__init__.py 47 KB
Newer Older
1
#! /usr/bin/env python
2
# -*- coding: utf-8 -*-
3
#
4
5
# graph_tool -- a general graph manipulation python module
#
Tiago Peixoto's avatar
Tiago Peixoto committed
6
# Copyright (C) 2007-2012 Tiago de Paula Peixoto <tiago@skewed.de>
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
``graph_tool.generation`` - Random graph generation
23
---------------------------------------------------
24
25
26
27
28
29
30
31
32
33
34
35

Summary
+++++++

.. autosummary::
   :nosignatures:

   random_graph
   random_rewire
   predecessor_tree
   line_graph
   graph_union
36
   triangulation
37
38
   lattice
   geometric_graph
39
   price_network
40
41
42

Contents
++++++++
43
44
"""

Tiago Peixoto's avatar
Tiago Peixoto committed
45
46
from .. dl_import import dl_import
dl_import("import libgraph_tool_generation")
47

48
from .. import Graph, GraphView, _check_prop_scalar, _prop, _limit_args, _gt_type
Tiago Peixoto's avatar
Tiago Peixoto committed
49
from .. stats import label_parallel_edges, label_self_loops
50
51
import inspect
import types
52
import sys, numpy, numpy.random
53

Tiago Peixoto's avatar
Tiago Peixoto committed
54
__all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph",
55
56
           "graph_union", "triangulation", "lattice", "geometric_graph",
           "price_network"]
57

Tiago Peixoto's avatar
Tiago Peixoto committed
58

59
def random_graph(N, deg_sampler, deg_corr=None, cache_probs=True, directed=True,
60
                 parallel_edges=False, self_loops=False, blockmodel=None,
61
                 block_type="int", degree_block=False,
62
                 random=True, mix_time=10, verbose=False):
Tiago Peixoto's avatar
Tiago Peixoto committed
63
64
65
66
67
68
69
70
71
72
73
74
75
    r"""
    Generate a random graph, with a given degree distribution and correlation.

    Parameters
    ----------
    N : int
        Number of vertices in the graph.
    deg_sampler : function
        A degree sampler function which is called without arguments, and returns
        a tuple of ints representing the in and out-degree of a given vertex (or
        a single int for undirected graphs, representing the out-degree). This
        function is called once per vertex, but may be called more times, if the
        degree sequence cannot be used to build a graph.
76

77
78
79
80
81
        Optionally, you can also pass a function which receives one or two
        arguments: If ``blockmodel == None``, the single argument passed will
        be the index of the vertex which will receive the degree.
        If ``blockmodel != None``, the first value passed will be the vertex
        index, and the second will be the block value of the vertex.
82
    deg_corr : function (optional, default: ``None``)
Tiago Peixoto's avatar
Tiago Peixoto committed
83
        A function which gives the degree correlation of the graph. It should be
Tiago Peixoto's avatar
Tiago Peixoto committed
84
85
86
87
88
        callable with two parameters: the in,out-degree pair of the source
        vertex an edge, and the in,out-degree pair of the target of the same
        edge (for undirected graphs, both parameters are single values). The
        function should return a number proportional to the probability of such
        an edge existing in the generated graph.
89
90
91

        If ``blockmodel != None``, the value passed to the function will be the
        block value of the respective vertices, not the in/out-degree pairs.
92
93
94
95
96
97
98
    cache_probs : bool (optional, default: ``True``)
        If ``True``, the probabilities returned by the ``deg_corr`` parameter
        will be cached internally. This is crucial for good performance, since
        in this case the supplied python function is called only a few times,
        and not at every attempted edge rewire move. However, in the case were
        the different parameter combinations to the probability function is very
        large, the memory requirements to keep the cache may be very large.
99
    directed : bool (optional, default: ``True``)
Tiago Peixoto's avatar
Tiago Peixoto committed
100
        Whether the generated graph should be directed.
101
102
103
104
105
106
107
108
109
110
111
112
113
    parallel_edges : bool (optional, default: ``False``)
        If ``True``, parallel edges are allowed.
    self_loops : bool (optional, default: ``False``)
        If ``True``, self-loops are allowed.
    blockmodel : list or :class:`~numpy.ndarray` or function (optional, default: ``None``)
        If supplied, the graph will be sampled from a blockmodel ensemble. If
        the value is a list or a :class:`~numpy.ndarray`, it must have
        ``len(block_model) == N``, and the values will define to which block
        each vertex belongs.

        If this value is a function, it will be used to sample the block
        types. It must be callable either with no arguments or with a single
        argument which will be the vertex index. In either case it must return
114
115
116
117
118
119
120
121
        a type compatible with the ``block_type`` parameter.
    block_type : string (optional, default: ``"int"``)
        Value type of block labels. Valid only if ``blockmodel != None``.
    degree_block : bool (optional, default: ``False``)
        If ``True``, the degree of each vertex will be appended to block labels
        when constructing the blockmodel, such that the resulting block type
        will be a pair :math:`(r, k)`, where :math:`r` is the original block
        label.
122
123
124
125
    random : bool (optional, default: ``True``)
        If ``True``, the returned graph is randomized. Otherwise a deterministic
        placement of the edges will be used.
    mix_time : int (optional, default: ``10``)
126
127
128
        Number of edge sweeps to perform in order to mix the graph. This value
        is ignored if ``parallel_edges == self_loops == True`` and
        ``strat != "probabilistic"``.
129
130
    verbose : bool (optional, default: ``False``)
        If ``True``, verbose information is displayed.
Tiago Peixoto's avatar
Tiago Peixoto committed
131
132
133

