Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.linalg.modularitymatrix

"""Modularity matrix of graphs.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
from __future__ import division
import networkx as nx
from networkx.utils import not_implemented_for
__author__ = "\n".join(['Aric Hagberg <aric.hagberg@gmail.com>',
'Pieter Swart (swart@lanl.gov)',
'Dan Schult (dschult@colgate.edu)',
'Jean-Gabriel Young (Jean.gabriel.young@gmail.com)'])
__all__ = ['modularity_matrix', 'directed_modularity_matrix']

[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def modularity_matrix(G, nodelist=None, weight=None):
r"""Returns the modularity matrix of G.

The modularity matrix is the matrix B = A - <A>, where A is the adjacency
matrix and <A> is the average adjacency matrix, assuming that the graph
is described by the configuration model.

More specifically, the element B_ij of B is defined as

.. math::
A_{ij} - {k_i k_j \over 2 m}

where k_i is the degree of node i, and where m is the number of edges
in the graph. When weight is set to a name of an attribute edge, Aij, k_i,
k_j and m are computed using its value.

Parameters
----------
G : Graph
A NetworkX graph

nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist.
If nodelist is None, then the ordering is produced by G.nodes().

weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used for
the edge weight.  If None then all edge weights are 1.

Returns
-------
B : Numpy matrix
The modularity matrix of G.

Examples
--------
>>> import networkx as nx
>>> k =[3, 2, 2, 1, 0]
>>> G = nx.havel_hakimi_graph(k)
>>> B = nx.modularity_matrix(G)

--------
to_numpy_matrix
laplacian_matrix
directed_modularity_matrix

References
----------
..  M. E. J. Newman, "Modularity and community structure in networks",
Proc. Natl. Acad. Sci. USA, vol. 103, pp. 8577-8582, 2006.
"""
if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight,
format='csr')
k = A.sum(axis=1)
m = k.sum() * 0.5
X = k * k.transpose() / (2 * m)
return A - X

[docs]@not_implemented_for('undirected')
@not_implemented_for('multigraph')
def directed_modularity_matrix(G, nodelist=None, weight=None):
"""Returns the directed modularity matrix of G.

The modularity matrix is the matrix B = A - <A>, where A is the adjacency
matrix and <A> is the expected adjacency matrix, assuming that the graph
is described by the configuration model.

More specifically, the element B_ij of B is defined as

.. math::
B_{ij} = A_{ij} - k_i^{out} k_j^{in} / m

where :math:k_i^{in} is the in degree of node i, and :math:k_j^{out} is the out degree
of node j, with m the number of edges in the graph. When weight is set
to a name of an attribute edge, Aij, k_i, k_j and m are computed using
its value.

Parameters
----------
G : DiGraph
A NetworkX DiGraph

nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist.
If nodelist is None, then the ordering is produced by G.nodes().

weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used for
the edge weight.  If None then all edge weights are 1.

Returns
-------
B : Numpy matrix
The modularity matrix of G.

Examples
--------
>>> import networkx as nx
>>> G = nx.DiGraph()
>>> G.add_edges_from(((1,2), (1,3), (3,1), (3,2), (3,5), (4,5), (4,6),
...                   (5,4), (5,6), (6,4)))
>>> B = nx.directed_modularity_matrix(G)

Notes
-----
NetworkX defines the element A_ij of the adjacency matrix as 1 if there
is a link going from node i to node j. Leicht and Newman use the opposite
definition. This explains the different expression for B_ij.

--------
to_numpy_matrix
laplacian_matrix
modularity_matrix

References
----------
..  E. A. Leicht, M. E. J. Newman,
"Community structure in directed networks",
Phys. Rev Lett., vol. 100, no. 11, p. 118703, 2008.
"""
if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight,
format='csr')
k_in = A.sum(axis=0)
k_out = A.sum(axis=1)
m = k_in.sum()