blockmodel.py 120 KB
Newer Older
1
2
3
4
5
#! /usr/bin/env python
# -*- coding: utf-8 -*-
#
# graph_tool -- a general graph manipulation python module
#
Tiago Peixoto's avatar
Tiago Peixoto committed
6
# Copyright (C) 2006-2015 Tiago de Paula Peixoto <tiago@skewed.de>
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
#
# 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/>.

from __future__ import division, absolute_import, print_function
import sys
if sys.version_info < (3,):
    range = xrange

26
27
from .. import _degree, _prop, Graph, GraphView, libcore, _get_rng, PropertyMap
from .. stats import label_self_loops
28
from .. spectral import adjacency
29
30
import random
from numpy import *
31
import numpy
32
33
from scipy.optimize import fsolve, fminbound
import scipy.special
34
from collections import defaultdict
35
36
import copy
import heapq
37
38
39
40

from .. dl_import import dl_import
dl_import("from . import libgraph_tool_community as libcommunity")

41
__test__ = False
42

43
44
45
46
47
48
49
50
def set_test(test):
    global __test__
    __test__ = test

def _bm_test():
    global __test__
    return __test__

51
52
53
54
55
56
57
58
59
60
61
62
def get_block_graph(g, B, b, vcount, ecount):
    cg, br, vcount, ecount = condensation_graph(g, b,
                                                vweight=vcount,
                                                eweight=ecount,
                                                self_loops=True)[:4]
    cg.vp["count"] = vcount
    cg.ep["count"] = ecount
    cg = Graph(cg, vorder=br)

    cg.add_vertex(B - cg.num_vertices())
    return cg

63
64
65
66
67
68
69
70
class BlockState(object):
    r"""This class encapsulates the block state of a given graph.

    This must be instantiated and used by functions such as :func:`mcmc_sweep`.

    Parameters
    ----------
    g : :class:`~graph_tool.Graph`
71
        Graph to be modelled.
72
    eweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
73
        Edge multiplicities (for multigraphs or block graphs).
74
    vweight : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
75
        Vertex multiplicities (for block graphs).
76
77
78
79
80
81
82
83
    b : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        Initial block labels on the vertices. If not supplied, it will be
        randomly sampled.
    B : ``int`` (optional, default: ``None``)
        Number of blocks. If not supplied it will be either obtained from the
        parameter ``b``, or set to the maximum possible value according to the
        minimum description length.
    clabel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
84
85
        Constraint labels on the vertices. If supplied, vertices with different
        label values will not be clustered in the same group.
86
87
88
    deg_corr : ``bool`` (optional, default: ``True``)
        If ``True``, the degree-corrected version of the blockmodel ensemble will
        be assumed, otherwise the traditional variant will be used.
89
90
91
92
    max_BE : ``int`` (optional, default: ``1000``)
        If the number of blocks exceeds this number, a sparse representation of
        the block graph is used, which is slightly less efficient, but uses less
        memory,
93
94
    """

95
96
    _state_ref_count = 0

97
    def __init__(self, g, eweight=None, vweight=None, b=None,
98
99
                 B=None, clabel=None, deg_corr=True,
                 max_BE=1000, **kwargs):
100
101
        BlockState._state_ref_count += 1

102
        # initialize weights to unity, if necessary
103
104
        if eweight is None:
            eweight = g.new_edge_property("int")
105
            eweight.fa = 1
106
107
108
109
        elif eweight.value_type() != "int32_t":
            eweight = eweight.copy(value_type="int32_t")
        if vweight is None:
            vweight = g.new_vertex_property("int")
110
            vweight.fa = 1
111
112
        elif vweight.value_type() != "int32_t":
            vweight = vweight.copy(value_type="int32_t")
113
114
115
        self.eweight = g.own_property(eweight)
        self.vweight = g.own_property(vweight)

116
117
118
119
        self.is_weighted = False
        if ((g.num_edges() > 0 and self.eweight.fa.max() > 1) or
            kwargs.get("force_weighted", False)):
            self.is_weighted = True
120
121
122

        # configure the main graph and block model parameters
        self.g = g
123

124
125
        self.E = int(self.eweight.fa.sum())
        self.N = int(self.vweight.fa.sum())
126
127
128

        self.deg_corr = deg_corr

129
        # ensure we have at most as many blocks as nodes
130
        if B is not None and b is None:
131
132
            B = min(B, self.g.num_vertices())

133
        if b is None:
134
            # create a random partition into B blocks.
135
136
            if B is None:
                B = get_max_B(self.N, self.E, directed=g.is_directed())
137
138
            B = min(B, self.g.num_vertices())
            ba = random.randint(0, B, self.g.num_vertices())
139
            ba[:B] = arange(B)        # avoid empty blocks
140
141
            if B < self.g.num_vertices():
                random.shuffle(ba)
142
            b = g.new_vertex_property("int")
143
            b.fa = ba
144
145
            self.b = b
        else:
146
147
148
149
150
151
            # if a partition is available, we will incorporate it.
            if isinstance(b, numpy.ndarray):
                self.b = g.new_vertex_property("int")
                self.b.fa = b
            else:
                self.b = b = g.own_property(b.copy(value_type="int"))
152
            if B is None:
153
154
155
156
                B = int(self.b.fa.max()) + 1

