__init__.py 19.8 KB
Newer Older
Tiago Peixoto's avatar
Tiago Peixoto committed
1
#! /usr/bin/env python
2
# -*- coding: utf-8 -*-
Tiago Peixoto's avatar
Tiago Peixoto committed
3
#
4
5
6
# graph_tool -- a general graph manipulation python module
#
# Copyright (C) 2007-2010 Tiago de Paula Peixoto <tiago@forked.de>
Tiago Peixoto's avatar
Tiago Peixoto committed
7
8
9
10
11
12
13
14
15
16
17
18
19
20
#
# 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/>.

21
"""
22
23
``graph_tool.centrality`` - Centrality measures
-----------------------------------------------
24
25

This module includes centrality-related algorithms.
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40

Summary
+++++++

.. autosummary::
   :nosignatures:

   pagerank
   betweenness
   central_point_dominance
   eigentrust
   absolute_trust

Contents
++++++++
41
42
"""

Tiago Peixoto's avatar
Tiago Peixoto committed
43
44
45
46
from .. dl_import import dl_import
dl_import("import libgraph_tool_centrality")

from .. core import _prop
47
import sys, numpy
Tiago Peixoto's avatar
Tiago Peixoto committed
48
49
50
51

__all__ = ["pagerank", "betweenness", "central_point_dominance", "eigentrust",
           "absolute_trust"]

52
def pagerank(g, damping=0.8, prop=None, epslon=1e-6, max_iter=None,
Tiago Peixoto's avatar
Tiago Peixoto committed
53
             ret_iter=False):
54
55
56
57
58
    r"""
    Calculate the PageRank of each vertex.

    Parameters
    ----------
59
    g : :class:`~graph_tool.Graph`
60
61
62
        Graph to be used.
    damping : float, optional (default: 0.8)
        Damping factor.
63
    prop : :class:`~graph_tool.PropertyMap`, optional (default: None)
64
65
66
67
68
69
70
71
72
73
74
        Vertex property map to store the PageRank values.
    epslon : 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
    -------
75
76
    pagerank : :class:`~graph_tool.PropertyMap`
        A vertex property map containing the PageRank values.
77
78
79
80
81
82
83
84
85

    See Also
    --------
    betweenness: betweenness centrality
    eigentrust: eigentrust centrality
    absolute_trust: absolute trust centrality

    Notes
    -----
86
    The value of PageRank [pagerank-wikipedia]_ of vertex v :math:`PR(v)` is
87
88
89
    given interactively by the relation:

    .. math::
90
91

        PR(v) = \frac{1-d}{N} + d \sum_{w \in \Gamma^{-}(v)}
92
                \frac{PR (w)}{d^{+}(w)}
93
94
95
96
97
98
99
100
101
102
103
104

    where :math:`\Gamma^{-}(v)` are the in-neighbours of v, :math:`d^{+}(w)` is
    the out-degree of w, and d is a damping factor.

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

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
105
106
    >>> from numpy.random import poisson, seed
    >>> seed(42)
107
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
108
    >>> pr = gt.pagerank(g)
109
    >>> print pr.a
Tiago Peixoto's avatar
Tiago Peixoto committed
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
    [ 0.63876901  1.13528868  0.31465963  0.55855277  0.2         0.75605741
      0.42628689  0.53066254  0.55004112  0.91717076  0.71164749  0.32015438
      0.67275227  1.08207389  1.14412231  0.9049167   1.32002     1.4692142
      0.76549771  0.71510277  0.23732927  0.40844911  0.2         0.27912876
      0.71309781  0.32015438  1.3376236   0.31352887  0.59346569  0.33381039
      0.67300081  0.73318264  0.65812653  0.73409673  0.93051993  0.83241145
      1.59816568  0.43979363  0.2512247   1.15663357  0.2         0.35977148
      0.72182022  1.01267711  0.76304859  0.49247376  0.49384283  1.8436647
      0.64312224  1.00778243  0.62287633  1.15215387  0.56176895  0.7166227
      0.56506109  0.67104337  0.95570565  0.27996953  0.79975983  0.33631497
      1.09471419  0.33631497  0.2512247   2.09126732  0.68157485  0.2
      0.37140185  0.65619459  1.27370737  0.48383225  1.36125161  0.2
      0.78300573  1.03427279  0.56904755  1.66077917  1.73302035  0.28749261
      0.83143045  1.04969728  0.70090048  0.55991433  0.68440994  0.2
      0.34018009  0.45485484  0.28        1.2015438   2.11850885  1.24990775
      0.59914308  0.59989185  0.73535564  0.78168417  0.55390281  0.38627667
      1.42274704  0.51105348  0.92550979  1.27968065]
127
128
129