    Returns
    -------
134
    random_graph : :class:`~graph_tool.Graph`
Tiago Peixoto's avatar
Tiago Peixoto committed
135
        The generated graph.
136
137
138
    blocks : :class:`~graph_tool.PropertyMap`
        A vertex property map with the block values. This is only returned if
        ``blockmodel != None``.
Tiago Peixoto's avatar
Tiago Peixoto committed
139
140
141
142
143
144
145

    See Also
    --------
    random_rewire: in place graph shuffling

    Notes
    -----
Tiago Peixoto's avatar
Tiago Peixoto committed
146
147
148
    The algorithm makes sure the degree sequence is graphical (i.e. realizable)
    and keeps re-sampling the degrees if is not. With a valid degree sequence,
    the edges are placed deterministically, and later the graph is shuffled with
149
150
    the :func:`~graph_tool.generation.random_rewire` function, with the
    ``mix_time`` parameter passed as ``n_iter``.
Tiago Peixoto's avatar
Tiago Peixoto committed
151

152
    The complexity is :math:`O(V + E)` if parallel edges are allowed, and
153
    :math:`O(V + E \times\text{mix-time})` if parallel edges are not allowed.
154
155
156
157
158
159
160
161
162
163
164
165
166


    .. note ::

        If ``parallel_edges == False`` this algorithm only guarantees that the
        returned graph will be a random sample from the desired ensemble if
        ``mix_time`` is sufficiently large. The algorithm implements an
        efficient Markov chain based on edge swaps, with a mixing time which
        depends on the degree distribution and correlations desired. If degree
        correlations are provided, the mixing time tends to be larger.

        If ``strat == "probabilistic"``, the Markov chain still needs to be
        mixed, even if parallel edges and self-loops are allowed. In this case
167
168
169
        the Markov chain is implemented using the Metropolis-Hastings
        [metropolis-equations-1953]_ [hastings-monte-carlo-1970]_
        acceptance/rejection algorithm.
Tiago Peixoto's avatar
Tiago Peixoto committed
170
171
172
173
174

    Examples
    --------
    >>> from numpy.random import randint, random, seed, poisson
    >>> from pylab import *
175
    >>> seed(43)
Tiago Peixoto's avatar
Tiago Peixoto committed
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196

    This is a degree sampler which uses rejection sampling to sample from the
    distribution :math:`P(k)\propto 1/k`, up to a maximum.

    >>> def sample_k(max):
    ...     accept = False
    ...     while not accept:
    ...         k = randint(1,max+1)
    ...         accept = random() < 1.0/k
    ...     return k
    ...

    The following generates a random undirected graph with degree distribution
    :math:`P(k)\propto 1/k` (with k_max=40) and an *assortative* degree
    correlation of the form:

    .. math::

        P(i,k) \propto \frac{1}{1+|i-k|}

    >>> g = gt.random_graph(1000, lambda: sample_k(40),
197
198
    ...                     lambda i, k: 1.0 / (1 + abs(i - k)), directed=False,
    ...                     mix_time=100)
Tiago Peixoto's avatar
Tiago Peixoto committed
199
    >>> gt.scalar_assortativity(g, "out")
Tiago Peixoto's avatar
Tiago Peixoto committed
200
    (0.6435658697163692, 0.010420519538259333)
Tiago Peixoto's avatar
Tiago Peixoto committed
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223

    The following samples an in,out-degree pair from the joint distribution:

    .. math::

        p(j,k) = \frac{1}{2}\frac{e^{-m_1}m_1^j}{j!}\frac{e^{-m_1}m_1^k}{k!} +
                 \frac{1}{2}\frac{e^{-m_2}m_2^j}{j!}\frac{e^{-m_2}m_2^k}{k!}

    with :math:`m_1 = 4` and :math:`m_2 = 20`.

    >>> def deg_sample():
    ...    if random() > 0.5:
    ...        return poisson(4), poisson(4)
    ...    else:
    ...        return poisson(20), poisson(20)
    ...

    The following generates a random directed graph with this distribution, and
    plots the combined degree correlation.

    >>> g = gt.random_graph(20000, deg_sample)
    >>>
    >>> hist = gt.combined_corr_hist(g, "in", "out")
224
225
    >>>
    >>> clf()
Tiago Peixoto's avatar
Tiago Peixoto committed
226
227
228
229
    >>> imshow(hist[0], interpolation="nearest")
    <...>
    >>> colorbar()
    <...>
230
    >>> xlabel("in-degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
231
    <...>
232
    >>> ylabel("out-degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
233
    <...>
234
    >>> savefig("combined-deg-hist.pdf")
Tiago Peixoto's avatar
Tiago Peixoto committed
235

236
    .. figure:: combined-deg-hist.*
Tiago Peixoto's avatar
Tiago Peixoto committed
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
        :align: center

        Combined degree histogram.

    A correlated directed graph can be build as follows. Consider the following
    degree correlation:

    .. math::

         P(j',k'|j,k)=\frac{e^{-k}k^{j'}}{j'!}
         \frac{e^{-(20-j)}(20-j)^{k'}}{k'!}

    i.e., the in->out correlation is "disassortative", the out->in correlation
    is "assortative", and everything else is uncorrelated.
    We will use a flat degree distribution in the range [1,20).