        # if B > self.N:
        #     raise ValueError("B > N!")
157

158
        if self.b.fa.max() >= B:
159
            raise ValueError("Maximum value of b is larger or equal to B! (%d vs %d)" % (self.b.fa.max(), B))
160
161

        # Construct block-graph
162
        self.bg = get_block_graph(g, B, self.b, self.vweight, self.eweight)
163
164
165
166
        self.bg.set_fast_edge_removal()

        self.mrs = self.bg.ep["count"]
        self.wr = self.bg.vp["count"]
167

168
169
170
171
172
173
174
175
176
        del self.bg.ep["count"]
        del self.bg.vp["count"]

        self.mrp = self.bg.degree_property_map("out", weight=self.mrs)

        if g.is_directed():
            self.mrm = self.bg.degree_property_map("in", weight=self.mrs)
        else:
            self.mrm = self.mrp
177
178
179

        self.vertices = libcommunity.get_vector(B)
        self.vertices.a = arange(B)
180
        self.B = B
181

182
183
        if clabel is not None:
            if isinstance(clabel, PropertyMap):
184
                self.clabel = self.g.own_property(clabel.copy())
185
186
187
188
            else:
                self.clabel = self.g.new_vertex_property("int")
                self.clabel.a = clabel
        else:
189
190
191
192
193
194
195
            self.clabel = self.g.new_vertex_property("int")

        self.emat = None
        if max_BE is None:
            max_BE = 1000
        self.max_BE = max_BE

196
197
        self.overlap = False

198
199
200
201
        # used by mcmc_sweep()
        self.egroups = None
        self.nsampler = None
        self.sweep_vertices = None
202
203
        self.overlap_stats = libcommunity.overlap_stats()
        self.partition_stats = libcommunity.partition_stats()
204
        self.edges_dl = False
205

206
        # computation cache
207
208
209
210
211
        E = g.num_edges()
        N = g.num_vertices()
        libcommunity.init_safelog(int(5 * max(E, N)))
        libcommunity.init_xlogx(int(5 * max(E, N)))
        libcommunity.init_lgamma(int(3 * max(E, N)))
212

213
    def __del__(self):
Tiago Peixoto's avatar
Tiago Peixoto committed
214
215
216
217
218
219
        try:
            BlockState._state_ref_count -= 1
            if BlockState._state_ref_count == 0:
                libcommunity.clear_safelog()
                libcommunity.clear_xlogx()
                libcommunity.clear_lgamma()
220
        except (ValueError, AttributeError, TypeError):
Tiago Peixoto's avatar
Tiago Peixoto committed
221
            pass
222

223
224
225
226
227
228
    def __repr__(self):
        return "<BlockState object with %d blocks,%s for graph %s, at 0x%x>" % \
            (self.B, " degree corrected," if self.deg_corr else "", str(self.g),
             id(self))


229
230
    def __init_partition_stats(self, empty=True, edges_dl=False):
        self.edges_dl = edges_dl
231
232
233
234
        if not empty:
            self.partition_stats = libcommunity.init_partition_stats(self.g._Graph__graph,
                                                                     _prop("v", self.g, self.b),
                                                                     _prop("e", self.g, self.eweight),
235
236
                                                                     self.N, self.B,
                                                                     edges_dl)
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
        else:
            self.partition_stats = libcommunity.partition_stats()



    def copy(self, b=None, B=None, deg_corr=None, clabel=None, overlap=False):
        r"""Copies the block state. The parameters override the state properties, and
         have the same meaning as in the constructor. If ``overlap=True`` an
         instance of :class:`~graph_tool.community.OverlapBlockState` is
         returned."""

        if not overlap:
            state = BlockState(self.g,
                               eweight=self.eweight,
                               vweight=self.vweight,
                               b=self.b.copy() if b is None else b,
                               B=(self.B if b is None else None) if B is None else B,
                               clabel=self.clabel if clabel is None else clabel,
                               deg_corr=self.deg_corr if deg_corr is None else deg_corr,
                               max_BE=self.max_BE)
        else:
            state = OverlapBlockState(self.g,
                                      b=b if b is not None else self.b,
                                      B=(self.B if b is None else None) if B is None else B,
                                      clabel=self.clabel if clabel is None else clabel,
                                      deg_corr=self.deg_corr if deg_corr is None else deg_corr,
                                      max_BE=self.max_BE)

        if not state.__check_clabel():
            b = state.b.a + state.clabel.a * state.B
            continuous_map(b)
            state = state.copy(b=b)

270
            if _bm_test():
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
                assert state.__check_clabel()

        return state


    def __getstate__(self):
        state = dict(g=self.g,
                     eweight=self.eweight,
                     vweight=self.vweight,
                     b=self.b,
                     B=self.B,
                     clabel=self.clabel,
                     deg_corr=self.deg_corr,
                     max_BE=self.max_BE)
        return state

    def __setstate__(self, state):
        self.__init__(**state)
        return state

