python-networkx 1.8.1-0ubuntu3 / usr / lib / python2.7 / dist-packages / networkx / algorithms / centrality / katz.py

"""
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 *
__author__ = "\n".join(['Aric Hagberg (hagberg@lanl.gov)',
                        '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')
def katz_centrality(G, alpha=0.1, beta=1.0,
                    max_iter=1000, tol=1.0e-6, nstart=None, normalized=True):
    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.

    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.

    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 = Alpha * Ax - Beta
        for n in x:
            for nbr in G[n]:
                x[n] += xlast[nbr] * G[n][nbr].get('weight',1)
            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."%(i+1))')

@not_implemented_for('multigraph')
def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True):
    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.

    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.

    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)
    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 numpy.linalg
    except:
        raise SkipTest("numpy not available")