    >>> p = scipy.stats.poisson
    >>> g = gt.random_graph(20000, lambda: (sample_k(19), sample_k(19)),
255
256
257
    ...                     lambda a,b: (p.pmf(a[0], b[1]) *
    ...                                  p.pmf(a[1], 20 - b[0])),
    ...                     mix_time=100)
Tiago Peixoto's avatar
Tiago Peixoto committed
258
259
260

    Lets plot the average degree correlations to check.

261
    >>> clf()
262
263
    >>> axes([0.1,0.15,0.63,0.8])
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
264
    >>> corr = gt.avg_neighbour_corr(g, "in", "in")
265
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
266
    ...         label=r"$\left<\text{in}\right>$ vs in")
267
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
268
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
269
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
270
    ...         label=r"$\left<\text{out}\right>$ vs in")
271
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
272
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
273
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
274
    ...          label=r"$\left<\text{in}\right>$ vs out")
275
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
276
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
277
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
278
    ...          label=r"$\left<\text{out}\right>$ vs out")
Tiago Peixoto's avatar
Tiago Peixoto committed
279
    <...>
280
281
282
    >>> legend(bbox_to_anchor=(1.01, 0.5), loc="center left", borderaxespad=0.)
    <...>
    >>> xlabel("Source degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
283
    <...>
284
    >>> ylabel("Average target degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
285
    <...>
286
    >>> savefig("deg-corr-dir.pdf")
Tiago Peixoto's avatar
Tiago Peixoto committed
287

288
    .. figure:: deg-corr-dir.*
Tiago Peixoto's avatar
Tiago Peixoto committed
289
290
291
        :align: center

        Average nearest neighbour correlations.
292
293
294
295
296


    **Blockmodels**


297
298
299
    The following example shows how a stochastic blockmodel
    [holland-stochastic-1983]_ [karrer-stochastic-2011]_ can be generated. We
    will consider a system of 10 blocks, which form communities. The connection
300
301
302
303
304
305
306
307
308
309
310
311
312
    probability will be given by

    >>> def corr(a, b):
    ...    if a == b:
    ...        return 0.999
    ...    else:
    ...        return 0.001

    The blockmodel can be generated as follows.

    >>> g, bm = gt.random_graph(1000, lambda: poisson(10), directed=False,
    ...                         blockmodel=lambda: randint(10), deg_corr=corr,
    ...                         mix_time=500)
313
    >>> gt.graph_draw(g, vertex_fill_color=bm, output="blockmodel.pdf")
314
315
    <...>

316
    .. figure:: blockmodel.*
317
318
319
320
321
322
323
324
325
326
327
328
329
330
        :align: center

        Simple blockmodel with 10 blocks.


    References
    ----------
    .. [metropolis-equations-1953]  Metropolis, N.; Rosenbluth, A.W.;
       Rosenbluth, M.N.; Teller, A.H.; Teller, E. "Equations of State
       Calculations by Fast Computing Machines". Journal of Chemical Physics 21
       (6): 1087–1092 (1953). :doi:`10.1063/1.1699114`
    .. [hastings-monte-carlo-1970] Hastings, W.K. "Monte Carlo Sampling Methods
       Using Markov Chains and Their Applications". Biometrika 57 (1): 97–109 (1970).
       :doi:`10.1093/biomet/57.1.97`
331
332
333
334
335
336
    .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, and
       Samuel Leinhardt, "Stochastic blockmodels: First steps," Social Networks
       5, no. 2: 109-13 (1983) :doi:`10.1016/0378-8733(83)90021-7`
    .. [karrer-stochastic-2011] Brian Karrer and M. E. J. Newman, "Stochastic
       blockmodels and community structure in networks," Physical Review E 83,
       no. 1: 016107 (2011) :doi:`10.1103/PhysRevE.83.016107` :arxiv:`1008.3926`
Tiago Peixoto's avatar
Tiago Peixoto committed
337
    """
338

339
    seed = numpy.random.randint(0, sys.maxint)
340
341
342
343
344
    g = Graph()
    if deg_corr == None:
        uncorrelated = True
    else:
        uncorrelated = False
345
346
347

    if (type(blockmodel) is types.FunctionType or
        type(blockmodel) is types.LambdaType):
348
349
        btype = block_type
        bm = []
350
351
        if len(inspect.getargspec(blockmodel)[0]) == 0:
            for i in xrange(N):
352
                bm.append(blockmodel())
353
354
        else:
            for i in xrange(N):
355
356
                bm.append(blockmodel(i))
        blockmodel = bm
Tiago Peixoto's avatar
Tiago Peixoto committed
357
    elif blockmodel is not None:
358
        btype = _gt_type(blockmodel[0])
359
360
361

    if len(inspect.getargspec(deg_sampler)[0]) > 0:
        if blockmodel is not None:
362
            sampler = lambda i: deg_sampler(i, blockmodel[i])
363
        else:
Tiago Peixoto's avatar
Tiago Peixoto committed
364
            sampler = deg_sampler
365
366
367
368
    else:
        sampler = lambda i: deg_sampler()

    libgraph_tool_generation.gen_graph(g._Graph__graph, N, sampler,
369
370
371
                                       uncorrelated, not parallel_edges,
                                       not self_loops, not directed,
                                       seed, verbose, True)
372
373
    g.set_directed(directed)