291
292
    def get_block_state(self, b=None, vweight=False, deg_corr=False,
                        overlap=False, **kwargs):
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
        r"""Returns a :class:`~graph_tool.community.BlockState`` corresponding to the
        block graph. The parameters have the same meaning as the in the constructor."""


        state = BlockState(self.bg, eweight=self.mrs,
                           vweight=self.wr if vweight else None,
                           b=self.bg.vertex_index.copy("int") if b is None else b,
                           clabel=self.get_bclabel(),
                           deg_corr=deg_corr,
                           max_BE=self.max_BE)
        if overlap:
            state = state.copy(overlap=True)
        n_map = self.b.copy()
        return state, n_map

    def get_bclabel(self):
        r"""Returns a :class:`~graph_tool.PropertyMap`` corresponding to constraint
        labels for the block graph."""

        bclabel = self.bg.new_vertex_property("int")
        reverse_map(self.b, bclabel)
        pmap(bclabel, self.clabel)
        return bclabel

    def __check_clabel(self):
        b = self.b.a + self.clabel.a * self.B
        continuous_map(b)
        b2 = self.b.copy()
        continuous_map(b2.a)
        return (b == b2.a).all()

324
325
326
327
    def __get_emat(self):
        if self.emat is None:
            self.__regen_emat()
        return self.emat
328
329

    def __regen_emat(self):
330
331
332
333
        if self.B <= self.max_BE:
            self.emat = libcommunity.create_emat(self.bg._Graph__graph)
        else:
            self.emat = libcommunity.create_ehash(self.bg._Graph__graph)
334

335
    def __build_egroups(self, empty=False):
336
337
        self.esrcpos = self.g.new_edge_property("int")
        self.etgtpos = self.g.new_edge_property("int")
338

339
        self.egroups = libcommunity.build_egroups(self.g._Graph__graph,
340
341
342
343
344
                                                  self.bg._Graph__graph,
                                                  _prop("v", self.g, self.b),
                                                  _prop("e", self.g, self.eweight),
                                                  _prop("e", self.g, self.esrcpos),
                                                  _prop("e", self.g, self.etgtpos),
345
                                                  self.is_weighted, empty)
346

347
    def __build_nsampler(self, empty=False):
348
        self.nsampler = libcommunity.init_neighbour_sampler(self.g._Graph__graph,
349
                                                            _prop("e", self.g, self.eweight),
350
                                                            True, empty)
351
352
353
354
355
356

    def __cleanup_bg(self):
        emask = self.bg.new_edge_property("bool")
        emask.a = self.mrs.a[:len(emask.a)] > 0
        self.bg.set_edge_filter(emask)
        self.bg.purge_edges()
357
        self.emat = None
358
359
360
361
362
363
364
365
366
367

    def get_blocks(self):
        r"""Returns the property map which contains the block labels for each vertex."""
        return self.b

    def get_bg(self):
        r"""Returns the block graph."""
        return self.bg

    def get_ers(self):
368
369
        r"""Returns the edge property map of the block graph which contains the :math:`e_{rs}` matrix entries.
        For undirected graphs, the diagonal values (self-loops) contain :math:`e_{rr}/2`."""
370
371
372
373
374
375
376
377
        return self.mrs

    def get_er(self):
        r"""Returns the vertex property map of the block graph which contains the number
        :math:`e_r` of half-edges incident on block :math:`r`. If the graph is
        directed, a pair of property maps is returned, with the number of
        out-edges :math:`e^+_r` and in-edges :math:`e^-_r`, respectively."""
        if self.bg.is_directed():
378
            return self.mrp, self.mrm
379
380
381
382
383
384
385
        else:
            return self.mrp

    def get_nr(self):
        r"""Returns the vertex property map of the block graph which contains the block sizes :math:`n_r`."""
        return self.wr

386
387
388
    def entropy(self, complete=True, dl=False, partition_dl=True,
                degree_dl=True, edges_dl=True, dense=False, multigraph=True,
                norm=False, dl_ent=False, **kwargs):
389
        r"""Calculate the entropy associated with the current block partition.
390
391
392
393
394
395
396
397

        Parameters
        ----------
        complete : ``bool`` (optional, default: ``False``)
            If ``True``, the complete entropy will be returned, including constant
            terms not relevant to the block partition.
        dl : ``bool`` (optional, default: ``False``)
            If ``True``, the full description length will be returned.
398
399
400
401
402
403
404
405
        partition_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the partition description length
            will be considered.
        edges_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the edge matrix description length
            will be considered.
        degree_dl : ``bool`` (optional, default: ``True``)
            If ``True``, and ``dl == True`` the degree sequence description
406
            length will be considered.
407
408
409
        dense : ``bool`` (optional, default: ``False``)
            If ``True``, the "dense" variant of the entropy will be computed.
        multigraph : ``bool`` (optional, default: ``False``)
410
411
412
413
414
415
416
            If ``True``, the multigraph entropy will be used.
        norm : ``bool`` (optional, default: ``True``)
            If ``True``, the entropy will be "normalized" by dividing by the
            number of edges.
        dl_ent : ``bool`` (optional, default: ``False``)
            If ``True``, the description length of the degree sequence will be
            approximated by its entropy.
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442

        Notes
        -----
        For the traditional blockmodel (``deg_corr == False``), the entropy is
        given by

        .. math::

          \mathcal{S}_t &\cong E - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
          \mathcal{S}^d_t &\cong E - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

        for undirected and directed graphs, respectively, where :math:`e_{rs}`
        is the number of edges from block :math:`r` to :math:`s` (or the number
        of half-edges for the undirected case when :math:`r=s`), and :math:`n_r`
        is the number of vertices in block :math:`r` .

