__init__.py 40.3 KB
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#! /usr/bin/env python
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# -*- coding: utf-8 -*-
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#
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# graph_tool -- a general graph manipulation python module
#
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# Copyright (C) 2006-2020 Tiago de Paula Peixoto <tiago@skewed.de>
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#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.

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"""
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``graph_tool.centrality`` - Centrality measures
-----------------------------------------------
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This module includes centrality-related algorithms.
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Summary
+++++++

.. autosummary::
   :nosignatures:

   pagerank
   betweenness
   central_point_dominance
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   closeness
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   eigenvector
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   katz
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   hits
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   eigentrust
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   trust_transitivity
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Contents
++++++++
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"""

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from .. dl_import import dl_import
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dl_import("from . import libgraph_tool_centrality")
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from .. import _prop, ungroup_vector_property, Vector_size_t
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from .. topology import shortest_distance
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import numpy
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import numpy.linalg
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__all__ = ["pagerank", "betweenness", "central_point_dominance", "closeness",
           "eigentrust", "eigenvector", "katz", "hits", "trust_transitivity"]
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def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6,
             max_iter=None, ret_iter=False):
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    r"""Calculate the PageRank of each vertex.
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    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    damping : float, optional (default: 0.85)
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        Damping factor.
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    pers : :class:`~graph_tool.VertexPropertyMap`, optional (default: None)
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        Personalization vector. If omitted, a constant value of :math:`1/N`
        will be used.
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    weight : :class:`~graph_tool.EdgePropertyMap`, optional (default: None)
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        Edge weights. If omitted, a constant value of 1 will be used.
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    prop : :class:`~graph_tool.VertexPropertyMap`, optional (default: None)
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        Vertex property map to store the PageRank values. If supplied, it will
        be used uninitialized.
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    epsilon : float, optional (default: 1e-6)
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        Convergence condition. The iteration will stop if the total delta of all
        vertices are below this value.
    max_iter : int, optional (default: None)
        If supplied, this will limit the total number of iterations.
    ret_iter : bool, optional (default: False)
        If true, the total number of iterations is also returned.

    Returns
    -------
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    pagerank : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the PageRank values.
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    See Also
    --------
    betweenness: betweenness centrality
    eigentrust: eigentrust centrality
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    eigenvector: eigenvector centrality
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    hits: authority and hub centralities
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    trust_transitivity: pervasive trust transitivity
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    Notes
    -----
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    The value of PageRank [pagerank-wikipedia]_ of vertex v, :math:`PR(v)`, is
    given iteratively by the relation:
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    .. math::
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        PR(v) = \frac{1-d}{N} + d \sum_{u \in \Gamma^{-}(v)}
                \frac{PR (u)}{d^{+}(u)}
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    where :math:`\Gamma^{-}(v)` are the in-neighbors of v, :math:`d^{+}(u)` is
    the out-degree of u, and d is a damping factor.
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    If a personalization property :math:`p(v)` is given, the definition becomes:

    .. math::

        PR(v) = (1-d)p(v) + d \sum_{u \in \Gamma^{-}(v)}
                \frac{PR (u)}{d^{+}(u)}

    If edge weights are also given, the equation is then generalized to:

    .. math::

        PR(v) = (1-d)p(v) + d \sum_{u \in \Gamma^{-}(v)}
                \frac{PR (u) w_{u\to v}}{d^{+}(u)}

    where :math:`d^{+}(u)=\sum_{y}A_{u,y}w_{u\to y}` is redefined to be the sum
    of the weights of the out-going edges from u.

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    If a node has out-degree zero, it is assumed to connect to every other node
    with a weight proportional to :math:`p(v)` or a constant if no
    personalization is given.