374
375
376
377
378
379
380
381
382
383
384
385
    if degree_block:
        if btype in ["object", "string"] or "vector" in btype:
            btype = "object"
        elif btype in ["int", "int32_t", "bool"]:
            btype = "vector<int32_t>"
        elif btype in ["long", "int64_t"]:
            btype = "vector<int64_t>"
        elif btype in ["double"]:
            btype = "vector<double>"
        elif btype in ["long double"]:
            btype = "vector<long double>"

386
387
388
389
    if blockmodel is not None:
        bm = g.new_vertex_property(btype)
        if btype in ["object", "string"] or "vector" in btype:
            for v in g.vertices():
390
391
392
393
394
395
396
397
                if not degree_block:
                    bm[v] = blockmodel[int(v)]
                else:
                    if g.is_directed():
                        bm[v] = (blockmodel[int(v)], v.in_degree(),
                                 v.out_degree())
                    else:
                        bm[v] = (blockmodel[int(v)], v.out_degree())
398
399
400
401
402
403
404
405
406
        else:
            try:
                bm.a = blockmodel
            except ValueError:
                bm = g.new_vertex_property("object")
                for v in g.vertices():
                    bm[v] = blockmodel[int(v)]
    else:
        bm = None
407

408
    if parallel_edges and self_loops and deg_corr is None:
409
        mix_time = 1
Tiago Peixoto's avatar
Tiago Peixoto committed
410
    if random:
411
412
        if deg_corr is not None:
            random_rewire(g, strat="probabilistic", n_iter=mix_time,
Tiago Peixoto's avatar
Tiago Peixoto committed
413
                          parallel_edges=parallel_edges, deg_corr=deg_corr,
414
415
                          cache_probs=cache_probs, self_loops=self_loops,
                          blockmodel=bm, verbose=verbose)
416
417
418
419
        else:
            random_rewire(g, parallel_edges=parallel_edges, n_iter=mix_time,
                          self_loops=self_loops, verbose=verbose)

420
421
422
423
    if bm is None:
        return g
    else:
        return g, bm
424

Tiago Peixoto's avatar
Tiago Peixoto committed
425

426
427
@_limit_args({"strat": ["erdos", "correlated", "uncorrelated",
                        "probabilistic"]})
428
429
def random_rewire(g, strat="uncorrelated", n_iter=1, edge_sweep=True,
                  parallel_edges=False, self_loops=False, deg_corr=None,
430
431
                  cache_probs=True, blockmodel=None, ret_fail=False,
                  verbose=False):
432
    r"""
433
434
435
436
437
438
439
440
    Shuffle the graph in-place.

    If ``strat != "erdos"``, the degrees (either in or out) of each vertex are
    always the same, but otherwise the edges are randomly placed. If
    ``strat == "correlated"``, the degree correlations are also maintained: The
    new source and target of each edge both have the same in and out-degree. If
    ``strat == "probabilistic"``, then edges are rewired according to the degree
    correlation given by the parameter ``deg_corr``.
441
442
443

    Parameters
    ----------
444
    g : :class:`~graph_tool.Graph`
445
        Graph to be shuffled. The graph will be modified.
446
447
448
449
    strat : string (optional, default: ``"uncorrelated"``)
        If ``strat == "erdos"``, the resulting graph will be entirely random. If
        ``strat == "uncorrelated"`` only the degrees of the vertices will be
        maintained, nothing else. If ``strat == "correlated"``, additionally the
450
        new source and target of each edge both have the same in and out-degree.
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
        If ``strat == "probabilistic"``, than edges are rewired according to the
        degree correlation given by the parameter ``deg_corr``.
    n_iter : int (optional, default: ``1``)
        Number of iterations. If ``edge_sweep == True``, each iteration
        corresponds to an entire "sweep" over all edges. Otherwise this
        corresponds to the total number of edges which are randomly chosen for a
        swap attempt (which may repeat).
    edge_sweep : bool (optional, default: ``True``)
        If ``True``, each iteration will perform an entire "sweep" over the
        edges, where each edge is visited once in random order, and a edge swap
        is attempted.
    parallel : bool (optional, default: ``False``)
        If ``True``, parallel edges are allowed.
    self_loops : bool (optional, default: ``False``)
        If ``True``, self-loops are allowed.
    deg_corr : function (optional, default: ``None``)
467
468
469
470
471
472
        A function which gives the degree correlation of the graph. It should be
        callable with two parameters: the in,out-degree pair of the source
        vertex an edge, and the in,out-degree pair of the target of the same
        edge (for undirected graphs, both parameters are single values). The
        function should return a number proportional to the probability of such
        an edge existing in the generated graph. This parameter is ignored,
473
        unless ``strat == "probabilistic"``.
474
475
476

        If ``blockmodel != None``, the value passed to the function will be the
        block value of the respective vertices, not the in/out-degree pairs.
477
478
479
480
481
482
483
    cache_probs : bool (optional, default: ``True``)
        If ``True``, the probabilities returned by the ``deg_corr`` parameter
        will be cached internally. This is crucial for good performance, since
        in this case the supplied python function is called only a few times,
        and not at every attempted edge rewire move. However, in the case were
        the different parameter combinations to the probability function is very
        large, the memory requirements to keep the cache may be very large.
484
485
486
487
    blockmodel : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
        If supplied, the graph will be rewired to conform to a blockmodel
        ensemble. The value must be a vertex property map which defines the
        block of each vertex.
488
489
490
491
492
493
494
495
496
497
498
    ret_fail : bool (optional, default: ``False``)
        If ``True``, the number of failed edge moves (due to parallel edges or
        self-loops) is returned.
    verbose : bool (optional, default: ``False``)
        If ``True``, verbose information is displayed.