        For the degree-corrected variant with "hard" degree constraints the
        equivalent expressions are

        .. math::

            \mathcal{S}_c &\cong -E -\sum_kN_k\ln k! - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
            \mathcal{S}^d_c &\cong -E -\sum_{k^+}N_{k^+}\ln k^+!  -\sum_{k^-}N_{k^-}\ln k^-! - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

        where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
        block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
443
        \sum_se_{sr}` are the numbers of out- and in-edges adjacent to block
444
445
        :math:`r`, respectively.

446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
        If ``dense == False`` and ``multigraph == True``, the entropy used will
        be of the "Poisson" model, with the additional term:

        .. math::

            {\mathcal{S}_{cm}^{(d)}} = \mathcal{S}_c^{(d)} + \sum_{i>j} \ln A_{ij}! + \sum_i \ln A_{ii}!!


        If ``dl == True``, the description length :math:`\mathcal{L}_t` of the
        model will be returned as well, as described in
        :func:`model_entropy`. Note that for the degree-corrected version the
        description length is

        .. math::

            \mathcal{L}_c = \mathcal{L}_t + \sum_r\min\left(\mathcal{L}^{(1)}_r, \mathcal{L}^{(2)}_r\right),

        with

        .. math::

              \mathcal{L}^{(1)}_r &= \ln{\left(\!\!{n_r \choose e_r}\!\!\right)}, \\
              \mathcal{L}^{(2)}_r &= \ln\Xi_r + \ln n_r! - \sum_k \ln n^r_k!,

        and :math:`\ln\Xi_r \simeq 2\sqrt{\zeta(2)e_r}`, where :math:`\zeta(x)`
        is the `Riemann zeta function
        <https://en.wikipedia.org/wiki/Riemann_zeta_function>`_, and
        :math:`n^r_k` is the number of nodes in block :math:`r` with degree
        :math:`k`. For directed graphs we have instead :math:`k \to (k^-, k^+)`,
        and :math:`\ln\Xi_r \to \ln\Xi^+_r + \ln\Xi^-_r` with :math:`\ln\Xi_r^+
        \simeq 2\sqrt{\zeta(2)e^+_r}` and :math:`\ln\Xi_r^- \simeq
        2\sqrt{\zeta(2)e^-_r}`.

        If ``dl_ent=True`` is passed, this will be approximated instead by
480
481
482

        .. math::

483
            \mathcal{L}_c \simeq \mathcal{L}_t - \sum_rn_r\sum_kp^r_k\ln p^r_k,
484

485
        where :math:`p^r_k = n^r_k / n_r`.
486

487
488
        If the "dense" entropies are requested (``dense=True``), they will be
        computed as
489
490
491
492
493
494
495
496
497
498
499
500
501

        .. math::

            \mathcal{S}_t  &= \sum_{r>s} \ln{\textstyle {n_rn_s \choose e_{rs}}} + \sum_r \ln{\textstyle {{n_r\choose 2}\choose e_{rr}/2}}\\
            \mathcal{S}^d_t  &= \sum_{rs} \ln{\textstyle {n_rn_s \choose e_{rs}}},

        for simple graphs, and

        .. math::

            \mathcal{S}_m  &= \sum_{r>s} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)} + \sum_r \ln{\textstyle \left(\!\!{\left(\!{n_r\choose 2}\!\right)\choose e_{rr}/2}\!\!\right)}\\
            \mathcal{S}^d_m  &= \sum_{rs} \ln{\textstyle \left(\!\!{n_rn_s \choose e_{rs}}\!\!\right)},

502
503
504
        for multigraphs (i.e. ``multigraph == True``). A dense entropy for the
        degree-corrected model is not available, and if requested will return a
        :exc:`NotImplementedError`.
505

506
507
        If ``complete == False`` constants in the above equations which do not
        depend on the partition of the nodes will be omitted.
508

509
510
        Note that in all cases if ``norm==True`` the value returned corresponds
        to the entropy `per edge`, i.e. :math:`(\mathcal{S}_{t/c}\; [\,+\,\mathcal{L}_{t/c}])/ E`.
511
512
        """

513
514
515
        xi_fast = kwargs.get("xi_fast", False)
        dl_deg_alt = kwargs.get("dl_deg_alt", True)

516
517
518
        E = self.E
        N = self.N

519
520
        if dense:
            if self.deg_corr:
521
                raise NotImplementedError('A degree-corrected "dense" entropy is not yet implemented')
522