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    The implemented algorithm progressively iterates the above equations, until
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    it no longer changes, according to the parameter epsilon. It has a
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    topology-dependent running time.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: pagerank

       import matplotlib

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    .. doctest:: pagerank

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> pr = gt.pagerank(g)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr,
       ...               vertex_size=gt.prop_to_size(pr, mi=5, ma=15),
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       ...               vorder=pr, vcmap=matplotlib.cm.gist_heat,
       ...               output="polblogs_pr.pdf")
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       <...>

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    .. figure:: polblogs_pr.png
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       :align: center
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       :width: 80%
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       PageRank values of the a political blogs network of [adamic-polblogs]_.
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    Now with a personalization vector, and edge weights:

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    .. doctest:: pagerank

       >>> d = g.degree_property_map("total")
       >>> periphery = d.a <= 2
       >>> p = g.new_vertex_property("double")
       >>> p.a[periphery] = 100
       >>> pr = gt.pagerank(g, pers=p)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr,
       ...               vertex_size=gt.prop_to_size(pr, mi=5, ma=15),
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       ...               vorder=pr, vcmap=matplotlib.cm.gist_heat,
       ...               output="polblogs_pr_pers.pdf")
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       <...>

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    .. testcleanup:: pagerank
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       conv_png("polblogs_pr.pdf")
       conv_png("polblogs_pr_pers.pdf")
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    .. figure:: polblogs_pr_pers.png
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       :align: center
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       :width: 80%
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       Personalized PageRank values of the a political blogs network of
       [adamic-polblogs]_, where vertices with very low degree are given
       artificially high scores.
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    References
    ----------
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    .. [pagerank-wikipedia] http://en.wikipedia.org/wiki/Pagerank
    .. [lawrence-pagerank-1998] P. Lawrence, B. Sergey, M. Rajeev, W. Terry,
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       "The pagerank citation ranking: Bringing order to the web", Technical
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       report, Stanford University, 1998
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    .. [Langville-survey-2005] A. N. Langville, C. D. Meyer, "A Survey of
       Eigenvector Methods for Web Information Retrieval", SIAM Review, vol. 47,
       no. 1, pp. 135-161, 2005, :DOI:`10.1137/S0036144503424786`
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """

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    if max_iter is None:
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        max_iter = 0
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    if prop is None:
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        prop = g.new_vertex_property("double")
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        N = len(prop.fa)
        prop.fa = pers.fa[:N] if pers is not None else 1. / g.num_vertices()
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    ic = libgraph_tool_centrality.\
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            get_pagerank(g._Graph__graph, _prop("v", g, prop),
                         _prop("v", g, pers), _prop("e", g, weight),
                         damping, epsilon, max_iter)
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    if ret_iter:
        return prop, ic
    else:
        return prop

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def betweenness(g, pivots=None, vprop=None, eprop=None, weight=None, norm=True):
    r"""Calculate the betweenness centrality for each vertex and edge.
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    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    pivots : list or :class:`~numpy.ndarray`, optional (default: None)
        If provided, the betweenness will be estimated using the vertices in
        this list as pivots. If the list contains all nodes (the default) the
        algorithm will be exact, and if the vertices are randomly chosen the
        result will be an unbiased estimator.
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    vprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: None)
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        Vertex property map to store the vertex betweenness values.
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    eprop : :class:`~graph_tool.EdgePropertyMap`, optional (default: None)
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        Edge property map to store the edge betweenness values.
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    weight : :class:`~graph_tool.EdgePropertyMap`, optional (default: None)
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        Edge property map corresponding to the weight value of each edge.
    norm : bool, optional (default: True)
        Whether or not the betweenness values should be normalized.

    Returns
    -------
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    vertex_betweenness : A vertex property map with the vertex betweenness values.
    edge_betweenness : An edge property map with the edge betweenness values.
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    See Also
    --------
    central_point_dominance: central point dominance of the graph
    pagerank: PageRank centrality
    eigentrust: eigentrust centrality
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    eigenvector: eigenvector centrality
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    hits: authority and hub centralities
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    trust_transitivity: pervasive trust transitivity
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    Notes
    -----
    Betweenness centrality of a vertex :math:`C_B(v)` is defined as,

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    .. math::

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        C_B(v)= \sum_{s \neq v \neq t \in V \atop s \neq t}
                \frac{\sigma_{st}(v)}{\sigma_{st}}

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    where :math:`\sigma_{st}` is the number of shortest paths from s to t, and
    :math:`\sigma_{st}(v)` is the number of shortest paths from s to t that pass
    through a vertex :math:`v`. This may be normalised by dividing through the
    number of pairs of vertices not including v, which is :math:`(n-1)(n-2)/2`,
    for undirected graphs, or :math:`(n-1)(n-2)` for directed ones.
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    The algorithm used here is defined in [brandes-faster-2001]_, and has a
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    complexity of :math:`O(VE)` for unweighted graphs and :math:`O(VE +
    V(V+E)\log V)` for weighted graphs. The space complexity is :math:`O(VE)`.

