Source code for networkx.algorithms.centrality.eigenvector

"""
Eigenvector centrality.
"""
#    Copyright (C) 2004-2014 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
__author__ = "\n".join(['Aric Hagberg (aric.hagberg@gmail.com)',
                        'Pieter Swart (swart@lanl.gov)',
                        'Sasha Gutfraind (ag362@cornell.edu)'])
__all__ = ['eigenvector_centrality',
           'eigenvector_centrality_numpy']

[docs]def eigenvector_centrality(G, max_iter=100, tol=1.0e-6, nstart=None,
                           weight='weight'):
    """Compute the eigenvector centrality for the graph G.

    Uses the power method to find the eigenvector for the
    largest eigenvalue of the adjacency matrix of G.

    Parameters
    ----------
    G : graph
      A networkx graph

    max_iter : interger, optional
      Maximum number of iterations in power method.

    tol : float, optional
      Error tolerance used to check convergence in power method iteration.

    nstart : dictionary, optional
      Starting value of eigenvector iteration for each node.

    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 eigenvector centrality as the value.

    Examples
    --------
    >>> G = nx.path_graph(4)
    >>> centrality = nx.eigenvector_centrality(G)
    >>> print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
    ['0 0.37', '1 0.60', '2 0.60', '3 0.37']

    Notes
    ------
    The eigenvector calculation is done by the power iteration method
    and has no guarantee of convergence.  The iteration will stop
    after max_iter iterations or an error tolerance of
    number_of_nodes(G)*tol has been reached.

    For directed graphs this is "left" eigevector centrality which corresponds
    to the in-edges in the graph.  For out-edges eigenvector centrality
    first reverse the graph with G.reverse().

    See Also
    --------
    eigenvector_centrality_numpy
    pagerank
    hits
    """
    from math import sqrt
    if type(G) == nx.MultiGraph or type(G) == nx.MultiDiGraph:
        raise nx.NetworkXException("Not defined for multigraphs.")

    if len(G) == 0:
        raise nx.NetworkXException("Empty graph.")

    if nstart is None:
        # choose starting vector with entries of 1/len(G)
        x = dict([(n,1.0/len(G)) for n in G])
    else:
        x = nstart
    # normalize starting vector
    s = 1.0/sum(x.values())
    for k in x: 
        x[k] *= s
    nnodes = G.number_of_nodes()
    # make up to max_iter iterations
    for i in range(max_iter):
        xlast = x
        x = dict.fromkeys(xlast, 0)
        # do the multiplication y^T = x^T A
        for n in x:
            for nbr in G[n]:
                x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
        # 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
        for n in x:
            x[n] *= s
        # check convergence
        err = sum([abs(x[n]-xlast[n]) for n in x])
        if err < nnodes*tol:
            return x

    raise nx.NetworkXError("""eigenvector_centrality():
power iteration failed to converge in %d iterations."%(i+1))""")


[docs]def eigenvector_centrality_numpy(G, weight='weight'):
    """Compute the eigenvector centrality for the graph G.

    Parameters
    ----------
    G : graph
      A networkx graph

    weight : None or string, optional
      The name of the edge attribute used as weight.
      If None, all edge weights are considered equal.


    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with eigenvector centrality as the value.

    Examples
    --------
    >>> G = nx.path_graph(4)
    >>> centrality = nx.eigenvector_centrality_numpy(G)
    >>> print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
    ['0 0.37', '1 0.60', '2 0.60', '3 0.37']

    Notes
    ------
    This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to
    find the largest eigenvalue/eigenvector pair.

    For directed graphs this is "left" eigevector centrality which corresponds
    to the in-edges in the graph.  For out-edges eigenvector centrality
    first reverse the graph with G.reverse().

    See Also
    --------
    eigenvector_centrality
    pagerank
    hits
    """
    import scipy as sp
    from scipy.sparse import linalg
    if len(G) == 0:
        raise nx.NetworkXException('Empty graph.')
    M = nx.to_scipy_sparse_matrix(G, nodelist=G.nodes(), weight=weight,
                                  dtype=float)
    eigenvalue, eigenvector = linalg.eigs(M.T, k=1, which='LR') 
    largest = eigenvector.flatten().real
    norm = sp.sign(largest.sum())*sp.linalg.norm(largest)
    centrality = dict(zip(G,map(float,largest/norm)))
    return centrality


# fixture for nose tests
def setup_module(module):
    from nose import SkipTest
    try:
        import scipy
    except:
        raise SkipTest("SciPy not available")