# Source code for graph_tool.centrality

#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2006-2021 Tiago de Paula Peixoto <tiago@skewed.de>
#
# This program is free software; you can redistribute it and/or modify it under
# 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 Lesser General Public License for more
# details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.

"""
graph_tool.centrality - Centrality measures
-----------------------------------------------

This module includes centrality-related algorithms.

Summary
+++++++

.. autosummary::
:nosignatures:

pagerank
betweenness
central_point_dominance
closeness
eigenvector
katz
hits
eigentrust
trust_transitivity

Contents
++++++++
"""

from .. dl_import import dl_import
dl_import("from . import libgraph_tool_centrality")

from .. import _prop, ungroup_vector_property, Vector_size_t
from .. topology import shortest_distance
import numpy
import numpy.linalg

__all__ = ["pagerank", "betweenness", "central_point_dominance", "closeness",
"eigentrust", "eigenvector", "katz", "hits", "trust_transitivity"]

[docs]def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6,
max_iter=None, ret_iter=False):
r"""Calculate the PageRank of each vertex.

Parameters
----------
g : :class:~graph_tool.Graph
Graph to be used.
damping : float, optional (default: 0.85)
Damping factor.
pers : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Personalization vector. If omitted, a constant value of :math:1/N
will be used.
weight : :class:~graph_tool.EdgePropertyMap, optional (default: None)
Edge weights. If omitted, a constant value of 1 will be used.
prop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Vertex property map to store the PageRank values. If supplied, 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.
ret_iter : bool, optional (default: False)
If true, the total number of iterations is also returned.

Returns
-------
pagerank : :class:~graph_tool.VertexPropertyMap
A vertex property map containing the PageRank values.

--------
betweenness: betweenness centrality
eigentrust: eigentrust centrality
eigenvector: eigenvector centrality
hits: authority and hub centralities
trust_transitivity: pervasive trust transitivity

Notes
-----
The value of PageRank [pagerank-wikipedia]_ of vertex v, :math:PR(v), is
given iteratively by the relation:

.. math::

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

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.

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.

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.

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

If enabled during compilation, this algorithm runs in parallel.

Examples
--------

.. testsetup:: pagerank

import matplotlib

.. 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),
...               vorder=pr, vcmap=matplotlib.cm.gist_heat,
...               output="polblogs_pr.pdf")
<...>

.. figure:: polblogs_pr.png
:align: center
:width: 80%

PageRank values of the a political blogs network of [adamic-polblogs]_.

Now with a personalization vector, and edge weights:

.. 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),
...               vorder=pr, vcmap=matplotlib.cm.gist_heat,
...               output="polblogs_pr_pers.pdf")
<...>

.. testcleanup:: pagerank

conv_png("polblogs_pr.pdf")
conv_png("polblogs_pr_pers.pdf")

.. figure:: polblogs_pr_pers.png
:align: center
:width: 80%

Personalized PageRank values of the a political blogs network of
[adamic-polblogs]_, where vertices with very low degree are given
artificially high scores.

References
----------
.. [pagerank-wikipedia] http://en.wikipedia.org/wiki/Pagerank
.. [lawrence-pagerank-1998] P. Lawrence, B. Sergey, M. Rajeev, W. Terry,
"The pagerank citation ranking: Bringing order to the web", Technical
report, Stanford University, 1998
.. [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
and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277
"""

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

[docs]def betweenness(g, pivots=None, vprop=None, eprop=None, weight=None, norm=True):
r"""Calculate the betweenness centrality for each vertex and edge.

Parameters
----------
g : :class:~graph_tool.Graph
Graph to be used.
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.
vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Vertex property map to store the vertex betweenness values.
eprop : :class:~graph_tool.EdgePropertyMap, optional (default: None)
Edge property map to store the edge betweenness values.
weight : :class:~graph_tool.EdgePropertyMap, optional (default: None)
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
-------
vertex_betweenness : A vertex property map with the vertex betweenness values.
edge_betweenness : An edge property map with the edge betweenness values.

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

Notes
-----
Betweenness centrality of a vertex :math:C_B(v) is defined as,

.. math::

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

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.

The algorithm used here is defined in [brandes-faster-2001]_, and has a
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.

If enabled during compilation, this algorithm runs in parallel.

Examples
--------

.. testsetup:: betweenness

import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=vp, output="polblogs_betweenness.pdf")
<...>

.. testcleanup:: betweenness

conv_png("polblogs_betweenness.pdf")

.. figure:: polblogs_betweenness.png
:align: center
:width: 80%

Betweenness values of the a political blogs network of [adamic-polblogs]_.

