Source code for networkx.linalg.modularitymatrix

"""Modularity matrix of graphs.
"""
#    Copyright (C) 2004-2018 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.
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): """Return 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 m \over 2} where k_i is the degree of node i, and were 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) See Also -------- to_numpy_matrix adjacency_matrix laplacian_matrix directed_modularity_matrix References ---------- .. [1] 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 # Expected adjacency matrix 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): """Return 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. See Also -------- to_numpy_matrix adjacency_matrix laplacian_matrix modularity_matrix References ---------- .. [1] 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() # Expected adjacency matrix X = k_out * k_in / m return A - X
# fixture for nose tests def setup_module(module): from nose import SkipTest try: import numpy import scipy except: raise SkipTest("NumPy not available")