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

Source code for networkx.algorithms.centrality.katz

"""
Katz centrality.
"""
#    Copyright (C) 2004-2013 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 not_implemented_for
__author__ = "\n".join(['Aric Hagberg (aric.hagberg@gmail.com)',
                        'Pieter Swart (swart@lanl.gov)',
                        'Sasha Gutfraind (ag362@cornell.edu)',
                        'Vincent Gauthier (vgauthier@luxbulb.org)'])

__all__ = ['katz_centrality',
           'katz_centrality_numpy']

@not_implemented_for('multigraph')
[docs]def katz_centrality(G, alpha=0.1, beta=1.0, max_iter=1000, tol=1.0e-6, nstart=None, normalized=True, weight = 'weight'): r"""Compute the Katz centrality for the nodes of the graph G. Katz centrality is related to eigenvalue centrality and PageRank. The Katz centrality for node `i` is .. math:: x_i = \alpha \sum_{j} A_{ij} x_j + \beta, where `A` is the adjacency matrix of the graph G with eigenvalues `\lambda`. The parameter `\beta` controls the initial centrality and .. math:: \alpha < \frac{1}{\lambda_{max}}. Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Extra weight can be provided to immediate neighbors through the parameter :math:`\beta`. Connections made with distant neighbors are, however, penalized by an attenuation factor `\alpha` which should be strictly less than the inverse largest eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information is provided in [1]_ . Parameters ---------- G : graph A NetworkX graph alpha : float Attenuation factor beta : scalar or dictionary, optional (default=1.0) Weight attributed to the immediate neighborhood. If not a scalar the dictionary must have an value for every node. max_iter : integer, optional (default=1000) Maximum number of iterations in power method. tol : float, optional (default=1.0e-6) Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of Katz iteration for each node. normalized : bool, optional (default=True) If True normalize the resulting values. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- nodes : dictionary Dictionary of nodes with Katz centrality as the value. Examples -------- >>> import math >>> G = nx.path_graph(4) >>> phi = (1+math.sqrt(5))/2.0 # largest eigenvalue of adj matrix >>> centrality = nx.katz_centrality(G,1/phi-0.01) >>> for n,c in sorted(centrality.items()): ... print("%d %0.2f"%(n,c)) 0 0.37 1 0.60 2 0.60 3 0.37 Notes ----- This algorithm it uses the power method to find the eigenvector corresponding to the largest eigenvalue of the adjacency matrix of G. The constant alpha should be strictly less than the inverse of largest eigenvalue of the adjacency matrix for the algorithm to converge. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached. When `\alpha = 1/\lambda_{max}` and `\beta=1` Katz centrality is the same as eigenvector centrality. For directed graphs this finds "left" eigenvectors which corresponds to the in-edges in the graph. For out-edges Katz centrality first reverse the graph with G.reverse(). References ---------- .. [1] M. Newman, Networks: An Introduction. Oxford University Press, USA, 2010, p. 720. See Also -------- katz_centrality_numpy eigenvector_centrality eigenvector_centrality_numpy pagerank hits """ from math import sqrt if len(G) == 0: return {} nnodes = G.number_of_nodes() if nstart is None: # choose starting vector with entries of 0 x = dict([(n,0) for n in G]) else: x = nstart try: b = dict.fromkeys(G,float(beta)) except (TypeError,ValueError): b = beta if set(beta) != set(G): raise nx.NetworkXError('beta dictionary ' 'must have a value for every node') # make up to max_iter iterations for i in range(max_iter): xlast = x x = dict.fromkeys(xlast, 0) # do the multiplication y^T = Alpha * x^T A - Beta for n in x: for nbr in G[n]: x[nbr] += xlast[n] * G[n][nbr].get(weight, 1) for n in x: x[n] = alpha*x[n] + b[n] # check convergence err = sum([abs(x[n]-xlast[n]) for n in x]) if err < nnodes*tol: if normalized: # normalize vector try: s = 1.0/sqrt(sum(v**2 for v in x.values())) # this should never be zero? except ZeroDivisionError: s = 1.0 else: s = 1 for n in x: x[n] *= s return x raise nx.NetworkXError('Power iteration failed to converge in ' '%d iterations.' % max_iter)
@not_implemented_for('multigraph')
[docs]def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight = 'weight'): r"""Compute the Katz centrality for the graph G. Katz centrality is related to eigenvalue centrality and PageRank. The Katz centrality for node `i` is .. math:: x_i = \alpha \sum_{j} A_{ij} x_j + \beta, where `A` is the adjacency matrix of the graph G with eigenvalues `\lambda`. The parameter `\beta` controls the initial centrality and .. math:: \alpha < \frac{1}{\lambda_{max}}. Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Extra weight can be provided to immediate neighbors through the parameter :math:`\beta`. Connections made with distant neighbors are, however, penalized by an attenuation factor `\alpha` which should be strictly less than the inverse largest eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information is provided in [1]_ . Parameters ---------- G : graph A NetworkX graph alpha : float Attenuation factor beta : scalar or dictionary, optional (default=1.0) Weight attributed to the immediate neighborhood. If not a scalar the dictionary must have an value for every node. normalized : bool If True normalize the resulting values. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- nodes : dictionary Dictionary of nodes with Katz centrality as the value. Examples -------- >>> import math >>> G = nx.path_graph(4) >>> phi = (1+math.sqrt(5))/2.0 # largest eigenvalue of adj matrix >>> centrality = nx.katz_centrality_numpy(G,1/phi) >>> for n,c in sorted(centrality.items()): ... print("%d %0.2f"%(n,c)) 0 0.37 1 0.60 2 0.60 3 0.37 Notes ------ This algorithm uses a direct linear solver to solve the above equation. The constant alpha should be strictly less than the inverse of largest eigenvalue of the adjacency matrix for there to be a solution. When `\alpha = 1/\lambda_{max}` and `\beta=1` Katz centrality is the same as eigenvector centrality. For directed graphs this finds "left" eigenvectors which corresponds to the in-edges in the graph. For out-edges Katz centrality first reverse the graph with G.reverse(). References ---------- .. [1] M. Newman, Networks: An Introduction. Oxford University Press, USA, 2010, p. 720. See Also -------- katz_centrality eigenvector_centrality_numpy eigenvector_centrality pagerank hits """ try: import numpy as np except ImportError: raise ImportError('Requires NumPy: http://scipy.org/') if len(G) == 0: return {} try: nodelist = beta.keys() if set(nodelist) != set(G): raise nx.NetworkXError('beta dictionary ' 'must have a value for every node') b = np.array(list(beta.values()), dtype=float) except AttributeError: nodelist = G.nodes() try: b = np.ones((len(nodelist),1))*float(beta) except (TypeError,ValueError): raise nx.NetworkXError('beta must be a number') A = nx.adj_matrix(G, nodelist=nodelist, weight=weight).todense().T n = np.array(A).shape[0] centrality = np.linalg.solve( np.eye(n,n) - (alpha * A) , b) if normalized: norm = np.sign(sum(centrality)) * np.linalg.norm(centrality) else: norm = 1.0 centrality = dict(zip(nodelist, map(float,centrality/norm))) return centrality # fixture for nose tests
def setup_module(module): from nose import SkipTest try: import numpy import scipy except: raise SkipTest("SciPy not available")