References
----------
.. [betweenness-wikipedia] http://en.wikipedia.org/wiki/Centrality#Betweenness_centrality
.. [brandes-faster-2001] U. Brandes, "A faster algorithm for betweenness
centrality", Journal of Mathematical Sociology, 2001, :doi:10.1080/0022250X.2001.9990249
.. [brandes-centrality-2007] U. Brandes, C. Pich, "Centrality estimation in
large networks", Int. J. Bifurcation Chaos 17, 2303 (2007).
:DOI:10.1142/S0218127407018403
and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277

"""
if vprop is None:
vprop = g.new_vertex_property("double")
if eprop is None:
eprop = g.new_edge_property("double")
if weight is not None and weight.value_type() != eprop.value_type():
nw = g.new_edge_property(eprop.value_type())
g.copy_property(weight, nw)
weight = nw
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
libgraph_tool_centrality.\
get_betweenness(g._Graph__graph, vpivots, _prop("e", g, weight),
_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))
return vprop, eprop

[docs]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.
weight : :class:~graph_tool.EdgePropertyMap, optional (default: None)
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.
vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
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
-------
vertex_closeness : :class:~graph_tool.VertexPropertyMap
A vertex property map with the vertex closeness values.

--------
central_point_dominance: central point dominance of the graph
pagerank: PageRank centrality
eigentrust: eigentrust centrality
eigenvector: eigenvector centrality
hits: authority and hub centralities
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
:math:V-1.

The algorithm complexity of :math:O(V(V + E)) for unweighted graphs and
:math:O(V(V+E) \log V) for weighted graphs. If the option source is
specified, this drops to :math:O(V + E) and :math:O((V+E)\log V)
respectively.

If enabled during compilation, this algorithm runs in parallel.

Examples
--------

.. testsetup:: closeness

import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=c, output="polblogs_closeness.pdf")
<...>

.. testcleanup:: closeness

conv_png("polblogs_closeness.pdf")

.. figure:: polblogs_closeness.png
:align: center
:width: 80%

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
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:
if vprop is None:
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
dist = shortest_distance(g, source=source, weights=weight,
max_dist=max_dist)
dists = dist.fa[(dist.fa < max_dist) * (dist.fa > 0)]
if harmonic:
c = (1. / dists).sum()
if norm:
c /= g.num_vertices() - 1
else:
c = 1. / dists.sum()
if norm:
c *= len(dists)
return c

[docs]def central_point_dominance(g, betweenness):
r"""Calculate the central point dominance of the graph, given the betweenness
centrality of each vertex.

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

Returns
-------
cp : float
The central point dominance.

--------
betweenness: betweenness centrality

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

.. math::

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
--------

>>> 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))
0.218286...

References
----------
.. [freeman-set-1977] Linton C. Freeman, "A Set of Measures of Centrality
Based on Betweenness", Sociometry, Vol. 40, No. 1, pp. 35-41, 1977,
:doi:10.2307/3033543
"""

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

[docs]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.
weight : :class:~graph_tool.EdgePropertyMap (optional, default: None)
Edge property map with the edge weights.
vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
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.

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

--------
betweenness: betweenness centrality
pagerank: PageRank centrality
hits: authority and hub centralities
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
--------

.. testsetup:: eigenvector

np.random.seed(42)
import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=x, output="polblogs_eigenvector.pdf")
<...>

.. testcleanup:: eigenvector

conv_png("polblogs_eigenvector.pdf")

.. figure:: polblogs_eigenvector.png
:align: center
:width: 80%

Eigenvector values of the a political blogs network of
[adamic-polblogs]_, with random weights attributed to the edges.

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
and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277

"""

if vprop is None:
vprop = g.new_vertex_property("double")
vprop.fa = 1. / g.num_vertices()
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

[docs]def katz(g, alpha=0.01, beta=None, weight=None, vprop=None, epsilon=1e-6,
max_iter=None, norm=True):
r"""
Calculate the Katz centrality of each vertex in the graph.

Parameters
----------
g : :class:~graph_tool.Graph
Graph to be used.
weight : :class:~graph_tool.EdgePropertyMap (optional, default: None)
Edge property map with the edge weights.
alpha : float, optional (default: 0.01)
Free parameter :math:\alpha. This must be smaller than the inverse of
the largest eigenvalue of the adjacency matrix.
beta : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Vertex property map where the local personalization values. If not
provided, the global value of 1 will be used.
vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
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.
norm : bool, optional (default: True)
Whether or not the centrality values should be normalized.

Returns
-------
centrality : :class:~graph_tool.VertexPropertyMap
A vertex property map containing the Katz centrality values.

--------
betweenness: betweenness centrality
pagerank: PageRank centrality
eigenvector: eigenvector centrality
hits: authority and hub centralities
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
--------
.. testsetup:: katz

np.random.seed(42)
import matplotlib

.. doctest:: katz

>>> 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))
>>> 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=x, output="polblogs_katz.pdf")
<...>

.. testcleanup:: katz

conv_png("polblogs_katz.pdf")

.. figure:: polblogs_katz.png
:align: center
:width: 80%

Katz centrality values of the a political blogs network of
[adamic-polblogs]_, with random weights attributed to the edges.

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
and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277
"""

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

[docs]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.
weight : :class:~graph_tool.EdgePropertyMap (optional, default: None)
Edge property map with the edge weights.
xprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Vertex property map where the authority centrality must be stored.
yprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
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.
x : :class:~graph_tool.VertexPropertyMap
A vertex property map containing the authority centrality values.
y : :class:~graph_tool.VertexPropertyMap
A vertex property map containing the hub centrality values.