    If the ``pivots`` parameter is given, the complexity will be instead
    :math:`O(PE)` for unweighted graphs and :math:`O(PE + P(V+E)\log V)` for
    weighted graphs, where :math:`P` is the number of pivot vertices.
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    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: betweenness

       import matplotlib

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    .. doctest:: betweenness

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> vp, ep = gt.betweenness(g)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=vp,
       ...               vertex_size=gt.prop_to_size(vp, mi=5, ma=15),
       ...               edge_pen_width=gt.prop_to_size(ep, mi=0.5, ma=5),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=vp, output="polblogs_betweenness.pdf")
       <...>

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    .. testcleanup:: betweenness
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       conv_png("polblogs_betweenness.pdf")
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    .. figure:: polblogs_betweenness.png
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       :align: center
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       :width: 80%
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       Betweenness values of the a political blogs network of [adamic-polblogs]_.
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    References
    ----------
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    .. [betweenness-wikipedia] http://en.wikipedia.org/wiki/Centrality#Betweenness_centrality
    .. [brandes-faster-2001] U. Brandes, "A faster algorithm for betweenness
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       centrality", Journal of Mathematical Sociology, 2001, :doi:`10.1080/0022250X.2001.9990249`
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    .. [brandes-centrality-2007] U. Brandes, C. Pich, "Centrality estimation in
       large networks", Int. J. Bifurcation Chaos 17, 2303 (2007).
       :DOI:`10.1142/S0218127407018403`
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """
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    if vprop is None:
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        vprop = g.new_vertex_property("double")
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    if eprop is None:
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        eprop = g.new_edge_property("double")
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    if weight is not None and weight.value_type() != eprop.value_type():
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        nw = g.new_edge_property(eprop.value_type())
        g.copy_property(weight, nw)
        weight = nw
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    if pivots is not None:
        pivots = numpy.asarray(pivots, dtype="uint64")
    else:
        pivots = g.get_vertices()
    vpivots = Vector_size_t(len(pivots))
    vpivots.a = pivots
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    libgraph_tool_centrality.\
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            get_betweenness(g._Graph__graph, vpivots, _prop("e", g, weight),
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                            _prop("e", g, eprop), _prop("v", g, vprop))
    if norm:
        libgraph_tool_centrality.\
            norm_betweenness(g._Graph__graph, vpivots, _prop("e", g, eprop),
                             _prop("v", g, vprop))
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    return vprop, eprop

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def closeness(g, weight=None, source=None, vprop=None, norm=True, harmonic=False):
    r"""
    Calculate the closeness centrality for each vertex.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
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    weight : :class:`~graph_tool.EdgePropertyMap`, optional (default: None)
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        Edge property map corresponding to the weight value of each edge.
    source : :class:`~graph_tool.Vertex`, optional (default: ``None``)
        If specified, the centrality is computed for this vertex alone.
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    vprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map to store the vertex centrality values.
    norm : bool, optional (default: ``True``)
        Whether or not the centrality values should be normalized.
    harmonic : bool, optional (default: ``False``)
        If true, the sum of the inverse of the distances will be computed,
        instead of the inverse of the sum.

    Returns
    -------
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    vertex_closeness : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map with the vertex closeness values.

    See Also
    --------
    central_point_dominance: central point dominance of the graph
    pagerank: PageRank centrality
    eigentrust: eigentrust centrality
    eigenvector: eigenvector centrality
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    hits: authority and hub centralities
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    trust_transitivity: pervasive trust transitivity

    Notes
    -----
    The closeness centrality of a vertex :math:`i` is defined as,

    .. math::

        c_i = \frac{1}{\sum_j d_{ij}}

    where :math:`d_{ij}` is the (possibly directed and/or weighted) distance
    from :math:`i` to :math:`j`. In case there is no path between the two
    vertices, here the distance is taken to be zero.

    If ``harmonic == True``, the definition becomes

    .. math::

        c_i = \sum_j\frac{1}{d_{ij}},

    but now, in case there is no path between the two vertices, we take
    :math:`d_{ij} \to\infty` such that :math:`1/d_{ij}=0`.

