# helper to rescale betweenness centrality
if normalized is True:
order=len(sc)
if order <=2:
scale=None
else:
scale=1.0/((order-1.0)**2-(order-1.0))
if scale is not None:
for v in sc:
sc[v] *= scale
return sc
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def communicability(G):
r"""Return communicability between all pairs of nodes in G.
The communicability between pairs of nodes in G is the sum of
closed walks of different lengths starting at node u and ending at node v.
Parameters
----------
G: graph
Returns
-------
comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability
as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability_centrality_exp:
Communicability centrality for each node of G using matrix exponential.
communicability_centrality:
Communicability centrality for each node in G using spectral
decomposition.
communicability:
Communicability between pairs of nodes in G.
Notes
-----
This algorithm uses a spectral decomposition of the adjacency matrix.
Let G=(V,E) be a simple undirected graph. Using the connection between
the powers of the adjacency matrix and the number of walks in the graph,
the communicability between nodes `u` and `v` based on the graph spectrum
is [1]_
.. math::
C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}},
where `\phi_{j}(u)` is the `u\rm{th}` element of the `j\rm{th}` orthonormal
eigenvector of the adjacency matrix associated with the eigenvalue
`\lambda_{j}`.
References
----------
.. [1] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
http://arxiv.org/abs/0707.0756
Examples
--------
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability(G)
"""
import numpy
import scipy.linalg
nodelist = G.nodes() # ordering of nodes in matrix
A = nx.to_numpy_matrix(G,nodelist)
# convert to 0-1 matrix
A[A!=0.0] = 1
w,vec = numpy.linalg.eigh(A)
expw = numpy.exp(w)
mapping = dict(zip(nodelist,range(len(nodelist))))
sc={}
# computing communicabilities
for u in G:
sc[u]={}
for v in G:
s = 0
p = mapping[u]
q = mapping[v]
for j in range(len(nodelist)):
s += vec[:,j][p,0]*vec[:,j][q,0]*expw[j]
sc[u][v] = float(s)
return sc
@not_implemented_for('directed')
@not_implemented_for('multigraph')
[docs]def communicability_exp(G):
r"""Return communicability between all pairs of nodes in G.
Communicability between pair of node (u,v) of node in G is the sum of
closed walks of different lengths starting at node u and ending at node v.
Parameters
----------
G: graph
Returns
-------
comm: dictionary of dictionaries
Dictionary of dictionaries keyed by nodes with communicability
as the value.
Raises
------
NetworkXError
If the graph is not undirected and simple.
See Also
--------
communicability_centrality_exp:
Communicability centrality for each node of G using matrix exponential.
communicability_centrality:
Communicability centrality for each node in G using spectral
decomposition.
communicability_exp:
Communicability between all pairs of nodes in G using spectral
decomposition.
Notes
-----
This algorithm uses matrix exponentiation of the adjacency matrix.
Let G=(V,E) be a simple undirected graph. Using the connection between
the powers of the adjacency matrix and the number of walks in the graph,
the communicability between nodes u and v is [1]_,
.. math::
C(u,v) = (e^A)_{uv},
where `A` is the adjacency matrix of G.
References
----------
.. [1] Ernesto Estrada, Naomichi Hatano,
"Communicability in complex networks",
Phys. Rev. E 77, 036111 (2008).
http://arxiv.org/abs/0707.0756
Examples
--------
>>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)])
>>> c = nx.communicability_exp(G)
"""
import scipy.linalg
nodelist = G.nodes() # ordering of nodes in matrix
A = nx.to_numpy_matrix(G,nodelist)
# convert to 0-1 matrix
A[A!=0.0] = 1
# communicability matrix
expA = scipy.linalg.expm(A)
mapping = dict(zip(nodelist,range(len(nodelist))))
sc = {}
for u in G:
sc[u]={}
for v in G:
sc[u][v] = float(expA[mapping[u],mapping[v]])
return sc
[docs]def estrada_index(G):
r"""Return the Estrada index of a the graph G.
Parameters
----------
G: graph
Returns
-------
estrada index: float
Raises
------
NetworkXError
If the graph is not undirected and simple.
See also
--------
estrada_index_exp
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
.. 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).
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(communicability_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")