__init__.py 25.4 KB
Newer Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
#! /usr/bin/env python
# graph_tool.py -- a general graph manipulation python module
#
# Copyright (C) 2007 Tiago de Paula Peixoto <tiago@forked.de>
#
# 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/>.

19
"""
20
``graph_tool.generation`` - Random graph generation
21
---------------------------------------------------
22
23
24
25
26
27
28
29
30
31
32
33

Summary
+++++++

.. autosummary::
   :nosignatures:

   random_graph
   random_rewire
   predecessor_tree
   line_graph
   graph_union
34
   triangulation
35
36
37

Contents
++++++++
38
39
"""

Tiago Peixoto's avatar
Tiago Peixoto committed
40
41
from .. dl_import import dl_import
dl_import("import libgraph_tool_generation")
42

43
from .. core import Graph, _check_prop_scalar, _prop, _limit_args
Tiago Peixoto's avatar
Tiago Peixoto committed
44
from .. stats import label_parallel_edges, label_self_loops
45
import sys, numpy, numpy.random
46

Tiago Peixoto's avatar
Tiago Peixoto committed
47
__all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph",
48
           "graph_union", "triangulation"]
49
50

def random_graph(N, deg_sampler, deg_corr=None, directed=True,
Tiago Peixoto's avatar
Tiago Peixoto committed
51
52
                 parallel_edges=False, self_loops=False, random=True,
                 verbose=False):
Tiago Peixoto's avatar
Tiago Peixoto committed
53
54
55
56
57
58
59
60
61
62
63
64
65
66
    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.
    deg_corr : function (optional, default: None)
Tiago Peixoto's avatar
Tiago Peixoto committed
67
        A function which gives the degree correlation of the graph. It should be
Tiago Peixoto's avatar
Tiago Peixoto committed
68
69
70
71
72
73
74
        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.
    directed : bool (optional, default: True)
        Whether the generated graph should be directed.
75
    parallel_edges : bool (optional, default: False)
Tiago Peixoto's avatar
Tiago Peixoto committed
76
77
78
        If True, parallel edges are allowed.
    self_loops : bool (optional, default: False)
        If True, self-loops are allowed.
Tiago Peixoto's avatar
Tiago Peixoto committed
79
80
    random : bool (optional, default: True)
        If True, the returned graph is randomized.
81
82
    verbose : bool (optional, default: False)
        If True, verbose information is displayed.
Tiago Peixoto's avatar
Tiago Peixoto committed
83
84
85

    Returns
    -------
86
    random_graph : :class:`~graph_tool.Graph`
Tiago Peixoto's avatar
Tiago Peixoto committed
87
88
89
90
91
92
93
94
        The generated graph.

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

    Notes
    -----
Tiago Peixoto's avatar
Tiago Peixoto committed
95
96
97
98
99
100
101
102
103
    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
    the :func:`~graph_tool.generation.random_rewire` function.

    The complexity is :math:`O(V+E)` if parallel edges are allowed, and
    :math:`O(V+E \times \log N_k)` if parallel edges are not allowed, where
    :math:`N_k < V` is the number of different degrees sampled (or in,out-degree
    pairs).
Tiago Peixoto's avatar
Tiago Peixoto committed
104

Tiago Peixoto's avatar
Tiago Peixoto committed
105
106
107
    References
    ----------
    [deg-sequence] http://en.wikipedia.org/wiki/Degree_%28graph_theory%29#Degree_sequence
Tiago Peixoto's avatar
Tiago Peixoto committed
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137

    Examples
    --------

    >>> from numpy.random import randint, random, seed, poisson
    >>> from pylab import *
    >>> seed(42)

    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),
    ...                     lambda i,k: 1.0/(1+abs(i-k)), directed=False)
    >>> gt.scalar_assortativity(g, "out")
Tiago Peixoto's avatar
Tiago Peixoto committed
138
    (0.63243885897121965, 0.011153551018567562)
Tiago Peixoto's avatar
Tiago Peixoto committed
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165

    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")
    >>> imshow(hist[0], interpolation="nearest")
    <...>
    >>> colorbar()
    <...>
166
    >>> xlabel("in-degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
167
    <...>
168
    >>> ylabel("out-degree")
Tiago Peixoto's avatar
Tiago Peixoto committed
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
    <...>
    >>> savefig("combined-deg-hist.png")

    .. figure:: combined-deg-hist.png
        :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)),
    ...                                     lambda a,b: (p.pmf(a[0],b[1])*
    ...                                                  p.pmf(a[1],20-b[0])))

    Lets plot the average degree correlations to check.