523
            S = libcommunity.entropy_dense(self.bg._Graph__graph,
524
525
526
                                            _prop("e", self.bg, self.mrs),
                                            _prop("v", self.bg, self.wr),
                                            multigraph)
527
528
        else:
            S = libcommunity.entropy(self.bg._Graph__graph,
529
530
531
532
533
                                      _prop("e", self.bg, self.mrs),
                                      _prop("v", self.bg, self.mrp),
                                      _prop("v", self.bg, self.mrm),
                                      _prop("v", self.bg, self.wr),
                                      self.deg_corr)
534

535
            if _bm_test():
536
537
538
                assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                random=random, dl=dl,
                                                                                                partition_dl=partition_dl,
539
                                                                                                edges_dl=edges_dl,
540
541
                                                                                                dense=dense, multigraph=multigraph,
                                                                                                norm=norm)))
542
543
544
545
546
            if self.deg_corr:
                S -= E
            else:
                S += E

547
548
            if complete:
                if self.deg_corr:
549
550
551
552
                    S += libcommunity.deg_entropy_term(self.g._Graph__graph,
                                                       libcore.any(),
                                                       self.overlap_stats,
                                                       self.N)
553

554
555
556
557
                if multigraph:
                    S += libcommunity.entropy_parallel(self.g._Graph__graph,
                                                       _prop("e", self.g, self.eweight))

558
                if _bm_test():
559
560
561
                    assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                    random=random, dl=dl,
                                                                                                    partition_dl=partition_dl,
562
                                                                                                    edges_dl=edges_dl,
563
564
                                                                                                    dense=dense, multigraph=multigraph,
                                                                                                    norm=norm)))
565
        if dl:
566
567
568
569
570
571
572
573
            if partition_dl:
                if self.partition_stats.is_enabled():
                    S += self.partition_stats.get_partition_dl()
                else:
                    self.__init_partition_stats(empty=False)
                    S += self.partition_stats.get_partition_dl()
                    self.__init_partition_stats(empty=True)

574
                if _bm_test():
575
576
577
                    assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                                    random=random, dl=dl,
                                                                                                    partition_dl=partition_dl,
578
                                                                                                    edges_dl=edges_dl,
579
580
                                                                                                    dense=dense, multigraph=multigraph,
                                                                                                    norm=norm)))
581
582
583
            if edges_dl:
                actual_B = (self.wr.a > 0).sum()
                S += model_entropy(actual_B, N, E, directed=self.g.is_directed(), nr=False)
584

585
            if self.deg_corr and degree_dl:
586
587
588
589
590
591
                if self.partition_stats.is_enabled():
                    S_seq = self.partition_stats.get_deg_dl(dl_ent, dl_deg_alt, xi_fast)
                else:
                    self.__init_partition_stats(empty=False)
                    S_seq = self.partition_stats.get_deg_dl(dl_ent, dl_deg_alt, xi_fast)
                    self.__init_partition_stats(empty=True)
592

593
                S += S_seq
594

595
                if _bm_test():
596
597
598
                    assert not isnan(S_seq) and not isinf(S_seq), "invalid entropy %g (%s) " % (S_seq, str(dict(complete=complete,
                                                                                                                random=random, dl=dl,
                                                                                                                partition_dl=partition_dl,
599
                                                                                                                edges_dl=edges_dl,
600
601
602
                                                                                                                dense=dense, multigraph=multigraph,
                                                                                                                norm=norm)))

603
        if _bm_test():
604
605
606
            assert not isnan(S) and not isinf(S), "invalid entropy %g (%s) " % (S, str(dict(complete=complete,
                                                                                            random=random, dl=dl,
                                                                                            partition_dl=partition_dl,
607
                                                                                            edges_dl=edges_dl,
608
609
610
611
612
613
614
                                                                                            dense=dense, multigraph=multigraph,
                                                                                            norm=norm)))

        if norm:
            return S / E
        else:
            return S
615

616
617
618
    def get_matrix(self):
        r"""Returns the block matrix (as a sparse :class:`~scipy.sparse.csr_matrix`),
        which contains the number of edges between each block pair.
619

620
621
622
623
624
625
        .. warning::

           This corresponds to the adjacency matrix of the block graph, which by
           convention includes twice the amount of edges in the diagonal entries
           if the graph is undirected.

626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
        Examples
        --------

        .. testsetup:: get_matrix

           gt.seed_rng(42)
           np.random.seed(42)
           from pylab import *

        .. doctest:: get_matrix

           >>> g = gt.collection.data["polbooks"]
           >>> state = gt.BlockState(g, B=5, deg_corr=True)
           >>> for i in range(1000):
           ...     ds, nmoves = gt.mcmc_sweep(state)
641
           >>> m = state.get_matrix()
642
643
           >>> figure()
           <...>
644
           >>> matshow(m.todense())
645
646
647
648
649
650
651
652
653
654
655
656
657
           <...>
           >>> savefig("bloc_mat.pdf")

        .. testcleanup:: get_matrix

           savefig("bloc_mat.png")

        .. figure:: bloc_mat.*
           :align: center

           A  5x5 block matrix.

       """
658

659
        return adjacency(self.bg, weight=self.mrs)
660
661


662
def model_entropy(B, N, E, directed=False, nr=None):
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
    r"""Computes the amount of information necessary for the parameters of the traditional blockmodel ensemble, for ``B`` blocks, ``N`` vertices, ``E`` edges, and either a directed or undirected graph.

