Source code for networkx.algorithms.communicability_alg

"""
Communicability.
"""
import networkx as nx
from networkx.utils import not_implemented_for

__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 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_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 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 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: 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