    References
    ----------
130
131
    .. [pagerank-wikipedia] http://en.wikipedia.org/wiki/Pagerank
    .. [lawrence-pagerank-1998] P. Lawrence, B. Sergey, M. Rajeev, W. Terry,
132
       "The pagerank citation ranking: Bringing order to the web", Technical
133
134
135
136
137
       report, Stanford University, 1998
    """

    if max_iter == None:
        max_iter = 0
Tiago Peixoto's avatar
Tiago Peixoto committed
138
139
140
141
142
143
144
145
146
147
    if prop == None:
        prop = g.new_vertex_property("double")
    ic = libgraph_tool_centrality.\
            get_pagerank(g._Graph__graph, _prop("v", g, prop), damping, epslon,
                         max_iter)
    if ret_iter:
        return prop, ic
    else:
        return prop

148
149
150
151
152
153
def betweenness(g, vprop=None, eprop=None, weight=None, norm=True):
    r"""
    Calculate the betweenness centrality for each vertex and edge.

    Parameters
    ----------
154
    g : :class:`~graph_tool.Graph`
155
        Graph to be used.
156
    vprop : :class:`~graph_tool.PropertyMap`, optional (default: None)
157
        Vertex property map to store the vertex betweenness values.
158
    eprop : :class:`~graph_tool.PropertyMap`, optional (default: None)
159
        Edge property map to store the edge betweenness values.
160
    weight : :class:`~graph_tool.PropertyMap`, optional (default: None)
161
162
163
164
165
166
        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
    -------
167
168
169
170
    vertex_betweenness : A vertex property map with the vertex betweenness
                         values.
    edge_betweenness : An edge property map with the edge betweenness
                       values.
171
172
173
174
175
176
177
178
179
180
181
182

    See Also
    --------
    central_point_dominance: central point dominance of the graph
    pagerank: PageRank centrality
    eigentrust: eigentrust centrality
    absolute_trust: absolute trust centrality

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

183
184
    .. math::

185
186
187
188
189
190
191
192
193
        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 geodesic paths from s to
    t, and :math:`\sigma_{st}(v)` is the number of shortest geodesic paths from
    s to t that pass through a vertex 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`.

194
    The algorithm used here is defined in [brandes-faster-2001]_, and has a
195
196
197
198
199
200
201
    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 enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
202
203
    >>> from numpy.random import poisson, seed
    >>> seed(42)
204
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
205
    >>> vb, eb = gt.betweenness(g)
206
    >>> print vb.a
Tiago Peixoto's avatar
Tiago Peixoto committed
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
    [ 0.03395047  0.07911989  0.00702948  0.02337119  0.          0.02930099
      0.01684377  0.02558675  0.03440095  0.02886187  0.03124262  0.00975953
      0.01307953  0.03938858  0.07266505  0.01313647  0.          0.06450598
      0.0575418   0.00525468  0.00466089  0.01803829  0.          0.00050161
      0.0085034   0.02362432  0.05620574  0.00097157  0.04006816  0.01301474
      0.02154916  0.          0.06009194  0.02780363  0.08963522  0.04049657
      0.06993559  0.02082698  0.00288318  0.03264322  0.          0.03641759
      0.01083859  0.03750864  0.04079359  0.02092599  0.          0.02153655
      0.          0.05674631  0.03861911  0.05473282  0.00904367  0.03249097
      0.00894043  0.0192741   0.03379204  0.02125998  0.0018321   0.0013495
      0.0336502   0.0210088   0.00125318  0.0489189   0.05254974  0.
      0.00432189  0.04866168  0.06444727  0.02508525  0.02533085  0.
      0.05308703  0.02539854  0.02270809  0.044889    0.04766016  0.0086368
      0.01501699  0.          0.03107868  0.0054221   0.          0.
      0.00596081  0.01183977  0.00159761  0.11435876  0.03988501  0.05128991
      0.04558135  0.02303469  0.05092032  0.04700221  0.00927644  0.00841903
      0.          0.03243633  0.04514374  0.05170213]
224
225
226