    A traditional blockmodel is defined as a set of :math:`N` vertices which can
    belong to one of :math:`B` blocks, and the matrix :math:`e_{rs}` describes
    the number of edges from block :math:`r` to :math:`s` (or twice that number
    if :math:`r=s` and the graph is undirected).

    For an undirected graph, the number of distinct :math:`e_{rs}` matrices is given by,

    .. math::

       \Omega_m = \left(\!\!{\left(\!{B \choose 2}\!\right) \choose E}\!\!\right)

    and for a directed graph,

    .. math::
       \Omega_m = \left(\!\!{B^2 \choose E}\!\!\right)


    where :math:`\left(\!{n \choose k}\!\right) = {n+k-1\choose k}` is the
    number of :math:`k` combinations with repetitions from a set of size :math:`n`.

    The total information necessary to describe the model is then,

    .. math::

689
690
       \mathcal{L}_t = \ln\Omega_m + \ln\left(\!\!{B \choose N}\!\!\right) + \ln N! - \sum_r \ln n_r!,

691

692
693
    where the remaining term is the information necessary to describe the
    block partition, where :math:`n_r` is the number of nodes in block :math:`r`.
694

695
696
    If ``nr`` is ``None``, it is assumed :math:`n_r=N/B`.

697
698
699
    References
    ----------

Tiago Peixoto's avatar
Tiago Peixoto committed
700
701
    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
Tiago Peixoto's avatar
Tiago Peixoto committed
702
703
704
    .. [peixoto-hierarchical-2014] Tiago P. Peixoto, "Hierarchical block structures and high-resolution
       model selection in large networks ", Phys. Rev. X 4, 011047 (2014), :doi:`10.1103/PhysRevX.4.011047`,
       :arxiv:`1310.4377`.
705
706
707

    """

708
709
710
711
    if directed:
        x = (B * B);
    else:
        x = (B * (B + 1)) / 2;
712
713
714
715
716
    if nr is False:
        L = lbinom(x + E - 1, E)
    else:
        L = lbinom(x + E - 1, E) + partition_entropy(B, N, nr)
    return L
717

718
def lbinom(n, k):
719
    return scipy.special.gammaln(float(n + 1)) - scipy.special.gammaln(float(n - k + 1)) - scipy.special.gammaln(float(k + 1))
720

721
722
723
724
725
726
def lbinom_careful(n, k):
    return libcommunity.lbinom_careful(n, k)

def lbinom_fast(n, k):
    return libcommunity.lbinom_fast(n, k)

727
728
729
730
def partition_entropy(B, N, nr=None):
    if nr is None:
        S = N * log(B) + log1p(-(1 - 1./B) ** N)
    else:
731
        S = lbinom(B + N - 1, N) + scipy.special.gammaln(N + 1) - scipy.special.gammaln(nr + 1).sum()
732
    return S
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752

def get_max_B(N, E, directed=False):
    r"""Return the maximum detectable number of blocks, obtained by minimizing:

    .. math::

        \mathcal{L}_t(B, N, E) - E\ln B

    where :math:`\mathcal{L}_t(B, N, E)` is the information necessary to
    describe a traditional blockmodel with `B` blocks, `N` nodes and `E`
    edges (see :func:`model_entropy`).

    Examples
    --------

    >>> gt.get_max_B(N=1e6, E=5e6)
    1572

    References
    ----------
Tiago Peixoto's avatar
Tiago Peixoto committed
753
754
    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
755
756
757
758


    """

759
760
761
    def Sdl(B, S, N, E, directed=False):
        return S + model_entropy(B, N, E, directed) / E

762
763
764
765
    B = fminbound(lambda B: Sdl(B, -log(B), N, E, directed), 1, E,
                  xtol=1e-6, maxfun=1500, disp=0)
    if isnan(B):
        B = 1
766
    return min(N, max(int(ceil(B)), 2))
767
768

def get_akc(B, I, N=float("inf"), directed=False):
Tiago Peixoto's avatar
Tiago Peixoto committed
769
    r"""Return the minimum value of the average degree of the network, so that some block structure with :math:`B` blocks can be detected, according to the minimum description length criterion.
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802

    This is obtained by solving

    .. math::

       \Sigma_b = \mathcal{L}_t(B, N, E) - E\mathcal{I}_{t/c} = 0,

    where :math:`\mathcal{L}_{t}` is the necessary information to describe the
    blockmodel parameters (see :func:`model_entropy`), and
    :math:`\mathcal{I}_{t/c}` characterizes the planted block structure, and is
    given by

    .. math::

        \mathcal{I}_t &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{w_rw_s}\right),\\
        \mathcal{I}_c &= \sum_{rs}m_{rs}\ln\left(\frac{m_{rs}}{m_rm_s}\right),

    where :math:`m_{rs} = e_{rs}/2E` (or :math:`m_{rs} = e_{rs}/E` for directed
    graphs) and :math:`w_r=n_r/N`. We note that :math:`\mathcal{I}_{t/c}\in[0,
    \ln B]`. In the case where :math:`E \gg B^2`, this simplifies to

    .. math::

       \left<k\right>_c &= \frac{2\ln B}{\mathcal{I}_{t/c}},\\
       \left<k^{-/+}\right>_c &= \frac{\ln B}{\mathcal{I}_{t/c}},

    for undirected and directed graphs, respectively. This limit is assumed if
    ``N == inf``.