    If ``norm == True``, the values of :math:`c_i` are normalized by
    :math:`n_i-1` where :math:`n_i` is the size of the (out-) component of
    :math:`i`. If ``harmonic == True``, they are instead simply normalized by
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    :math:`V-1`.
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    The algorithm complexity of :math:`O(V(V + E))` for unweighted graphs and
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    :math:`O(V(V+E) \log V)` for weighted graphs. If the option ``source`` is
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    specified, this drops to :math:`O(V + E)` and :math:`O((V+E)\log V)`
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    respectively.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------

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    .. testsetup:: closeness

       import matplotlib

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    .. doctest:: closeness

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> c = gt.closeness(g)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=c,
       ...               vertex_size=gt.prop_to_size(c, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=c, output="polblogs_closeness.pdf")
       <...>

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    .. testcleanup:: closeness
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       conv_png("polblogs_closeness.pdf")
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    .. figure:: polblogs_closeness.png
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       :align: center
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       :width: 80%
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       Closeness values of the a political blogs network of [adamic-polblogs]_.

    References
    ----------
    .. [closeness-wikipedia] https://en.wikipedia.org/wiki/Closeness_centrality
    .. [opsahl-node-2010] Opsahl, T., Agneessens, F., Skvoretz, J., "Node
       centrality in weighted networks: Generalizing degree and shortest
       paths". Social Networks 32, 245-251, 2010 :DOI:`10.1016/j.socnet.2010.03.006`
    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`

    """
    if source is None:
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        if vprop is None:
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            vprop = g.new_vertex_property("double")
        libgraph_tool_centrality.\
            closeness(g._Graph__graph, _prop("e", g, weight),
                      _prop("v", g, vprop), harmonic, norm)
        return vprop
    else:
        max_dist = g.num_vertices() + 1
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        dist = shortest_distance(g, source=source, weights=weight,
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                                 max_dist=max_dist)
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        dists = dist.fa[(dist.fa < max_dist) * (dist.fa > 0)]
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        if harmonic:
            c = (1. / dists).sum()
            if norm:
                c /= g.num_vertices() - 1
        else:
            c = 1. / dists.sum()
            if norm:
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                c *= len(dists)
        return c
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def central_point_dominance(g, betweenness):
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    r"""Calculate the central point dominance of the graph, given the betweenness
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    centrality of each vertex.

    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    betweenness : :class:`~graph_tool.VertexPropertyMap`
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        Vertex property map with the betweenness centrality values. The values
        must be normalized.

    Returns
    -------
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    cp : float
        The central point dominance.
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    See Also
    --------
    betweenness: betweenness centrality

    Notes
    -----
    Let :math:`v^*` be the vertex with the largest relative betweenness
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    centrality; then, the central point dominance [freeman-set-1977]_ is defined
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    as:

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    .. math::

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        C'_B = \frac{1}{|V|-1} \sum_{v} C_B(v^*) - C_B(v)

    where :math:`C_B(v)` is the normalized betweenness centrality of vertex
    v. The value of :math:`C_B` lies in the range [0,1].

    The algorithm has a complexity of :math:`O(V)`.

    Examples
    --------
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    >>> g = gt.collection.data["polblogs"]
    >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
    >>> vp, ep = gt.betweenness(g)
    >>> print(gt.central_point_dominance(g, vp))
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    0.218286...
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    References
    ----------
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    .. [freeman-set-1977] Linton C. Freeman, "A Set of Measures of Centrality
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       Based on Betweenness", Sociometry, Vol. 40, No. 1, pp. 35-41, 1977,
       :doi:`10.2307/3033543`
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    """

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    return libgraph_tool_centrality.\
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           get_central_point_dominance(g._Graph__graph,
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                                       _prop("v", g, betweenness))

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def eigenvector(g, weight=None, vprop=None, epsilon=1e-6, max_iter=None):
    r"""
    Calculate the eigenvector centrality of each vertex in the graph, as well as
    the largest eigenvalue.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
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    weight : :class:`~graph_tool.EdgePropertyMap` (optional, default: ``None``)
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        Edge property map with the edge weights.
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    vprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the values of eigenvector must be stored. If
        provided, it will be used uninitialized.
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    epsilon : float, optional (default: ``1e-6``)
        Convergence condition. The iteration will stop if the total delta of all
        vertices are below this value.
    max_iter : int, optional (default: ``None``)
        If supplied, this will limit the total number of iterations.