    References
    ----------
227
228
    .. [betweenness-wikipedia] http://en.wikipedia.org/wiki/Centrality#Betweenness_centrality
    .. [brandes-faster-2001] U. Brandes, "A faster algorithm for betweenness
229
230
       centrality",  Journal of Mathematical Sociology, 2001
    """
Tiago Peixoto's avatar
Tiago Peixoto committed
231
232
233
234
235
236
237
238
239
240
241
242
243
244
    if vprop == None:
        vprop = g.new_vertex_property("double")
    if eprop == None:
        eprop = g.new_edge_property("double")
    if weight != None and weight.value_type() != eprop.value_type():
        nw = g.new_edge_property(eprop.value_type())
        g.copy_property(weight, nw)
        weight = nw
    libgraph_tool_centrality.\
            get_betweenness(g._Graph__graph, _prop("e", g, weight),
                            _prop("e", g, eprop), _prop("v", g, vprop), norm)
    return vprop, eprop

def central_point_dominance(g, betweenness):
245
246
247
248
249
250
    r"""
    Calculate the central point dominance of the graph, given the betweenness
    centrality of each vertex.

    Parameters
    ----------
251
    g : :class:`~graph_tool.Graph`
252
        Graph to be used.
253
    betweenness : :class:`~graph_tool.PropertyMap`
254
255
256
257
258
        Vertex property map with the betweenness centrality values. The values
        must be normalized.

    Returns
    -------
259
260
    cp : float
        The central point dominance.
261
262
263
264
265
266
267
268

    See Also
    --------
    betweenness: betweenness centrality

    Notes
    -----
    Let :math:`v^*` be the vertex with the largest relative betweenness
269
    centrality; then, the central point dominance [freeman-set-1977]_ is defined
270
271
    as:

272
273
    .. math::

274
275
276
277
278
279
280
281
282
        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
    --------
283
284
    >>> from numpy.random import poisson, seed
    >>> seed(42)
285
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
286
287
    >>> vb, eb = gt.betweenness(g)
    >>> print gt.central_point_dominance(g, vb)
Tiago Peixoto's avatar
Tiago Peixoto committed
288
    0.0884414811909
289
290
291

    References
    ----------
292
    .. [freeman-set-1977] Linton C. Freeman, "A Set of Measures of Centrality
293
294
295
       Based on Betweenness", Sociometry, Vol. 40, No. 1,  pp. 35-41 (1977)
    """

Tiago Peixoto's avatar
Tiago Peixoto committed
296
    return libgraph_tool_centrality.\
297
           get_central_point_dominance(g._Graph__graph,
Tiago Peixoto's avatar
Tiago Peixoto committed
298
299
                                       _prop("v", g, betweenness))

300
301

def eigentrust(g, trust_map, vprop=None, norm=False, epslon=1e-6, max_iter=0,
Tiago Peixoto's avatar
Tiago Peixoto committed
302
               ret_iter=False):
303
304
305
306
307
    r"""
    Calculate the eigentrust centrality of each vertex in the graph.

    Parameters
    ----------
308
    g : :class:`~graph_tool.Graph`
309
        Graph to be used.
310
    trust_map : :class:`~graph_tool.PropertyMap`
311
        Edge property map with the values of trust associated with each
312
        edge. The values must lie in the range [0,1].
313
314
315
316
317
318
319
320
321
322
323
324
325
326
    vprop : PropertyMap, 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.
    epslon : 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
    -------
327
    eigentrust : A vertex property map containing the eigentrust values.
328
329
330
331
332
333
334
335
336

    See Also
    --------
    betweenness: betweenness centrality
    pagerank: PageRank centrality
    absolute_trust: absolute trust centrality

    Notes
    -----
337
    The eigentrust [kamvar-eigentrust-2003]_ values :math:`t_i` correspond the
338
339
    following limit