196
197
198
199
    >>> figure(figsize=(6,3))
    <...>
    >>> axes([0.1,0.15,0.63,0.8])
    <...>
Tiago Peixoto's avatar
Tiago Peixoto committed
200
    >>> corr = gt.avg_neighbour_corr(g, "in", "in")
201
202
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...         label=r"$\left<\text{in}\right>$ vs in")
Tiago Peixoto's avatar
Tiago Peixoto committed
203
204
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
205
206
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...         label=r"$\left<\text{out}\right>$ vs in")
Tiago Peixoto's avatar
Tiago Peixoto committed
207
208
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
209
210
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{in}\right>$ vs out")
Tiago Peixoto's avatar
Tiago Peixoto committed
211
212
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
213
214
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{out}\right>$ vs out")
Tiago Peixoto's avatar
Tiago Peixoto committed
215
    (...)
216
    >>> legend(loc=(1.05,0.5))
Tiago Peixoto's avatar
Tiago Peixoto committed
217
218
219
220
221
222
223
224
225
226
227
228
    <...>
    >>> xlabel("source degree")
    <...>
    >>> ylabel("average target degree")
    <...>
    >>> savefig("deg-corr-dir.png")

    .. figure:: deg-corr-dir.png
        :align: center

        Average nearest neighbour correlations.
    """
229
    seed = numpy.random.randint(0, sys.maxint)
230
231
232
233
234
    g = Graph()
    if deg_corr == None:
        uncorrelated = True
    else:
        uncorrelated = False
235
236
    libgraph_tool_generation.gen_random_graph(g._Graph__graph, N, deg_sampler,
                                              uncorrelated, not parallel_edges,
237
238
239
                                              not self_loops, not directed,
                                              seed, verbose)
    g.set_directed(directed)
Tiago Peixoto's avatar
Tiago Peixoto committed
240
241
    if random:
        random_rewire(g, parallel_edges = parallel_edges,
242
                      self_loops = self_loops, verbose = verbose)
Tiago Peixoto's avatar
Tiago Peixoto committed
243
244
245
246
        if deg_corr != None:
            random_rewire(g, strat = "probabilistic",
                          parallel_edges = parallel_edges, deg_corr = deg_corr,
                          self_loops = self_loops, verbose = verbose)
247
    return g
248

249
@_limit_args({"strat":["erdos", "correlated", "uncorrelated", "probabilistic"]})
250
def random_rewire(g, strat= "uncorrelated", parallel_edges = False,
251
                  self_loops = False, deg_corr = None, verbose = False):
252
    r"""
253
254
    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
255
    randomly placed. If `strat` = "correlated", the degree correlations are
256
    also maintained: The new source and target of each edge both have the same
Tiago Peixoto's avatar
Tiago Peixoto committed
257
258
    in and out-degree. If `strat` = "probabilistic", than edges are rewired
    according to the degree correlation given by the parameter `deg_corr`.
259
260
261

    Parameters
    ----------
262
    g : :class:`~graph_tool.Graph`
263
264
        Graph to be shuffled. The graph will be modified.
    strat : string (optional, default: "uncorrelated")
265
266
267
268
        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
        new source and target of each edge both have the same in and out-degree.
269
270
        If `strat` == "probabilistic", than edges are rewired according to the
        degree correlation given by the parameter `deg_corr`.
271
272
273
274
    parallel : bool (optional, default: False)
        If True, parallel edges are allowed.
    self_loops : bool (optional, default: False)
        If True, self-loops are allowed.
275
276
277
278
279
280
281
282
283
284
    deg_corr : function (optional, default: None)
        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,
        unless `strat` = "probabilistic".
    verbose : bool (optional, default: False)
        If True, verbose information is displayed.
285
286
287
288
289
290
291

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

    Notes
    -----
Tiago Peixoto's avatar
Tiago Peixoto committed
292
293
294
295
296
297
298
299
300
    This algorithm iterates through all the edges in the network and tries to
    swap its target our edge with another edge.