    Returns
    -------
    eigenvalue : float
        The largest eigenvalue of the (weighted) adjacency matrix.
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    eigenvector : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the eigenvector values.

    See Also
    --------
    betweenness: betweenness centrality
    pagerank: PageRank centrality
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    hits: authority and hub centralities
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    trust_transitivity: pervasive trust transitivity

    Notes
    -----

    The eigenvector centrality :math:`\mathbf{x}` is the eigenvector of the
    (weighted) adjacency matrix with the largest eigenvalue :math:`\lambda`,
    i.e. it is the solution of

    .. math::

        \mathbf{A}\mathbf{x} = \lambda\mathbf{x},


    where :math:`\mathbf{A}` is the (weighted) adjacency matrix and
    :math:`\lambda` is the largest eigenvalue.

    The algorithm uses the power method which has a topology-dependent complexity of
    :math:`O\left(N\times\frac{-\log\epsilon}{\log|\lambda_1/\lambda_2|}\right)`,
    where :math:`N` is the number of vertices, :math:`\epsilon` is the ``epsilon``
    parameter, and :math:`\lambda_1` and :math:`\lambda_2` are the largest and
    second largest eigenvalues of the (weighted) adjacency matrix, respectively.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: eigenvector

       np.random.seed(42)
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       import matplotlib
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    .. doctest:: eigenvector

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> w = g.new_edge_property("double")
       >>> w.a = np.random.random(len(w.a)) * 42
       >>> ee, x = gt.eigenvector(g, w)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x,
       ...               vertex_size=gt.prop_to_size(x, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=x, output="polblogs_eigenvector.pdf")
       <...>

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    .. testcleanup:: eigenvector
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       conv_png("polblogs_eigenvector.pdf")
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    .. figure:: polblogs_eigenvector.png
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       :align: center
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       :width: 80%
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       Eigenvector values of the a political blogs network of
       [adamic-polblogs]_, with random weights attributed to the edges.
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    References
    ----------

    .. [eigenvector-centrality] http://en.wikipedia.org/wiki/Centrality#Eigenvector_centrality
    .. [power-method] http://en.wikipedia.org/wiki/Power_iteration
    .. [langville-survey-2005] A. N. Langville, C. D. Meyer, "A Survey of
       Eigenvector Methods for Web Information Retrieval", SIAM Review, vol. 47,
       no. 1, pp. 135-161, 2005, :DOI:`10.1137/S0036144503424786`
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """

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    if vprop is None:
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        vprop = g.new_vertex_property("double")
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        vprop.fa = 1. / g.num_vertices()
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    if max_iter is None:
        max_iter = 0
    ee = libgraph_tool_centrality.\
         get_eigenvector(g._Graph__graph, _prop("e", g, weight),
                         _prop("v", g, vprop), epsilon, max_iter)
    return ee, vprop


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def katz(g, alpha=0.01, beta=None, weight=None, vprop=None, epsilon=1e-6,
         max_iter=None, norm=True):
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    r"""
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    Calculate the Katz centrality of each vertex in the graph.
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    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
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    weight : :class:`~graph_tool.EdgePropertyMap` (optional, default: ``None``)
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        Edge property map with the edge weights.
    alpha : float, optional (default: ``0.01``)
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        Free parameter :math:`\alpha`. This must be smaller than the inverse of
        the largest eigenvalue of the adjacency matrix.
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    beta : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the local personalization values. If not
        provided, the global value of 1 will be used.
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    vprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the values of eigenvector must be stored. If
        provided, it will be used uninitialized.
    epsilon : float, optional (default: ``1e-6``)
        Convergence condition. The iteration will stop if the total delta of all
        vertices are below this value.
    max_iter : int, optional (default: ``None``)
        If supplied, this will limit the total number of iterations.
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    norm : bool, optional (default: ``True``)
        Whether or not the centrality values should be normalized.
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    Returns
    -------
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    centrality : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the Katz centrality values.