--------
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
--------

.. testsetup:: hits

import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=y, output="polblogs_hits_hubs.pdf")
<...>

.. testcleanup:: hits

conv_png("polblogs_hits_auths.pdf")
conv_png("polblogs_hits_hubs.pdf")

.. figure:: polblogs_hits_auths.png
:align: center
:width: 80%

HITS authority values of the a political blogs network of

.. figure:: polblogs_hits_hubs.png
:align: center
:width: 80%

HITS hub values of the a political blogs network of [adamic-polblogs]_.

References
----------

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

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

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

Parameters
----------
g : :class:~graph_tool.Graph
Graph to be used.
trust_map : :class:~graph_tool.EdgePropertyMap
Edge property map with the values of trust associated with each
edge. The values must lie in the range [0,1].
vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None)
Vertex property map where the values of eigentrust must be stored.
norm : bool, optional (default:  False)
Norm eigentrust values so that the total sum equals 1.
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.
ret_iter : bool, optional (default: False)
If true, the total number of iterations is also returned.

Returns
-------
eigentrust : :class:~graph_tool.VertexPropertyMap
A vertex property map containing the eigentrust values.

--------
betweenness: betweenness centrality
pagerank: PageRank centrality
trust_transitivity: pervasive trust transitivity

Notes
-----
The eigentrust [kamvar-eigentrust-2003]_ values :math:t_i correspond the
following limit

.. math::

\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:

.. math::

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
--------

.. testsetup:: eigentrust

np.random.seed(42)
import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=t, output="polblogs_eigentrust.pdf")
<...>

.. testcleanup:: eigentrust

conv_png("polblogs_eigentrust.pdf")

.. figure:: polblogs_eigentrust.png
:align: center
:width: 80%

Eigentrust values of the a political blogs network of
[adamic-polblogs]_, with random weights attributed to the edges.

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

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

if ret_iter:
return vprop, i
else:
return vprop

[docs]def trust_transitivity(g, trust_map, source=None, target=None, vprop=None):
r"""
Calculate the pervasive trust transitivity between chosen (or all) vertices
in the graph.

Parameters
----------
g : :class:~graph_tool.Graph
Graph to be used.
trust_map : :class:~graph_tool.EdgePropertyMap
Edge property map with the values of trust associated with each
edge. The values must lie in the range [0,1].
source : :class:~graph_tool.Vertex (optional, default: None)
Source vertex. All trust values are computed relative to this vertex.
If left unspecified, the trust values for all sources are computed.
target : :class:~graph_tool.Vertex (optional, default: None)
The only target for which the trust value will be calculated. If left
unspecified, the trust values for all targets are computed.
vprop : :class:~graph_tool.VertexPropertyMap (optional, default: None)
A vertex property map where the values of transitive trust must be
stored.

Returns
-------
trust_transitivity : :class:~graph_tool.VertexPropertyMap or float
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.

--------
eigentrust: eigentrust centrality
betweenness: betweenness centrality
pagerank: PageRank centrality

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

.. math::

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)}

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

.. math::

w_G(i\to j) = \prod_{e\in i\to j} c_e.

The algorithm measures the transitive trust by finding the paths with
maximum weight, using Dijkstra's algorithm, to all in-neighbors of a given
target. This search needs to be performed repeatedly for every target, since
it needs to be removed from the graph first. For each given source, the
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
:math:k is the in-degree of the target. Thus, the complexity for obtaining
the complete trust matrix is :math:O(EV\log V), where :math:E is the
number of edges in the network.

If enabled during compilation, this algorithm runs in parallel.

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

np.random.seed(42)
import matplotlib

.. 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),
...               vcmap=matplotlib.cm.gist_heat,
...               vorder=t, output="polblogs_trust_transitivity.pdf")
<...>

.. testcleanup:: trust_transitivity

conv_png("polblogs_trust_transitivity.pdf")

.. figure:: polblogs_trust_transitivity.png
:align: center
:width: 80%

Trust transitivity values from source vertex 42 of the a political blogs
network of [adamic-polblogs]_, with random weights attributed to the
edges.

References
----------
.. [richters-trust-2010] Oliver Richters and Tiago P. Peixoto, "Trust
Transitivity in Social Networks," PLoS ONE 6, no. 4:
e1838 (2011), :doi:10.1371/journal.pone.0018384
and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the
Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277

"""

if vprop is None:
vprop = g.new_vertex_property("vector<double>")

if target is None:
target = -1
else:
target = g.vertex_index[target]

if source is None:
source = -1
else:
source = g.vertex_index[source]

libgraph_tool_centrality.\
get_trust_transitivity(g._Graph__graph, source, target,
_prop("e", g, trust_map),
_prop("v", g, vprop))
if target != -1 or source != -1:
vprop = ungroup_vector_property(vprop, )
if target != -1 and source != -1:
return vprop.a[target]
return vprop