    Returns
    -------
    fail_count : int
        Number of failed edge moves (due to parallel edges or self-loops).
499
500
501
502
503
504
505

    See Also
    --------
    random_graph: random graph generation

    Notes
    -----
Tiago Peixoto's avatar
Tiago Peixoto committed
506
    This algorithm iterates through all the edges in the network and tries to
507
    swap its target or source with the target or source of another edge.
Tiago Peixoto's avatar
Tiago Peixoto committed
508
509

    .. note::
510

511
512
513
514
515
516
517
518
519
520
        If ``parallel_edges = False``, parallel edges are not placed during
        rewiring. In this case, the returned graph will be a uncorrelated sample
        from the desired ensemble only if ``n_iter`` is sufficiently large. The
        algorithm implements an efficient Markov chain based on edge swaps, with
        a mixing time which depends on the degree distribution and correlations
        desired. If degree probabilistic correlations are provided, the mixing
        time tends to be larger.

        If ``strat == "probabilistic"``, the Markov chain still needs to be
        mixed, even if parallel edges and self-loops are allowed. In this case
521
522
523
        the Markov chain is implemented using the Metropolis-Hastings
        [metropolis-equations-1953]_ [hastings-monte-carlo-1970]_
        acceptance/rejection algorithm.
524

Tiago Peixoto's avatar
Tiago Peixoto committed
525

526
    Each edge is tentatively swapped once per iteration, so the overall
527
528
    complexity is :math:`O(V + E \times \text{n-iter})`. If ``edge_sweep ==
    False``, the complexity becomes :math:`O(V + E + \text{n-iter})`.
529

530
531
532
533
534
    Examples
    --------

    Some small graphs for visualization.

535
    >>> from numpy.random import random, seed
536
    >>> from pylab import *
537
    >>> seed(43)
538
    >>> g, pos = gt.triangulation(random((1000,2)))
539
540
    >>> pos = gt.arf_layout(g)
    >>> gt.graph_draw(g, pos=pos, output="rewire_orig.pdf", output_size=(200, 200))
541
    <...>
542
    >>> gt.random_rewire(g, "correlated")
543
544
    >>> pos = gt.arf_layout(g)
    >>> gt.graph_draw(g, pos=pos, output="rewire_corr.pdf", output_size=(200, 200))
545
    <...>
546
    >>> gt.random_rewire(g)
547
548
    >>> pos = gt.arf_layout(g)
    >>> gt.graph_draw(g, pos=pos, output="rewire_uncorr.pdf", output_size=(200, 200))
549
    <...>
550
    >>> gt.random_rewire(g, "erdos")
551
552
    >>> pos = gt.arf_layout(g)
    >>> gt.graph_draw(g, pos=pos, output="rewire_erdos.pdf", output_size=(200, 200))
553
    <...>
554

555
    Some `ridiculograms <http://www.youtube.com/watch?v=YS-asmU3p_4>`_ :
556

557
558
559
560
    .. image:: rewire_orig.*
    .. image:: rewire_corr.*
    .. image:: rewire_uncorr.*
    .. image:: rewire_erdos.*
561

562
563
    **From left to right**: Original graph; Shuffled graph, with degree correlations;
    Shuffled graph, without degree correlations; Shuffled graph, with random degrees.
564

565
    We can try with larger graphs to get better statistics, as follows.
566

567
568
    >>> figure()
    <...>
569
    >>> g = gt.random_graph(30000, lambda: sample_k(20),
570
571
    ...                     lambda i, j: exp(abs(i-j)), directed=False,
    ...                     mix_time=100)
572
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
573
574
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="Original")
    <...>
575
576
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
577
578
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="*", label="Correlated")
    <...>
579
580
    >>> gt.random_rewire(g)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
581
582
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="Uncorrelated")
    <...>
583
584
    >>> gt.random_rewire(g, "erdos")
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
585
586
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label=r"Erd\H{o}s")
    <...>
587
588
589
590
591
592
    >>> xlabel("$k$")
    <...>
    >>> ylabel(r"$\left<k_{nn}\right>$")
    <...>
    >>> legend(loc="best")
    <...>
593
    >>> savefig("shuffled-stats.pdf")
594

595
    .. figure:: shuffled-stats.*
596
597
598
599
600
601
602
603
604
605
606
        :align: center

        Average degree correlations for the different shuffled and non-shuffled
        graphs. The shuffled graph with correlations displays exactly the same
        correlation as the original graph.

    Now let's do it for a directed graph. See
    :func:`~graph_tool.generation.random_graph` for more details.

    >>> p = scipy.stats.poisson
    >>> g = gt.random_graph(20000, lambda: (sample_k(19), sample_k(19)),
607
608
    ...                     lambda a, b: (p.pmf(a[0], b[1]) * p.pmf(a[1], 20 - b[0])),
    ...                     mix_time=100)
609
    >>> figure()
610
611
612
    <...>
    >>> axes([0.1,0.15,0.6,0.8])
    <...>
613
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
614
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
615
    ...          label=r"$\left<\text{o}\right>$ vs i")
616
    <...>
617
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
618
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
619
    ...          label=r"$\left<\text{i}\right>$ vs o")
620
    <...>
621
622
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
623
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
624
    ...          label=r"$\left<\text{o}\right>$ vs i, corr.")
625
    <...>
626
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
627
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
628
    ...          label=r"$\left<\text{i}\right>$ vs o, corr.")
629
    <...>
630
631
    >>> gt.random_rewire(g, "uncorrelated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
632
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
633
    ...          label=r"$\left<\text{o}\right>$ vs i, uncorr.")
634
    <...>
635
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
636
    >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",
637
    ...          label=r"$\left<\text{i}\right>$ vs o, uncorr.")
638
    <...>
639
640
641
    >>> legend(bbox_to_anchor=(1.01, 0.5), loc="center left", borderaxespad=0.)
    <...>
    >>> xlabel("Source degree")
642
    <...>
643
    >>> ylabel("Average target degree")
644
    <...>
645
    >>> savefig("shuffled-deg-corr-dir.pdf")
646

