1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
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
138
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
166
167
168
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
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
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
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
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
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
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
443
444
445
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
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
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
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
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
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
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
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
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
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
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
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
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
927
928
929
930
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
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
|
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>networkx.algorithms.minors.contraction — NetworkX 3.0rc2.dev0 documentation</title>
<script data-cfasync="false">
document.documentElement.dataset.mode = localStorage.getItem("mode") || "light";
document.documentElement.dataset.theme = localStorage.getItem("theme") || "light";
</script>
<!-- Loaded before other Sphinx assets -->
<link href="../../../../_static/styles/theme.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../../_static/styles/bootstrap.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../../_static/styles/pydata-sphinx-theme.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link href="../../../../_static/vendor/fontawesome/6.1.2/css/all.min.css?digest=796348d33e8b1d947c94" rel="stylesheet">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-solid-900.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-brands-400.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin href="../../../../_static/vendor/fontawesome/6.1.2/webfonts/fa-regular-400.woff2">
<link rel="stylesheet" type="text/css" href="../../../../_static/pygments.css" />
<link rel="stylesheet" type="text/css" href="../../../../_static/custom.css" />
<link rel="stylesheet" type="text/css" href="../../../../_static/sg_gallery.css" />
<link rel="stylesheet" type="text/css" href="../../../../_static/sg_gallery-binder.css" />
<link rel="stylesheet" type="text/css" href="../../../../_static/sg_gallery-dataframe.css" />
<link rel="stylesheet" type="text/css" href="../../../../_static/sg_gallery-rendered-html.css" />
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../../../../_static/scripts/bootstrap.js?digest=796348d33e8b1d947c94">
<link rel="preload" as="script" href="../../../../_static/scripts/pydata-sphinx-theme.js?digest=796348d33e8b1d947c94">
<script data-url_root="../../../../" id="documentation_options" src="../../../../_static/documentation_options.js"></script>
<script src="../../../../_static/jquery.js"></script>
<script src="../../../../_static/underscore.js"></script>
<script src="../../../../_static/_sphinx_javascript_frameworks_compat.js"></script>
<script src="../../../../_static/doctools.js"></script>
<script src="../../../../_static/sphinx_highlight.js"></script>
<script src="../../../../_static/copybutton.js"></script>
<script>DOCUMENTATION_OPTIONS.pagename = '_modules/networkx/algorithms/minors/contraction';</script>
<link rel="canonical" href="https://networkx.org/documentation/stable/_modules/networkx/algorithms/minors/contraction.html" />
<link rel="search" type="application/opensearchdescription+xml"
title="Search within NetworkX 3.0rc2.dev0 documentation"
href="../../../../_static/opensearch.xml"/>
<link rel="index" title="Index" href="../../../../genindex.html" />
<link rel="search" title="Search" href="../../../../search.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="en">
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="180" data-default-mode="light">
<a class="skip-link" href="#main-content">Skip to main content</a>
<div class="container-fluid version-alert devbar">
<div class="row no-gutters">
<div class="col-12 text-center">
This page is documentation for a DEVELOPMENT / PRE-RELEASE version.