    See Also
    --------
    betweenness: betweenness centrality
    pagerank: PageRank centrality
    eigenvector: eigenvector centrality
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    hits: authority and hub centralities
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    trust_transitivity: pervasive trust transitivity

    Notes
    -----

    The Katz centrality :math:`\mathbf{x}` is the solution of the nonhomogeneous
    linear system

    .. math::

        \mathbf{x} = \alpha\mathbf{A}\mathbf{x} + \mathbf{\beta},


    where :math:`\mathbf{A}` is the (weighted) adjacency matrix and
    :math:`\mathbf{\beta}` is the personalization vector (if not supplied,
    :math:`\mathbf{\beta} = \mathbf{1}` is assumed).

    The algorithm uses successive iterations of the equation above, which has a
    topology-dependent convergence complexity.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: katz

       np.random.seed(42)
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       import matplotlib
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    .. doctest:: katz

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> w = g.new_edge_property("double")
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       >>> w.a = np.random.random(len(w.a))
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       >>> x = gt.katz(g, weight=w)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x,
       ...               vertex_size=gt.prop_to_size(x, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=x, output="polblogs_katz.pdf")
       <...>

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    .. testcleanup:: katz
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       conv_png("polblogs_katz.pdf")
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    .. figure:: polblogs_katz.png
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       :align: center
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       :width: 80%
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       Katz centrality values of the a political blogs network of
       [adamic-polblogs]_, with random weights attributed to the edges.
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    References
    ----------

    .. [katz-centrality] http://en.wikipedia.org/wiki/Katz_centrality
    .. [katz-new] L. Katz, "A new status index derived from sociometric analysis",
       Psychometrika 18, Number 1, 39-43, 1953, :DOI:`10.1007/BF02289026`
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """

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    if vprop is None:
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        vprop = g.new_vertex_property("double")
    if max_iter is None:
        max_iter = 0
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    libgraph_tool_centrality.\
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         get_katz(g._Graph__graph, _prop("e", g, weight), _prop("v", g, vprop),
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                  _prop("v", g, beta), float(alpha), epsilon, max_iter)
    if norm:
        vprop.fa = vprop.fa / numpy.linalg.norm(vprop.fa)
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    return vprop


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def hits(g, weight=None, xprop=None, yprop=None, epsilon=1e-6, max_iter=None):
    r"""
    Calculate the authority and hub centralities of each vertex in the graph.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
        Graph to be used.
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    weight : :class:`~graph_tool.EdgePropertyMap` (optional, default: ``None``)
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        Edge property map with the edge weights.
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    xprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the authority centrality must be stored.
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    yprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the hub centrality must be stored.
    epsilon : float, optional (default: ``1e-6``)
        Convergence condition. The iteration will stop if the total delta of all
        vertices are below this value.
    max_iter : int, optional (default: ``None``)
        If supplied, this will limit the total number of iterations.

    Returns
    -------
    eig : `float`
        The largest eigenvalue of the cocitation matrix.
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    x : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the authority centrality values.
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    y : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the hub centrality values.

    See Also
    --------
    betweenness: betweenness centrality
    eigenvector: eigenvector centrality
    pagerank: PageRank centrality
    trust_transitivity: pervasive trust transitivity

    Notes
    -----

    The Hyperlink-Induced Topic Search (HITS) centrality assigns hub
    (:math:`\mathbf{y}`) and authority (:math:`\mathbf{x}`) centralities to the
    vertices, following:

    .. math::

        \begin{align}
            \mathbf{x} &= \alpha\mathbf{A}\mathbf{y} \\
            \mathbf{y} &= \beta\mathbf{A}^T\mathbf{x}
        \end{align}


    where :math:`\mathbf{A}` is the (weighted) adjacency matrix and
    :math:`\lambda = 1/(\alpha\beta)` is the largest eigenvalue of the
    cocitation matrix, :math:`\mathbf{A}\mathbf{A}^T`. (Without loss of
    generality, we set :math:`\beta=1` in the algorithm.)