647
    .. figure:: shuffled-deg-corr-dir.*
648
649
650
651
652
653
        :align: center

        Average degree correlations for the different shuffled and non-shuffled
        directed graphs. The shuffled graph with correlations displays exactly
        the same correlation as the original graph.

654
655
656
657
658
659
660
661
662
    References
    ----------
    .. [metropolis-equations-1953]  Metropolis, N.; Rosenbluth, A.W.;
       Rosenbluth, M.N.; Teller, A.H.; Teller, E. "Equations of State
       Calculations by Fast Computing Machines". Journal of Chemical Physics 21
       (6): 1087–1092 (1953). :doi:`10.1063/1.1699114`
    .. [hastings-monte-carlo-1970] Hastings, W.K. "Monte Carlo Sampling Methods
       Using Markov Chains and Their Applications". Biometrika 57 (1): 97–109 (1970).
       :doi:`10.1093/biomet/57.1.97`
663
664
665
666
667
668
    .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, and
       Samuel Leinhardt, "Stochastic blockmodels: First steps," Social Networks
       5, no. 2: 109-13 (1983) :doi:`10.1016/0378-8733(83)90021-7`
    .. [karrer-stochastic-2011] Brian Karrer and M. E. J. Newman, "Stochastic
       blockmodels and community structure in networks," Physical Review E 83,
       no. 1: 016107 (2011) :doi:`10.1103/PhysRevE.83.016107` :arxiv:`1008.3926`
669
670

    """
671
    seed = numpy.random.randint(0, sys.maxint)
672

Tiago Peixoto's avatar
Tiago Peixoto committed
673
674
675
676
677
678
679
680
681
682
683
684
685
    if not parallel_edges:
        p = label_parallel_edges(g)
        if p.a.max() != 0:
            raise ValueError("Parallel edge detected. Can't rewire " +
                             "graph without parallel edges if it " +
                             "already contains parallel edges!")
    if not self_loops:
        l = label_self_loops(g)
        if l.a.max() != 0:
            raise ValueError("Self-loop detected. Can't rewire graph " +
                             "without self-loops if it already contains" +
                             " self-loops!")

686
    if (deg_corr is not None and not g.is_directed()) and blockmodel is None:
Tiago Peixoto's avatar
Tiago Peixoto committed
687
        corr = lambda i, j: deg_corr(i[1], j[1])
688
689
690
    else:
        corr = deg_corr

691
692
    if strat != "probabilistic":
        g = GraphView(g, reversed=False)
693
694
    elif blockmodel is not None:
        strat = "blockmodel"
695
696
697
    pcount = libgraph_tool_generation.random_rewire(g._Graph__graph, strat,
                                                    n_iter, not edge_sweep,
                                                    self_loops, parallel_edges,
698
                                                    corr, _prop("v", g, blockmodel),
699
                                                    cache_probs,
700
                                                    seed, verbose)
701
702
    if ret_fail:
        return pcount
Tiago Peixoto's avatar
Tiago Peixoto committed
703

Tiago Peixoto's avatar
Tiago Peixoto committed
704

Tiago Peixoto's avatar
Tiago Peixoto committed
705
def predecessor_tree(g, pred_map):
Tiago Peixoto's avatar
Tiago Peixoto committed
706
    """Return a graph from a list of predecessors given by the ``pred_map`` vertex property."""
Tiago Peixoto's avatar
Tiago Peixoto committed
707
708
709
710
711
712
713

    _check_prop_scalar(pred_map, "pred_map")
    pg = Graph()
    libgraph_tool_generation.predecessor_graph(g._Graph__graph,
                                               pg._Graph__graph,
                                               _prop("v", g, pred_map))
    return pg
714

Tiago Peixoto's avatar
Tiago Peixoto committed
715

716
def line_graph(g):
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
    """Return the line graph of the given graph `g`.

    Notes
    -----
    Given an undirected graph G, its line graph L(G) is a graph such that

        * each vertex of L(G) represents an edge of G; and
        * two vertices of L(G) are adjacent if and only if their corresponding
          edges share a common endpoint ("are adjacent") in G.

    For a directed graph, the second criterion becomes:

       * Two vertices representing directed edges from u to v and from w to x in
         G are connected by an edge from uv to wx in the line digraph when v =
         w.

    References
    ----------
    .. [line-wiki] http://en.wikipedia.org/wiki/Line_graph
    """
737
738
739
740
741
742
743
744
    lg = Graph(directed=g.is_directed())

    vertex_map = lg.new_vertex_property("int64_t")

    libgraph_tool_generation.line_graph(g._Graph__graph,
                                        lg._Graph__graph,
                                        _prop("v", lg, vertex_map))
    return lg, vertex_map
Tiago Peixoto's avatar
Tiago Peixoto committed
745

Tiago Peixoto's avatar
Tiago Peixoto committed
746

747
def graph_union(g1, g2, intersection=None, props=None, include=False):
748
749
750
751
752
753
754
755
756
    """Return the union of graphs g1 and g2, composed of all edges and vertices
    of g1 and g2, without overlap.

