Source code for networkx.algorithms.communicability_alg

# -*- coding: utf-8 -*-
"""
Communicability.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    Previously coded as communicability centrality
import networkx as nx
from networkx.utils import *
__author__ = "\n".join(['Aric Hagberg (hagberg@lanl.gov)',
'Franck Kalala (franckkalala@yahoo.fr'])
__all__ = ['communicability',
'communicability_exp',
]

[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def communicability(G):
r"""Returns 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.

--------
communicability_exp:
Communicability between all pairs of nodes in G  using spectral
decomposition.
communicability_betweenness_centrality:
Communicability betweeness centrality for each node 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).
https://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 = list(G)  # ordering of nodes in matrix
A = nx.to_numpy_array(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))))
c = {}
# computing communicabilities
for u in G:
c[u] = {}
for v in G:
s = 0
p = mapping[u]
q = mapping[v]
for j in range(len(nodelist)):
s += vec[:, j][p] * vec[:, j][q] * expw[j]
c[u][v] = float(s)
return c

[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def communicability_exp(G):
r"""Returns 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.

--------
communicability:
Communicability between pairs of nodes in G.
communicability_betweenness_centrality:
Communicability betweeness centrality for each node in G.

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).
https://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 = list(G)  # ordering of nodes in matrix
A = nx.to_numpy_array(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))))
c = {}
for u in G:
c[u] = {}
for v in G:
c[u][v] = float(expA[mapping[u], mapping[v]])
return c

# fixture for pytest
def setup_module(module):
import pytest
numpy = pytest.importorskip('numpy')
scipy = pytest.importorskip('scipy')