<a
class="btn version-stable font-weight-bold ml-3 my-3 align-baseline"
href="https://networkx.org/documentation/stable/"
>Switch to stable version</a
>
</div>
</div>
</div>
<input type="checkbox" class="sidebar-toggle" name="__primary" id="__primary">
<label class="overlay overlay-primary" for="__primary"></label>
<input type="checkbox" class="sidebar-toggle" name="__secondary" id="__secondary">
<label class="overlay overlay-secondary" for="__secondary"></label>
<div class="search-button__wrapper">
<div class="search-button__overlay"></div>
<div class="search-button__search-container">
<form class="bd-search d-flex align-items-center" action="../../../../search.html" method="get">
<i class="fa-solid fa-magnifying-glass"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search the docs ..." aria-label="Search the docs ..." autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false">
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
</form>
</div>
</div>
<nav class="bd-header navbar navbar-expand-lg bd-navbar" id="navbar-main"><div class="bd-header__inner bd-page-width">
<label class="sidebar-toggle primary-toggle" for="__primary">
<span class="fa-solid fa-bars"></span>
</label>
<div id="navbar-start">
<a class="navbar-brand logo" href="../../../../index.html">
<img src="../../../../_static/networkx_banner.svg" class="logo__image only-light" alt="Logo image">
<img src="../../../../_static/networkx_banner.svg" class="logo__image only-dark" alt="Logo image">
</a>
</div>
<div class="col-lg-9 navbar-header-items">
<div id="navbar-center" class="mr-auto">
<div class="navbar-center-item">
<nav class="navbar-nav">
<p class="sidebar-header-items__title" role="heading" aria-level="1" aria-label="Site Navigation">
Site Navigation
</p>
<ul id="navbar-main-elements" class="navbar-nav">
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../install.html">
Install
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../tutorial.html">
Tutorial
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../reference/index.html">
Reference
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../auto_examples/index.html">
Gallery
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../developer/index.html">
Developer
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../release/index.html">
Releases
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-external" href="https://networkx.org/nx-guides/">
Guides
</a>
</li>
</ul>
</nav>
</div>
</div>
<div id="navbar-end">
<div class="navbar-end-item navbar-persistent--container">
<button class="btn btn-sm navbar-btn search-button search-button__button" title="Search" aria-label="Search" data-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
</button>
</div>
<div class="navbar-end-item">
<button class="theme-switch-button btn btn-sm btn-outline-primary navbar-btn rounded-circle" title="light/dark" aria-label="light/dark" data-toggle="tooltip">
<span class="theme-switch" data-mode="light"><i class="fa-solid fa-sun"></i></span>
<span class="theme-switch" data-mode="dark"><i class="fa-solid fa-moon"></i></span>
<span class="theme-switch" data-mode="auto"><i class="fa-solid fa-circle-half-stroke"></i></span>
</button>
</div>
<div class="navbar-end-item">
<ul id="navbar-icon-links" class="navbar-nav" aria-label="Icon Links">
<li class="nav-item">
<a href="https://networkx.org" title="Home Page" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fas fa-home"></i></span>
<label class="sr-only">Home Page</label></a>
</li>
<li class="nav-item">
<a href="https://github.com/networkx/networkx" title="GitHub" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fab fa-github-square"></i></span>
<label class="sr-only">GitHub</label></a>
</li>
</ul>
</div>
<div class="navbar-end-item">
<ul class="navbar-nav">
<li class="mr-2 dropdown">
<button
type="button"
class="btn btn-version btn-sm navbar-btn dropdown-toggle"
id="dLabelMore"
data-toggle="dropdown"
>
v3.0rc2.dev0
<span class="caret"></span>
</button>
<ul class="dropdown-menu" aria-labelledby="dLabelMore">
<li>
<a href="https://networkx.org/documentation/latest/index.html"
>devel (latest)</a
>
</li>
<li>
<a href="https://networkx.org/documentation/stable/index.html"
>current (stable)</a
>
</li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div class="navbar-persistent--mobile">
<button class="btn btn-sm navbar-btn search-button search-button__button" title="Search" aria-label="Search" data-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass"></i>
</button>
</div>
<label class="sidebar-toggle secondary-toggle" for="__secondary">
<span class="fa-solid fa-outdent"></span>
</label>
</div>
</nav>
<div class="bd-container">
<div class="bd-container__inner bd-page-width">