    Parameters
    ----------
    g1 : :class:`~graph_tool.Graph`
       First graph in the union.
    g2 : :class:`~graph_tool.Graph`
       Second graph in the union.
757
758
759
760
761
    intersection : :class:`~graph_tool.PropertyMap` (optional, default: ``None``)
       Vertex property map owned by `g1` which maps each of each of its vertices
       to vertex indexes belonging to `g2`. Negative values mean no mapping
       exists, and thus both vertices in `g1` and `g2` will be present in the
       union graph.
762
    props : list of tuples of :class:`~graph_tool.PropertyMap` (optional, default: ``[]``)
763
764
765
766
       Each element in this list must be a tuple of two PropertyMap objects. The
       first element must be a property of `g1`, and the second of `g2`. The
       values of the property maps are propagated into the union graph, and
       returned.
767
    include : bool (optional, default: ``False``)
768
769
770
771
772
773
774
775
776
777
       If true, graph `g2` is inserted into `g1` which is modified. If false, a
       new graph is created, and both graphs remain unmodified.

    Returns
    -------
    ug : :class:`~graph_tool.Graph`
        The union graph
    props : list of :class:`~graph_tool.PropertyMap` objects
        List of propagated properties.  This is only returned if `props` is not
        empty.
778
779
780
781
782
783
784
785
786

    Examples
    --------

    >>> from numpy.random import random, seed
    >>> seed(42)
    >>> g = gt.triangulation(random((300,2)))[0]
    >>> ug = gt.graph_union(g, g)
    >>> uug = gt.graph_union(g, ug)
787
788
    >>> pos = gt.arf_layout(g)
    >>> gt.graph_draw(g, pos=pos, output_size=(300,300), output="graph_original.pdf")
789
    <...>
790
791
    >>> pos = gt.arf_layout(ug)
    >>> gt.graph_draw(ug, pos=pos, output_size=(300,300), output="graph_union.pdf")
792
    <...>
793
794
    >>> pos = gt.arf_layout(uug)
    >>> gt.graph_draw(uug, pos=pos, output_size=(300,300), output="graph_union2.pdf")
795
796
    <...>

797
798
799
    .. image:: graph_original.*
    .. image:: graph_union.*
    .. image:: graph_union2.*
800

801
    """
Tiago Peixoto's avatar
Tiago Peixoto committed
802
803
    if props == None:
        props = []
Tiago Peixoto's avatar
Tiago Peixoto committed
804
805
    if not include:
        g1 = Graph(g1)
806
807
808
809
810
811
812
813
    if intersection is None:
        intersection = g1.new_vertex_property("int32_t")
        intersection.a = 0
    else:
        intersection = intersection.copy("int32_t")
        intersection.a[intersection.a >= 0] += 1
        intersection.a[intersection.a < 0] = 0

Tiago Peixoto's avatar
Tiago Peixoto committed
814
815
816
817
818
819
820
821
    g1.stash_filter(directed=True)
    g1.set_directed(True)
    g2.stash_filter(directed=True)
    g2.set_directed(True)
    n_props = []

    try:
        vmap, emap = libgraph_tool_generation.graph_union(g1._Graph__graph,
822
823
824
825
                                                          g2._Graph__graph,
                                                          _prop("v", g1,
                                                                intersection))
        for p1, p2 in props:
Tiago Peixoto's avatar
Tiago Peixoto committed
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
            if not include:
                p1 = g1.copy_property(p1)
            if p2.value_type() != p1.value_type():
                p2 = g2.copy_property(p2, value_type=p1.value_type())
            if p1.key_type() == 'v':
                libgraph_tool_generation.\
                      vertex_property_union(g1._Graph__graph, g2._Graph__graph,
                                            vmap, emap,
                                            _prop(p1.key_type(), g1, p1),
                                            _prop(p2.key_type(), g2, p2))
            else:
                libgraph_tool_generation.\
                      edge_property_union(g1._Graph__graph, g2._Graph__graph,
                                          vmap, emap,
                                          _prop(p1.key_type(), g1, p1),
                                          _prop(p2.key_type(), g2, p2))
            n_props.append(p1)
    finally:
        g1.pop_filter(directed=True)
        g2.pop_filter(directed=True)

    if len(n_props) > 0:
        return g1, n_props
    else:
        return g1
851

Tiago Peixoto's avatar
Tiago Peixoto committed
852
853

@_limit_args({"type": ["simple", "delaunay"]})
854
def triangulation(points, type="simple", periodic=False):
855
856
857
858
859
860
861
862
    r"""
    Generate a 2D or 3D triangulation graph from a given point set.

    Parameters
    ----------
    points : :class:`~numpy.ndarray`
        Point set for the triangulation. It may be either a N x d array, where N
        is the number of points, and d is the space dimension (either 2 or 3).
863
    type : string (optional, default: ``'simple'``)
864
        Type of triangulation. May be either 'simple' or 'delaunay'.
865
866
867
    periodic : bool (optional, default: ``False``)
        If ``True``, periodic boundary conditions will be used. This is
        parameter is valid only for type="delaunay", and is otherwise ignored.
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882

    Returns
    -------
    triangulation_graph : :class:`~graph_tool.Graph`
        The generated graph.
    pos : :class:`~graph_tool.PropertyMap`
        Vertex property map with the Cartesian coordinates.