    The algorithm uses the power method which has a topology-dependent complexity of
    :math:`O\left(N\times\frac{-\log\epsilon}{\log|\lambda_1/\lambda_2|}\right)`,
    where :math:`N` is the number of vertices, :math:`\epsilon` is the ``epsilon``
    parameter, and :math:`\lambda_1` and :math:`\lambda_2` are the largest and
    second largest eigenvalues of the (weighted) cocitation matrix, respectively.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: hits

       import matplotlib

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    .. doctest:: hits

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> ee, x, y = gt.hits(g)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x,
       ...               vertex_size=gt.prop_to_size(x, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=x, output="polblogs_hits_auths.pdf")
       <...>
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=y,
       ...               vertex_size=gt.prop_to_size(y, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=y, output="polblogs_hits_hubs.pdf")
       <...>

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    .. testcleanup:: hits
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       conv_png("polblogs_hits_auths.pdf")
       conv_png("polblogs_hits_hubs.pdf")
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    .. figure:: polblogs_hits_auths.png
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       :align: center
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       :width: 80%
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       HITS authority values of the a political blogs network of
       [adamic-polblogs]_.

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    .. figure:: polblogs_hits_hubs.png
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       :align: center
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       :width: 80%
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       HITS hub values of the a political blogs network of [adamic-polblogs]_.
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    References
    ----------

    .. [hits-algorithm] http://en.wikipedia.org/wiki/HITS_algorithm
    .. [kleinberg-authoritative] J. Kleinberg, "Authoritative sources in a
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       hyperlinked environment", Journal of the ACM 46 (5): 604-632, 1999,
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       :DOI:`10.1145/324133.324140`.
    .. [power-method] http://en.wikipedia.org/wiki/Power_iteration
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """

    if xprop is None:
        xprop = g.new_vertex_property("double")
    if yprop is None:
        yprop = g.new_vertex_property("double")
    if max_iter is None:
        max_iter = 0
    l = libgraph_tool_centrality.\
         get_hits(g._Graph__graph, _prop("e", g, weight), _prop("v", g, xprop),
                  _prop("v", g, yprop), epsilon, max_iter)
    return 1. / l, xprop, yprop


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def eigentrust(g, trust_map, vprop=None, norm=False, epsilon=1e-6, max_iter=0,
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               ret_iter=False):
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    r"""
    Calculate the eigentrust centrality of each vertex in the graph.

    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    trust_map : :class:`~graph_tool.EdgePropertyMap`
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        Edge property map with the values of trust associated with each
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        edge. The values must lie in the range [0,1].
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    vprop : :class:`~graph_tool.VertexPropertyMap`, optional (default: ``None``)
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        Vertex property map where the values of eigentrust must be stored.
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    norm : bool, optional (default:  ``False``)
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        Norm eigentrust values so that the total sum equals 1.
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    epsilon : float, optional (default: ``1e-6``)
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        Convergence condition. The iteration will stop if the total delta of all
        vertices are below this value.
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    max_iter : int, optional (default: ``None``)
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        If supplied, this will limit the total number of iterations.
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    ret_iter : bool, optional (default: ``False``)
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        If true, the total number of iterations is also returned.

    Returns
    -------
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    eigentrust : :class:`~graph_tool.VertexPropertyMap`
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        A vertex property map containing the eigentrust values.
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    See Also
    --------
    betweenness: betweenness centrality
    pagerank: PageRank centrality
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    trust_transitivity: pervasive trust transitivity
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    Notes
    -----
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    The eigentrust [kamvar-eigentrust-2003]_ values :math:`t_i` correspond the
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    following limit

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    .. math::

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        \mathbf{t} = \lim_{n\to\infty} \left(C^T\right)^n \mathbf{c}

    where :math:`c_i = 1/|V|` and the elements of the matrix :math:`C` are the
    normalized trust values:

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    .. math::

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        c_{ij} = \frac{\max(s_{ij},0)}{\sum_{j} \max(s_{ij}, 0)}

    The algorithm has a topology-dependent complexity.

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: eigentrust

       np.random.seed(42)
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       import matplotlib
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    .. doctest:: eigentrust

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> w = g.new_edge_property("double")
       >>> w.a = np.random.random(len(w.a)) * 42
       >>> t = gt.eigentrust(g, w)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t,
       ...               vertex_size=gt.prop_to_size(t, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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       ...               vorder=t, output="polblogs_eigentrust.pdf")
       <...>

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    .. testcleanup:: eigentrust
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       conv_png("polblogs_eigentrust.pdf")
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    .. figure:: polblogs_eigentrust.png
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       :align: center
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       :width: 80%
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       Eigentrust values of the a political blogs network of
       [adamic-polblogs]_, with random weights attributed to the edges.