<div class="bd-sidebar-primary bd-sidebar">
<div class="sidebar-header-items sidebar-primary__section">
<div class="sidebar-header-items__center">
<div class="navbar-center-item">
<nav class="navbar-nav">
<p class="sidebar-header-items__title" role="heading" aria-level="1" aria-label="Site Navigation">
Site Navigation
</p>
<ul id="navbar-main-elements" class="navbar-nav">
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../install.html">
Install
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../tutorial.html">
Tutorial
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../reference/index.html">
Reference
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../auto_examples/index.html">
Gallery
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../developer/index.html">
Developer
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-internal" href="../../../../release/index.html">
Releases
</a>
</li>
<li class="nav-item">
<a class="nav-link nav-external" href="https://networkx.org/nx-guides/">
Guides
</a>
</li>
</ul>
</nav>
</div>
</div>
<div class="sidebar-header-items__end">
<div class="navbar-end-item">
<button class="theme-switch-button btn btn-sm btn-outline-primary navbar-btn rounded-circle" title="light/dark" aria-label="light/dark" data-toggle="tooltip">
<span class="theme-switch" data-mode="light"><i class="fa-solid fa-sun"></i></span>
<span class="theme-switch" data-mode="dark"><i class="fa-solid fa-moon"></i></span>
<span class="theme-switch" data-mode="auto"><i class="fa-solid fa-circle-half-stroke"></i></span>
</button>
</div>
<div class="navbar-end-item">
<ul id="navbar-icon-links" class="navbar-nav" aria-label="Icon Links">
<li class="nav-item">
<a href="https://networkx.org" title="Home Page" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fas fa-home"></i></span>
<label class="sr-only">Home Page</label></a>
</li>
<li class="nav-item">
<a href="https://github.com/networkx/networkx" title="GitHub" class="nav-link" rel="noopener" target="_blank" data-toggle="tooltip"><span><i class="fab fa-github-square"></i></span>
<label class="sr-only">GitHub</label></a>
</li>
</ul>
</div>
<div class="navbar-end-item">
<ul class="navbar-nav">
<li class="mr-2 dropdown">
<button
type="button"
class="btn btn-version btn-sm navbar-btn dropdown-toggle"
id="dLabelMore"
data-toggle="dropdown"
>
v3.0rc2.dev0
<span class="caret"></span>
</button>
<ul class="dropdown-menu" aria-labelledby="dLabelMore">
<li>
<a href="https://networkx.org/documentation/latest/index.html"
>devel (latest)</a
>
</li>
<li>
<a href="https://networkx.org/documentation/stable/index.html"
>current (stable)</a
>
</li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div class="sidebar-start-items sidebar-primary__section">
<div class="sidebar-start-items__item">
</div>
</div>
<div class="sidebar-end-items sidebar-primary__section">
<div class="sidebar-end-items__item">
</div>
</div>
<div id="rtd-footer-container"></div>
</div>
<main id="main-content" class="bd-main">
<div class="bd-content">
<div class="bd-article-container">
<div class="bd-header-article">
</div>
<article class="bd-article" role="main">
<h1>Source code for networkx.algorithms.minors.contraction</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Provides functions for computing minors of a graph."""</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span><span class="p">,</span> <span class="n">combinations</span><span class="p">,</span> <span class="n">permutations</span><span class="p">,</span> <span class="n">product</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx</span> <span class="kn">import</span> <span class="n">density</span>
<span class="kn">from</span> <span class="nn">networkx.exception</span> <span class="kn">import</span> <span class="n">NetworkXException</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">arbitrary_element</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"contracted_edge"</span><span class="p">,</span>
<span class="s2">"contracted_nodes"</span><span class="p">,</span>
<span class="s2">"equivalence_classes"</span><span class="p">,</span>
<span class="s2">"identified_nodes"</span><span class="p">,</span>
<span class="s2">"quotient_graph"</span><span class="p">,</span>
<span class="p">]</span>
<span class="n">chaini</span> <span class="o">=</span> <span class="n">chain</span><span class="o">.</span><span class="n">from_iterable</span>
<div class="viewcode-block" id="equivalence_classes"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.minors.equivalence_classes.html#networkx.algorithms.minors.equivalence_classes">[docs]</a><span class="k">def</span> <span class="nf">equivalence_classes</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">relation</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns equivalence classes of `relation` when applied to `iterable`.</span>