    See Also
    --------
    random_graph: random graph generation

    Notes
    -----

Tiago Peixoto's avatar
Tiago Peixoto committed
883
    A triangulation [cgal-triang]_ is a division of the convex hull of a point
884
    set into triangles, using only that set as triangle vertices.
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903

    In simple triangulations (`type="simple"`), the insertion of a point is done
    by locating a face that contains the point, and splitting this face into
    three new faces (the order of insertion is therefore important). If the
    point falls outside the convex hull, the triangulation is restored by
    flips. Apart from the location, insertion takes a time O(1). This bound is
    only an amortized bound for points located outside the convex hull.

    Delaunay triangulations (`type="delaunay"`) have the specific empty sphere
    property, that is, the circumscribing sphere of each cell of such a
    triangulation does not contain any other vertex of the triangulation in its
    interior. These triangulations are uniquely defined except in degenerate
    cases where five points are co-spherical. Note however that the CGAL
    implementation computes a unique triangulation even in these cases.

    Examples
    --------
    >>> from numpy.random import seed, random
    >>> seed(42)
904
    >>> points = random((500, 2)) * 4
905
    >>> g, pos = gt.triangulation(points)
906
907
908
909
910
911
912
    >>> weight = g.new_edge_property("double") # Edge weights corresponding to
    ...                                        # Euclidean distances
    >>> for e in g.edges():
    ...    weight[e] = sqrt(sum((array(pos[e.source()]) -
    ...                          array(pos[e.target()]))**2))
    >>> b = gt.betweenness(g, weight=weight)
    >>> b[1].a *= 100
913
914
    >>> gt.graph_draw(g, pos=pos, output_size=(300,300), vertex_fill_color=b[0],
    ...               edge_pen_width=b[1], output="triang.pdf")
915
916
    <...>
    >>> g, pos = gt.triangulation(points, type="delaunay")
917
918
919
920
921
922
    >>> weight = g.new_edge_property("double")
    >>> for e in g.edges():
    ...    weight[e] = sqrt(sum((array(pos[e.source()]) -
    ...                          array(pos[e.target()]))**2))
    >>> b = gt.betweenness(g, weight=weight)
    >>> b[1].a *= 120
923
924
    >>> gt.graph_draw(g, pos=pos, output_size=(300,300), vertex_fill_color=b[0],
    ...               edge_pen_width=b[1], output="triang-delaunay.pdf")
925
926
927
928
    <...>

    2D triangulation of random points:

929
930
    .. image:: triang.*
    .. image:: triang-delaunay.*
931

932
933
934
    *Left:* Simple triangulation. *Right:* Delaunay triangulation. The vertex
    colors and the edge thickness correspond to the weighted betweenness
    centrality.
935
936
937

    References
    ----------
Tiago Peixoto's avatar
Tiago Peixoto committed
938
    .. [cgal-triang] http://www.cgal.org/Manual/last/doc_html/cgal_manual/Triangulation_3/Chapter_main.html
939
940
941

    """

Tiago Peixoto's avatar
Tiago Peixoto committed
942
    if points.shape[1] not in [2, 3]:
943
944
945
946
947
948
949
950
951
952
        raise ValueError("points array must have shape N x d, with d either 2 or 3.")
    # copy points to ensure continuity and correct data type
    points = numpy.array(points, dtype='float64')
    if points.shape[1] == 2:
        npoints = numpy.zeros((points.shape[0], 3))
        npoints[:,:2] = points
        points = npoints
    g = Graph(directed=False)
    pos = g.new_vertex_property("vector<double>")
    libgraph_tool_generation.triangulation(g._Graph__graph, points,
953
                                           _prop("v", g, pos), type, periodic)
954
    return g, pos
955
956
957
958
959
960
961
962
963
964


def lattice(shape, periodic=False):
    r"""
    Generate a N-dimensional square lattice.

    Parameters
    ----------
    shape : list or :class:`~numpy.ndarray`
        List of sizes in each dimension.
965
    periodic : bool (optional, default: ``False``)
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
        If ``True``, periodic boundary conditions will be used.

    Returns
    -------
    lattice_graph : :class:`~graph_tool.Graph`
        The generated graph.

    See Also
    --------
    triangulation: 2D or 3D triangulation
    random_graph: random graph generation

    Examples
    --------
    >>> g = gt.lattice([10,10])
981
982
    >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, cooling_step=0.99, epsilon=1e-3),
    ...               output_size=(300,300), output="lattice.pdf")
983
984
    <...>
    >>> g = gt.lattice([10,20], periodic=True)
985
986
    >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, cooling_step=0.99, epsilon=1e-3, multilevel=True),
    ...               output_size=(300,300), output="lattice_periodic.pdf")
987
988
    <...>
    >>> g = gt.lattice([10,10,10])
989
990
    >>> gt.graph_draw(g, pos=gt.sfdp_layout(g, cooling_step=0.99, epsilon=1e-3, multilevel=True),
    ...               output_size=(300,300), output="lattice_3d.pdf")
991
992
    <...>

993
994
995
    .. image:: lattice.*
    .. image:: lattice_periodic.*
    .. image:: lattice_3d.*
996
997
998
999
1000

    *Left:* 10x10 2D lattice. *Middle:* 10x20 2D periodic lattice (torus).
    *Right:* 10x10x10 3D lattice.

    References
For faster browsing, not all history is shown. View entire blame