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    References
    ----------
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    .. [kamvar-eigentrust-2003] S. D. Kamvar, M. T. Schlosser, H. Garcia-Molina
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       "The eigentrust algorithm for reputation management in p2p networks",
       Proceedings of the 12th international conference on World Wide Web,
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       Pages: 640 - 651, 2003, :doi:`10.1145/775152.775242`
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    .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere
       and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
       Weblogging Ecosystem (2005). :DOI:`10.1145/1134271.1134277`
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    """

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    if vprop is None:
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        vprop = g.new_vertex_property("double")
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    i = libgraph_tool_centrality.\
           get_eigentrust(g._Graph__graph, _prop("e", g, trust_map),
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                          _prop("v", g, vprop), epsilon, max_iter)
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    if norm:
        vprop.get_array()[:] /= sum(vprop.get_array())

    if ret_iter:
        return vprop, i
    else:
        return vprop

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def trust_transitivity(g, trust_map, source=None, target=None, vprop=None):
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    r"""
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    Calculate the pervasive trust transitivity between chosen (or all) vertices
    in the graph.
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    Parameters
    ----------
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    g : :class:`~graph_tool.Graph`
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        Graph to be used.
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    trust_map : :class:`~graph_tool.EdgePropertyMap`
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        Edge property map with the values of trust associated with each
        edge. The values must lie in the range [0,1].
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    source : :class:`~graph_tool.Vertex` (optional, default: None)
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        Source vertex. All trust values are computed relative to this vertex.
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        If left unspecified, the trust values for all sources are computed.
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    target : :class:`~graph_tool.Vertex` (optional, default: None)
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        The only target for which the trust value will be calculated. If left
        unspecified, the trust values for all targets are computed.
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    vprop : :class:`~graph_tool.VertexPropertyMap` (optional, default: None)
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        A vertex property map where the values of transitive trust must be
        stored.
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    Returns
    -------
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    trust_transitivity : :class:`~graph_tool.VertexPropertyMap` or float
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        A vertex vector property map containing, for each source vertex, a
        vector with the trust values for the other vertices. If only one of
        `source` or `target` is specified, this will be a single-valued vertex
        property map containing the trust vector from/to the source/target
        vertex to/from the rest of the network. If both `source` and `target`
        are specified, the result is a single float, with the corresponding
        trust value for the target.
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    See Also
    --------
    eigentrust: eigentrust centrality
    betweenness: betweenness centrality
    pagerank: PageRank centrality

    Notes
    -----
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    The pervasive trust transitivity between vertices i and j is defined as
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    .. math::

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        t_{ij} = \frac{\sum_m A_{m,j} w^2_{G\setminus\{j\}}(i\to m)c_{m,j}}
                 {\sum_m A_{m,j} w_{G\setminus\{j\}}(i\to m)}
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    where :math:`A_{ij}` is the adjacency matrix, :math:`c_{ij}` is the direct
    trust from i to j, and :math:`w_G(i\to j)` is the weight of the path with
    maximum weight from i to j, computed as
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    .. math::

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       w_G(i\to j) = \prod_{e\in i\to j} c_e.
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    The algorithm measures the transitive trust by finding the paths with
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    maximum weight, using Dijkstra's algorithm, to all in-neighbors of a given
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    target. This search needs to be performed repeatedly for every target, since
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    it needs to be removed from the graph first. For each given source, the
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    resulting complexity is therefore :math:`O(V^2\log V)` for all targets, and
    :math:`O(V\log V)` for a single target. For a given target, the complexity
    for obtaining the trust from all given sources is :math:`O(kV\log V)`, where
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    :math:`k` is the in-degree of the target. Thus, the complexity for obtaining
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    the complete trust matrix is :math:`O(EV\log V)`, where :math:`E` is the
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    number of edges in the network.
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    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
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    .. testsetup:: trust_transitivity

       np.random.seed(42)
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       import matplotlib
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    .. doctest:: trust_transitivity

       >>> g = gt.collection.data["polblogs"]
       >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
       >>> g = gt.Graph(g, prune=True)
       >>> w = g.new_edge_property("double")
       >>> w.a = np.random.random(len(w.a))
       >>> t = gt.trust_transitivity(g, w, source=g.vertex(42))
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t,
       ...               vertex_size=gt.prop_to_size(t, mi=5, ma=15),
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       ...               vcmap=matplotlib.cm.gist_heat,
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