    .. note::
        If `parallel_edges` = False, parallel edges are not placed during
        rewiring. In this case, for some special graphs it may be necessary to
        call the function more than once to obtain a graph which corresponds to
        a uniform sample from the ensemble. But typically, if the graph is
        sufficiently large, a single call should be enough.
301
302

    Each edge gets swapped at least once, so the overall complexity is
Tiago Peixoto's avatar
Tiago Peixoto committed
303
304
305
306
    :math:`O(E)`. If `strat` = "probabilistic" the complexity is
    :math:`O(E\log N_k)`,  where :math:`N_k < V` is the number of different
    degrees (or in,out-degree pairs).

307
308
309
310
311
312

    Examples
    --------

    Some small graphs for visualization.

313
    >>> from numpy.random import random, seed
314
315
    >>> from pylab import *
    >>> seed(42)
316
    >>> g, pos = gt.triangulation(random((1000,2)))
317
    >>> gt.graph_draw(g, layout="arf", output="rewire_orig.png", size=(6,6))
318
    <...>
319
    >>> gt.random_rewire(g, "correlated")
320
    >>> gt.graph_draw(g, layout="arf", output="rewire_corr.png", size=(6,6))
321
    <...>
322
    >>> gt.random_rewire(g)
323
    >>> gt.graph_draw(g, layout="arf", output="rewire_uncorr.png", size=(6,6))
324
    <...>
325
326
327
    >>> gt.random_rewire(g, "erdos")
    >>> gt.graph_draw(g, layout="arf", output="rewire_erdos.png", size=(6,6))
    <...>
328

329
    Some `ridiculograms <http://www.youtube.com/watch?v=YS-asmU3p_4>`_ :
330

331
332
333
    .. image:: rewire_orig.png
    .. image:: rewire_corr.png
    .. image:: rewire_uncorr.png
334
    .. image:: rewire_erdos.png
335

336
337
338
    *From left to right:* Original graph; Shuffled graph, with degree
    correlations; Shuffled graph, without degree correlations; Shuffled graph,
    with random degrees.
339
340
341

    We can try some larger graphs to get better statistics.

342
343
    >>> figure()
    <...>
344
    >>> g = gt.random_graph(30000, lambda: sample_k(20),
345
346
    ...                     lambda i,j: exp(abs(i-j)), directed=False)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
347
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="original")
348
349
350
    (...)
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
351
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="*", label="correlated")
352
353
354
    (...)
    >>> gt.random_rewire(g)
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
355
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="uncorrelated")
356
    (...)
357
358
359
360
    >>> gt.random_rewire(g, "erdos")
    >>> corr = gt.avg_neighbour_corr(g, "out", "out")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="Erdos")
    (...)
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
    >>> xlabel("$k$")
    <...>
    >>> ylabel(r"$\left<k_{nn}\right>$")
    <...>
    >>> legend(loc="best")
    <...>
    >>> savefig("shuffled-stats.png")

    .. figure:: shuffled-stats.png
        :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)),
Tiago Peixoto's avatar
Tiago Peixoto committed
381
    ...                     lambda a,b: (p.pmf(a[0],b[1])*p.pmf(a[1],20-b[0])))
382
383
384
385
    >>> figure(figsize=(6,3))
    <...>
    >>> axes([0.1,0.15,0.6,0.8])
    <...>
386
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
387
388
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{o}\right>$ vs i")
389
390
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
391
392
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
    ...          label=r"$\left<\text{i}\right>$ vs o")
393
394
395
396
    (...)
    >>> gt.random_rewire(g, "correlated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
397
    ...          label=r"$\left<\text{o}\right>$ vs i, corr.")
398
399
400
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
401
    ...          label=r"$\left<\text{i}\right>$ vs o, corr.")
402
403
404
405
    (...)
    >>> gt.random_rewire(g, "uncorrelated")
    >>> corr = gt.avg_neighbour_corr(g, "in", "out")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
406
    ...          label=r"$\left<\text{o}\right>$ vs i, uncorr.")
407
408
409
    (...)
    >>> corr = gt.avg_neighbour_corr(g, "out", "in")
    >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",
410
    ...          label=r"$\left<\text{i}\right>$ vs o, uncorr.")
411
    (...)
412
    >>> legend(loc=(1.05,0.45))
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
    <...>
    >>> xlabel("source degree")
    <...>
    >>> ylabel("average target degree")
    <...>
    >>> savefig("shuffled-deg-corr-dir.png")