<span class="sd"> The equivalence classes, or blocks, consist of objects from `iterable`</span>
<span class="sd"> which are all equivalent. They are defined to be equivalent if the</span>
<span class="sd"> `relation` function returns `True` when passed any two objects from that</span>
<span class="sd"> class, and `False` otherwise. To define an equivalence relation the</span>
<span class="sd"> function must be reflexive, symmetric and transitive.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> iterable : list, tuple, or set</span>
<span class="sd"> An iterable of elements/nodes.</span>
<span class="sd"> relation : function</span>
<span class="sd"> A Boolean-valued function that implements an equivalence relation</span>
<span class="sd"> (reflexive, symmetric, transitive binary relation) on the elements</span>
<span class="sd"> of `iterable` - it must take two elements and return `True` if</span>
<span class="sd"> they are related, or `False` if not.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> set of frozensets</span>
<span class="sd"> A set of frozensets representing the partition induced by the equivalence</span>
<span class="sd"> relation function `relation` on the elements of `iterable`. Each</span>
<span class="sd"> member set in the return set represents an equivalence class, or</span>
<span class="sd"> block, of the partition.</span>
<span class="sd"> Duplicate elements will be ignored so it makes the most sense for</span>
<span class="sd"> `iterable` to be a :class:`set`.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This function does not check that `relation` represents an equivalence</span>
<span class="sd"> relation. You can check that your equivalence classes provide a partition</span>
<span class="sd"> using `is_partition`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> Let `X` be the set of integers from `0` to `9`, and consider an equivalence</span>
<span class="sd"> relation `R` on `X` of congruence modulo `3`: this means that two integers</span>
<span class="sd"> `x` and `y` in `X` are equivalent under `R` if they leave the same</span>
<span class="sd"> remainder when divided by `3`, i.e. `(x - y) mod 3 = 0`.</span>
<span class="sd"> The equivalence classes of this relation are `{0, 3, 6, 9}`, `{1, 4, 7}`,</span>
<span class="sd"> `{2, 5, 8}`: `0`, `3`, `6`, `9` are all divisible by `3` and leave zero</span>
<span class="sd"> remainder; `1`, `4`, `7` leave remainder `1`; while `2`, `5` and `8` leave</span>
<span class="sd"> remainder `2`. We can see this by calling `equivalence_classes` with</span>
<span class="sd"> `X` and a function implementation of `R`.</span>
<span class="sd"> >>> X = set(range(10))</span>
<span class="sd"> >>> def mod3(x, y): return (x - y) % 3 == 0</span>
<span class="sd"> >>> equivalence_classes(X, mod3) # doctest: +SKIP</span>
<span class="sd"> {frozenset({1, 4, 7}), frozenset({8, 2, 5}), frozenset({0, 9, 3, 6})}</span>
<span class="sd"> """</span>
<span class="c1"># For simplicity of implementation, we initialize the return value as a</span>
<span class="c1"># list of lists, then convert it to a set of sets at the end of the</span>
<span class="c1"># function.</span>
<span class="n">blocks</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># Determine the equivalence class for each element of the iterable.</span>
<span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">iterable</span><span class="p">:</span>
<span class="c1"># Each element y must be in *exactly one* equivalence class.</span>
<span class="c1">#</span>
<span class="c1"># Each block is guaranteed to be non-empty</span>
<span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">arbitrary_element</span><span class="p">(</span><span class="n">block</span><span class="p">)</span>
<span class="k">if</span> <span class="n">relation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="n">block</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="k">break</span>
<span class="k">else</span><span class="p">:</span>
<span class="c1"># If the element y is not part of any known equivalence class, it</span>
<span class="c1"># must be in its own, so we create a new singleton equivalence</span>
<span class="c1"># class for it.</span>
<span class="n">blocks</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">y</span><span class="p">])</span>
<span class="k">return</span> <span class="p">{</span><span class="nb">frozenset</span><span class="p">(</span><span class="n">block</span><span class="p">)</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">}</span></div>