340
341
    .. math::

342
343
344
345
346
        \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:

347
348
    .. math::

349
350
351
352
353
354
355
356
357
358
        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
    --------
    >>> from numpy.random import poisson, random, seed
    >>> seed(42)
359
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
360
361
    >>> trust = g.new_edge_property("double")
    >>> trust.get_array()[:] = random(g.num_edges())*42
362
    >>> t = gt.eigentrust(g, trust, norm=True)
363
    >>> print t.get_array()
Tiago Peixoto's avatar
Tiago Peixoto committed
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
    [  5.51422638e-03   1.12397965e-02   2.34959294e-04   6.32738574e-03
       0.00000000e+00   6.34804836e-03   2.67885424e-03   4.02497751e-03
       1.67943467e-02   6.46196106e-03   1.92402451e-02   9.04032352e-04
       9.70843104e-03   1.40319816e-02   1.04995777e-02   2.86712231e-02
       2.47285894e-02   2.38394469e-02   7.06936059e-03   9.45794717e-03
       2.09970054e-05   1.64768298e-03   0.00000000e+00   1.19346706e-03
       6.88434371e-03   5.36337333e-03   2.08428677e-02   2.85813783e-03
       1.10564670e-02   3.16345060e-04   5.25737238e-03   5.43761445e-03
       7.98048389e-03   7.95939648e-03   2.23891858e-02   5.68630666e-03
       2.09300588e-02   4.28902068e-03   1.70833078e-03   2.37814042e-02
       0.00000000e+00   1.20805010e-03   1.29713483e-02   5.73021992e-03
       8.71093674e-03   7.77661067e-03   8.76489806e-04   2.38519385e-02
       3.53225723e-03   8.46948906e-03   5.09874234e-03   2.44547150e-02
       1.32342629e-02   1.80085559e-03   4.37189381e-03   1.18195253e-02
       1.62748861e-02   1.83200678e-04   1.09745025e-02   1.47544090e-03
       3.34512926e-02   1.58885132e-03   1.13128910e-03   3.04944830e-02
       4.22684975e-03   0.00000000e+00   9.89654274e-04   4.25927156e-03
       2.34516214e-02   4.91370905e-03   2.29366664e-02   0.00000000e+00
       6.83407601e-03   1.60508753e-02   1.62762068e-03   3.94324856e-02
       2.84109571e-02   8.81167727e-04   2.16999908e-02   1.28688125e-02
       1.10825963e-02   2.64915564e-03   2.88711928e-03   0.00000000e+00
       4.24392252e-03   9.38398819e-03   0.00000000e+00   1.74508371e-02
       3.26594153e-02   4.07188867e-02   3.20678152e-03   6.35046287e-03
       8.07061556e-03   5.08505374e-03   3.27300367e-03   3.30989070e-03
       2.30651195e-02   4.20338525e-03   5.04332662e-03   3.58731532e-02]
389
390
391

    References
    ----------
392
    .. [kamvar-eigentrust-2003] S. D. Kamvar, M. T. Schlosser, H. Garcia-Molina
393
394
395
396
397
       "The eigentrust algorithm for reputation management in p2p networks",
       Proceedings of the 12th international conference on World Wide Web,
       Pages: 640 - 651, 2003
    """

Tiago Peixoto's avatar
Tiago Peixoto committed
398
399
    if vprop == None:
        vprop = g.new_vertex_property("double")
400
401
402
403
404
405
406
407
408
409
410
    i = libgraph_tool_centrality.\
           get_eigentrust(g._Graph__graph, _prop("e", g, trust_map),
                          _prop("v", g, vprop), epslon, max_iter)
    if norm:
        vprop.get_array()[:] /= sum(vprop.get_array())

    if ret_iter:
        return vprop, i
    else:
        return vprop

411
def absolute_trust(g, trust_map, source, target = None, vprop=None):
412
    r"""
413
414
    Calculate the absolute trust centrality of each vertex in the graph, from a
    given source.
415
416
417