    Examples
    --------

    >>> gt.get_akc(10, log(10) / 100, N=100)
803
    2.414413200430159
804
805
806

    References
    ----------
Tiago Peixoto's avatar
Tiago Peixoto committed
807
808
    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks",
       Phys. Rev. Lett. 110, 148701 (2013), :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
809
810
811
812

    """
    if N != float("inf"):
        if directed:
813
            get_dl = lambda ak: model_entropy(B, N, N * ak, directed) / N * ak - N * ak * I
814
        else:
815
            get_dl = lambda ak: model_entropy(B, N, N * ak / 2., directed) * 2 / (N * ak)  - N * ak * I / 2.
816
817
818
819
820
821
822
823
        ak = fsolve(lambda ak: get_dl(ak), 10)
        ak = float(ak)
    else:
        ak = 2 * log(B) / S
        if directed:
            ak /= 2
    return ak

824
825
def mcmc_sweep(state, beta=1., c=1., niter=1, dl=False, dense=False,
               multigraph=False, node_coherent=False, confine_layers=False,
826
               sequential=True, parallel=False, vertices=None,
Tiago Peixoto's avatar
Tiago Peixoto committed
827
               target_blocks=None, verbose=False, **kwargs):
828
    r"""Performs a Markov chain Monte Carlo sweep on the network, to sample the block partition according to a probability :math:`\propto e^{-\beta \mathcal{S}_{t/c}}`, where :math:`\mathcal{S}_{t/c}` is the blockmodel entropy.
829
830
831

    Parameters
    ----------
832
    state : :class:`~graph_tool.community.BlockState`, :class:`~graph_tool.community.OverlapBlockState` or :class:`~graph_tool.community.CovariateBlockState`
833
        The block state.
834
    beta : ``float`` (optional, default: `1.0`)
835
        The inverse temperature parameter :math:`\beta`.
836
837
838
839
840
    c : ``float`` (optional, default: ``1.0``)
        This parameter specifies how often fully random moves are attempted,
        instead of more likely moves based on the inferred block partition.
        For ``c == 0``, no fully random moves are attempted, and for ``c == inf``
        they are always attempted.
841
842
    niter : ``int`` (optional, default: ``1``)
        Number of sweeps to perform.
843
844
845
    dl : ``bool`` (optional, default: ``False``)
        If ``True``, the change in the whole description length will be
        considered after each vertex move, not only the entropy.
846
847
848
849
850
    dense : ``bool`` (optional, default: ``False``)
        If ``True``, the "dense" variant of the entropy will be computed.
    multigraph : ``bool`` (optional, default: ``False``)
        If ``True``, the multigraph entropy will be used. Only has an effect
        if ``dense == True``.
851
852
853
854
    node_coherent : ``bool`` (optional, default: ``False``)
        If ``True``, and if the ``state`` is an instance of
        :class:`~graph_tool.community.OverlapBlockState`, then all half-edges
        incident on the same node are moved simultaneously.
855
856
857
858
859
    confine_layers : ``bool`` (optional, default: ``False``)
        If ``True``, and if the ``state`` is an instance of
        :class:`~graph_tool.community.CovariateBlockState`, with an
        *overlapping* partition, the half edges will only be moved in such a way
         that inside each layer the group membership remains non-overlapping.
860
861
862
863
864
    sequential : ``bool`` (optional, default: ``True``)
        If ``True``, the move attempts on the vertices are done in sequential
        random order. Otherwise a total of `N` moves attempts are made, where
        `N` is the number of vertices, where each vertex can be selected with
        equal probability.
865
866
867
868
869
870
871
872
873
874
    parallel : ``bool`` (optional, default: ``False``)
        If ``True``, the updates are performed in parallel (multiple
        threads).

        .. warning::

            If this is used, the Markov Chain is not guaranteed to be sampled with
            the correct probabilities. This is better used in conjunction with
            ``beta=float('inf')``, where this is not an issue.

Tiago Peixoto's avatar
Tiago Peixoto committed
875
    vertices : ``list of ints`` (optional, default: ``None``)
876
877
        A list of vertices which will be attempted to be moved. If ``None``, all
        vertices will be attempted.
878
879
880
    target_blocks : ``list of ints`` (optional, default: ``None``)
        A list of groups to which the corresponding vertices will to be forcibly
        moved. If ``None``, the standard MCMC rules will be applied.
881
882
883
884
885
886
    verbose : ``bool`` (optional, default: ``False``)
        If ``True``, verbose information is displayed.

    Returns
    -------

887
    dS : ``float``
888
       The entropy difference (in nats) after the sweeps.
889
890
891
892
893
894
895
    nmoves : ``int``
       The number of accepted block membership moves.