    .. figure:: shuffled-deg-corr-dir.png
        :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.
    """

428
    seed = numpy.random.randint(0, sys.maxint)
429

Tiago Peixoto's avatar
Tiago Peixoto committed
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
    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!")

    if deg_corr != None and  not g.is_directed():
        corr = lambda i,j: deg_corr(i[1], j[1])
445
446
447
    else:
        corr = deg_corr

Tiago Peixoto's avatar
Tiago Peixoto committed
448
449
    if corr == None:
        g.stash_filter(reversed=True)
450
451
    try:
        libgraph_tool_generation.random_rewire(g._Graph__graph, strat,
452
453
                                               self_loops, parallel_edges,
                                               corr, seed, verbose)
454
    finally:
Tiago Peixoto's avatar
Tiago Peixoto committed
455
456
        if corr == None:
            g.pop_filter(reversed=True)
Tiago Peixoto's avatar
Tiago Peixoto committed
457
458
459
460
461
462
463
464
465
466
467

def predecessor_tree(g, pred_map):
    """Return a graph from a list of predecessors given by
    the 'pred_map' vertex property."""

    _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
468
469

def line_graph(g):
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
    """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
    """
490
491
492
493
494
495
496
497
    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
498
499

def graph_union(g1, g2, props=[], include=False):
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
    """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.
    props : list of tuples of :class:`~graph_tool.PropertyMap` (optional, default: [])
       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.
    include : bool (optional, default: False)
       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.
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544

    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)
    >>> gt.graph_draw(g, layout="arf", size=(8,8), output="graph_original.png")
    <...>
    >>> gt.graph_draw(ug, layout="arf", size=(8,8), output="graph_union.png")
    <...>
    >>> gt.graph_draw(uug, layout="arf", size=(8,8), output="graph_union2.png")
    <...>

    .. image:: graph_original.png
    .. image:: graph_union.png
    .. image:: graph_union2.png

545
    """
Tiago Peixoto's avatar
Tiago Peixoto committed
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
    if not include:
        g1 = Graph(g1)
    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,
                                                          g2._Graph__graph)
        for p in props:
            p1, p2 = p
            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
584
585

@_limit_args({"type":["simple", "delaunay"]})
586
def triangulation(points, type="simple", periodic=False):
587
588
589
590
591
592
593
594
595
596
    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).
    type : string (optional, default: 'simple')
        Type of triangulation. May be either 'simple' or 'delaunay'.
597
598
599
    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.
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614

    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
615
    A triangulation [cgal-triang]_ is a division of the convex hull of a point
616
    set into triangles, using only that set as triangle vertices.
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635

    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)
636
    >>> points = random((500,2))*4
637
    >>> g, pos = gt.triangulation(points)
638
639
640
641
642
643
644
645
646
    >>> 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
    >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0],
    ...               eprops={"penwidth":b[1]}, output="triang.png")
647
648
    <...>
    >>> g, pos = gt.triangulation(points, type="delaunay")
649
650
651
652
653
654
655
656
    >>> 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
    >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0],
    ...               eprops={"penwidth":b[1]}, output="triang-delaunay.png")
657
658
659
660
661
662
663
    <...>

    2D triangulation of random points:

    .. image:: triang.png
    .. image:: triang-delaunay.png

664
665
666
    *Left:* Simple triangulation. *Right:* Delaunay triangulation. The vertex
    colors and the edge thickness correspond to the weighted betweenness
    centrality.
667
668
669

    References
    ----------
Tiago Peixoto's avatar
Tiago Peixoto committed
670
    .. [cgal-triang] http://www.cgal.org/Manual/last/doc_html/cgal_manual/Triangulation_3/Chapter_main.html
671
672
673
674
675
676
677
678
679
680
681
682
683
684

    """

    if points.shape[1] not in [2,3]:
        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,
685
                                           _prop("v", g, pos), type, periodic)
686
687
    return g, pos