    Parameters
    ----------
418
    g : :class:`~graph_tool.Graph`
419
        Graph to be used.
420
    trust_map : :class:`~graph_tool.PropertyMap`
421
422
        Edge property map with the values of trust associated with each
        edge. The values must lie in the range [0,1].
423
    source : Vertex
424
        Source vertex. All trust values are computed relative to this vertex.
425
    target : Vertex (optional, default: None)
426
427
428
        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.PropertyMap` (optional, default: None)
429
        A vertex property map where the values of trust for each source
430
        must be stored.
431
432
433

    Returns
    -------
434
    absolute_trust : :class:`~graph_tool.PropertyMap` or float
435
        A vertex property map containing the absolute trust vector from the
436
437
438
        source vertex to the rest of the network. If `target` is specified, the
        result is a single float, with the corresponding trust value for the
        target.
439

440
441
442
443
444
445
446
447
448
449
    See Also
    --------
    eigentrust: eigentrust centrality
    betweenness: betweenness centrality
    pagerank: PageRank centrality

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

450
451
    .. math::

452
453
        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)}
454

455
456
457
    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
Tiago Peixoto's avatar
Tiago Peixoto committed
458

459
460
    .. math::

461
       w_G(i\to j) = \prod_{e\in i\to j} c_e.
462

463
464
465
466
467
468
    The algorithm measures the absolute trust by finding the paths with maximum
    weight, using Dijkstra's algorithm, to all in-neighbours of a given
    target. This search needs to be performed repeatedly for every target, since
    it needs to be removed from the graph first. The resulting complexity is
    therefore :math:`O(N^2\log N)` for all targets, and :math:`O(N\log N)` for a
    single target.
469
470
471
472
473
474
475

    If enabled during compilation, this algorithm runs in parallel.

    Examples
    --------
    >>> from numpy.random import poisson, random, seed
    >>> seed(42)
476
    >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3)))
477
    >>> trust = g.new_edge_property("double")
478
    >>> trust.a = random(g.num_edges())
479
480
    >>> t = gt.absolute_trust(g, trust, source=g.vertex(0))
    >>> print t.a
Tiago Peixoto's avatar
Tiago Peixoto committed
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
    [ 0.16260667  0.04129912  0.13735376  0.19146125  0.          0.09147461
      0.10371912  0.12465511  0.24631221  0.0603916   0.2375385   0.06637879
      0.08897662  0.0800988   0.05250601  0.66759022  0.09368793  0.08275437
      0.13674709  0.15553915  0.01376162  0.417068    0.          0.06096886
      0.08746817  0.39380693  0.09215297  0.09575144  0.15594162  0.04008874
      0.05483972  0.05691086  0.13571077  0.32376012  0.22477937  0.06347962
      0.10445085  0.19447845  0.38007043  0.13810585  0.          0.08451096
      0.06648153  0.18479174  0.13003649  0.14850631  0.00320603  0.1074644
      0.12088162  0.06792678  0.08472666  0.2002143   0.25963204  0.37838425
      0.03089371  0.18389694  0.39420339  0.03348093  0.11483196  0.0656204
      0.14206403  0.07066434  0.25168986  0.07040126  0.04870569  0.
      0.09861349  0.03882069  0.1105267   0.07951823  0.08748441  0.
      0.08393443  0.11121719  0.21903223  0.25529628  0.0414386   0.03695558
      0.17664854  0.05143033  0.11735779  0.06525968  0.19600919  0.          0.1220922
      0.33330041  0.          0.28595961  0.14526678  0.12514885  0.089524
      0.40738962  0.03719195  0.54409979  0.06247424  0.10660136  0.11674385
      0.13218144  0.02214988  0.23215937]
498
    """
Tiago Peixoto's avatar
Tiago Peixoto committed
499
500

    if vprop == None:
501
        vprop = g.new_vertex_property("double")
502

503
    source = g.vertex_index[source]
504

505
506
507
508
    if target == None:
        target = -1
    else:
        target = g.vertex_index[target]
509

510
511
512
513
514
    libgraph_tool_centrality.\
            get_absolute_trust(g._Graph__graph, source, target,
                               _prop("e", g, trust_map), _prop("v", g, vprop))
    if target != -1:
        return vprop.a[target]
515
    return vprop
Tiago Peixoto's avatar
Tiago Peixoto committed
516