    Notes
    -----

896
    This algorithm performs a Markov chain Monte Carlo sweep on the network,
897
898
    where the block memberships are randomly moved, and either accepted or
    rejected, so that after sufficiently many sweeps the partitions are sampled
899
    with probability proportional to :math:`e^{-\beta\mathcal{S}_{t/c}}`, where
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
    :math:`\mathcal{S}_{t/c}` is the blockmodel entropy, given by

    .. math::

      \mathcal{S}_t &\cong - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right), \\
      \mathcal{S}^d_t &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{n_rn_s}\right),

    for undirected and directed traditional blockmodels (``deg_corr == False``),
    respectively, where :math:`e_{rs}` is the number of edges from block
    :math:`r` to :math:`s` (or the number of half-edges for the undirected case
    when :math:`r=s`), and :math:`n_r` is the number of vertices in block
    :math:`r`, and constant terms which are independent of the block partition
    were dropped (see :meth:`BlockState.entropy` for the complete entropy). For
    the degree-corrected variant with "hard" degree constraints the equivalent
    expressions are

    .. math::

       \mathcal{S}_c &\cong  - \frac{1}{2} \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e_re_s}\right), \\
       \mathcal{S}^d_c &\cong - \sum_{rs}e_{rs}\ln\left(\frac{e_{rs}}{e^+_re^-_s}\right),

    where :math:`e_r = \sum_se_{rs}` is the number of half-edges incident on
    block :math:`r`, and :math:`e^+_r = \sum_se_{rs}` and :math:`e^-_r =
    \sum_se_{sr}` are the number of out- and in-edges adjacent to block
    :math:`r`, respectively.

    The Monte Carlo algorithm employed attempts to improve the mixing time of
927
928
    the Markov chain by proposing membership moves :math:`r\to s` with
    probability :math:`p(r\to s|t) \propto e_{ts} + c`, where :math:`t` is the
929
    block label of a random neighbour of the vertex being moved. See
930
    [peixoto-efficient-2014]_ for more details.
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968

    This algorithm has a complexity of :math:`O(E)`, where :math:`E` is the
    number of edges in the network.

    Examples
    --------
    .. testsetup:: mcmc

       gt.seed_rng(42)
       np.random.seed(42)

    .. doctest:: mcmc

       >>> g = gt.collection.data["polbooks"]
       >>> state = gt.BlockState(g, B=3, deg_corr=True)
       >>> pv = None
       >>> for i in range(1000):        # remove part of the transient
       ...     ds, nmoves = gt.mcmc_sweep(state)
       >>> for i in range(1000):
       ...     ds, nmoves = gt.mcmc_sweep(state)
       ...     pv = gt.collect_vertex_marginals(state, pv)
       >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.pdf")
       <...>

    .. testcleanup:: mcmc

       gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie", vertex_pie_fractions=pv, output="polbooks_blocks_soft.png")

    .. figure:: polbooks_blocks_soft.*
       :align: center

       "Soft" block partition of a political books network with :math:`B=3`.

     References
    ----------

    .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey,
       Samuel Leinhardt, "Stochastic blockmodels: First steps",
969
970
       Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.,
       :doi:`10.1016/0378-8733(83)90021-7`
971
972
    .. [faust-blockmodels-1992] Katherine Faust, and Stanley
       Wasserman. "Blockmodels: Interpretation and Evaluation." Social Networks
973
       14, no. 1-2 (1992): 5-61. :doi:`10.1016/0378-8733(92)90013-W`
974
975
976
977
    .. [karrer-stochastic-2011] Brian Karrer, and M. E. J. Newman. "Stochastic
       Blockmodels and Community Structure in Networks." Physical Review E 83,
       no. 1 (2011): 016107. :doi:`10.1103/PhysRevE.83.016107`.
    .. [peixoto-entropy-2012] Tiago P. Peixoto "Entropy of Stochastic Blockmodel
978
979
980
981
982
       Ensembles." Physical Review E 85, no. 5 (2012): 056122.
       :doi:`10.1103/PhysRevE.85.056122`, :arxiv:`1112.6028`.
    .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module
       inference in large networks", Phys. Rev. Lett. 110, 148701 (2013),
       :doi:`10.1103/PhysRevLett.110.148701`, :arxiv:`1212.4794`.
983
    .. [peixoto-efficient-2014] Tiago P. Peixoto, "Efficient Monte Carlo and greedy
984
985
986
       heuristic for the inference of stochastic block models", Phys. Rev. E 89,
       012804 (2014), :doi:`10.1103/PhysRevE.89.012804`, :arxiv:`1310.4378`.
    .. [peixoto-model-2015] Tiago P. Peixoto, "Model selection and hypothesis
987
       testing for large-scale network models with overlapping groups",
988
       Phys. Rev. X 5, 011033 (2015), :doi:`10.1103/PhysRevX.5.011033`,
989
       :arxiv:`1409.3059`.
990
991
    """

992
993
994
995
996
    nmerges = kwargs.get("nmerges", 0)
    merge_map = kwargs.get("merge_map", None)
    coherent_merge = kwargs.get("coherent_merge", False)
    edges_dl = kwargs.get("edges_dl", False)

997
    if state.B == 1:
998
999
        return 0., 0

1000
    if vertices is not None:
For faster browsing, not all history is shown. View entire blame