<div class="viewcode-block" id="quotient_graph"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.minors.quotient_graph.html#networkx.algorithms.minors.quotient_graph">[docs]</a><span class="k">def</span> <span class="nf">quotient_graph</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">partition</span><span class="p">,</span>
<span class="n">edge_relation</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">node_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">edge_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">relabel</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">create_using</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the quotient graph of `G` under the specified equivalence</span>
<span class="sd"> relation on nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> The graph for which to return the quotient graph with the</span>
<span class="sd"> specified node relation.</span>
<span class="sd"> partition : function, or dict or list of lists, tuples or sets</span>
<span class="sd"> If a function, this function must represent an equivalence</span>
<span class="sd"> relation on the nodes of `G`. It must take two arguments *u*</span>
<span class="sd"> and *v* and return True exactly when *u* and *v* are in the</span>
<span class="sd"> same equivalence class. The equivalence classes form the nodes</span>
<span class="sd"> in the returned graph.</span>
<span class="sd"> If a dict of lists/tuples/sets, the keys can be any meaningful</span>
<span class="sd"> block labels, but the values must be the block lists/tuples/sets</span>
<span class="sd"> (one list/tuple/set per block), and the blocks must form a valid</span>
<span class="sd"> partition of the nodes of the graph. That is, each node must be</span>
<span class="sd"> in exactly one block of the partition.</span>
<span class="sd"> If a list of sets, the list must form a valid partition of</span>
<span class="sd"> the nodes of the graph. That is, each node must be in exactly</span>
<span class="sd"> one block of the partition.</span>
<span class="sd"> edge_relation : Boolean function with two arguments</span>
<span class="sd"> This function must represent an edge relation on the *blocks* of</span>
<span class="sd"> the `partition` of `G`. It must take two arguments, *B* and *C*,</span>
<span class="sd"> each one a set of nodes, and return True exactly when there should be</span>
<span class="sd"> an edge joining block *B* to block *C* in the returned graph.</span>
<span class="sd"> If `edge_relation` is not specified, it is assumed to be the</span>
<span class="sd"> following relation. Block *B* is related to block *C* if and</span>
<span class="sd"> only if some node in *B* is adjacent to some node in *C*,</span>
<span class="sd"> according to the edge set of `G`.</span>
<span class="sd"> edge_data : function</span>
<span class="sd"> This function takes two arguments, *B* and *C*, each one a set</span>
<span class="sd"> of nodes, and must return a dictionary representing the edge</span>
<span class="sd"> data attributes to set on the edge joining *B* and *C*, should</span>
<span class="sd"> there be an edge joining *B* and *C* in the quotient graph (if</span>
<span class="sd"> no such edge occurs in the quotient graph as determined by</span>
<span class="sd"> `edge_relation`, then the output of this function is ignored).</span>
<span class="sd"> If the quotient graph would be a multigraph, this function is</span>
<span class="sd"> not applied, since the edge data from each edge in the graph</span>
<span class="sd"> `G` appears in the edges of the quotient graph.</span>
<span class="sd"> node_data : function</span>
<span class="sd"> This function takes one argument, *B*, a set of nodes in `G`,</span>
<span class="sd"> and must return a dictionary representing the node data</span>
<span class="sd"> attributes to set on the node representing *B* in the quotient graph.</span>
<span class="sd"> If None, the following node attributes will be set:</span>
<span class="sd"> * 'graph', the subgraph of the graph `G` that this block</span>
<span class="sd"> represents,</span>
<span class="sd"> * 'nnodes', the number of nodes in this block,</span>
<span class="sd"> * 'nedges', the number of edges within this block,</span>
<span class="sd"> * 'density', the density of the subgraph of `G` that this</span>
<span class="sd"> block represents.</span>
<span class="sd"> relabel : bool</span>
<span class="sd"> If True, relabel the nodes of the quotient graph to be</span>
<span class="sd"> nonnegative integers. Otherwise, the nodes are identified with</span>
<span class="sd"> :class:`frozenset` instances representing the blocks given in</span>
<span class="sd"> `partition`.</span>
<span class="sd"> create_using : NetworkX graph constructor, optional (default=nx.Graph)</span>
<span class="sd"> Graph type to create. If graph instance, then cleared before populated.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> NetworkX graph</span>
<span class="sd"> The quotient graph of `G` under the equivalence relation</span>
<span class="sd"> specified by `partition`. If the partition were given as a</span>
<span class="sd"> list of :class:`set` instances and `relabel` is False,</span>
<span class="sd"> each node will be a :class:`frozenset` corresponding to the same</span>
<span class="sd"> :class:`set`.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXException</span>
<span class="sd"> If the given partition is not a valid partition of the nodes of</span>
<span class="sd"> `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> The quotient graph of the complete bipartite graph under the "same</span>
<span class="sd"> neighbors" equivalence relation is `K_2`. Under this relation, two nodes</span>
<span class="sd"> are equivalent if they are not adjacent but have the same neighbor set.</span>
<span class="sd"> >>> G = nx.complete_bipartite_graph(2, 3)</span>
<span class="sd"> >>> same_neighbors = lambda u, v: (</span>
<span class="sd"> ... u not in G[v] and v not in G[u] and G[u] == G[v]</span>
<span class="sd"> ... )</span>
<span class="sd"> >>> Q = nx.quotient_graph(G, same_neighbors)</span>
<span class="sd"> >>> K2 = nx.complete_graph(2)</span>
<span class="sd"> >>> nx.is_isomorphic(Q, K2)</span>
<span class="sd"> True</span>
<span class="sd"> The quotient graph of a directed graph under the "same strongly connected</span>
<span class="sd"> component" equivalence relation is the condensation of the graph (see</span>
<span class="sd"> :func:`condensation`). This example comes from the Wikipedia article</span>
<span class="sd"> *`Strongly connected component`_*.</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> edges = [</span>
<span class="sd"> ... "ab",</span>
<span class="sd"> ... "be",</span>
<span class="sd"> ... "bf",</span>
<span class="sd"> ... "bc",</span>
<span class="sd"> ... "cg",</span>
<span class="sd"> ... "cd",</span>
<span class="sd"> ... "dc",</span>
<span class="sd"> ... "dh",</span>
<span class="sd"> ... "ea",</span>
<span class="sd"> ... "ef",</span>
<span class="sd"> ... "fg",</span>
<span class="sd"> ... "gf",</span>
<span class="sd"> ... "hd",</span>
<span class="sd"> ... "hf",</span>
<span class="sd"> ... ]</span>
<span class="sd"> >>> G.add_edges_from(tuple(x) for x in edges)</span>
<span class="sd"> >>> components = list(nx.strongly_connected_components(G))</span>
<span class="sd"> >>> sorted(sorted(component) for component in components)</span>
<span class="sd"> [['a', 'b', 'e'], ['c', 'd', 'h'], ['f', 'g']]</span>
<span class="sd"> >>></span>
<span class="sd"> >>> C = nx.condensation(G, components)</span>
<span class="sd"> >>> component_of = C.graph["mapping"]</span>
<span class="sd"> >>> same_component = lambda u, v: component_of[u] == component_of[v]</span>
<span class="sd"> >>> Q = nx.quotient_graph(G, same_component)</span>
<span class="sd"> >>> nx.is_isomorphic(C, Q)</span>
<span class="sd"> True</span>
<span class="sd"> Node identification can be represented as the quotient of a graph under the</span>
<span class="sd"> equivalence relation that places the two nodes in one block and each other</span>
<span class="sd"> node in its own singleton block.</span>
<span class="sd"> >>> K24 = nx.complete_bipartite_graph(2, 4)</span>
<span class="sd"> >>> K34 = nx.complete_bipartite_graph(3, 4)</span>
<span class="sd"> >>> C = nx.contracted_nodes(K34, 1, 2)</span>
<span class="sd"> >>> nodes = {1, 2}</span>
<span class="sd"> >>> is_contracted = lambda u, v: u in nodes and v in nodes</span>
<span class="sd"> >>> Q = nx.quotient_graph(K34, is_contracted)</span>
<span class="sd"> >>> nx.is_isomorphic(Q, C)</span>
<span class="sd"> True</span>
<span class="sd"> >>> nx.is_isomorphic(Q, K24)</span>
<span class="sd"> True</span>
<span class="sd"> The blockmodeling technique described in [1]_ can be implemented as a</span>
<span class="sd"> quotient graph.</span>
<span class="sd"> >>> G = nx.path_graph(6)</span>
<span class="sd"> >>> partition = [{0, 1}, {2, 3}, {4, 5}]</span>
<span class="sd"> >>> M = nx.quotient_graph(G, partition, relabel=True)</span>
<span class="sd"> >>> list(M.edges())</span>
<span class="sd"> [(0, 1), (1, 2)]</span>
<span class="sd"> Here is the sample example but using partition as a dict of block sets.</span>
<span class="sd"> >>> G = nx.path_graph(6)</span>
<span class="sd"> >>> partition = {0: {0, 1}, 2: {2, 3}, 4: {4, 5}}</span>
<span class="sd"> >>> M = nx.quotient_graph(G, partition, relabel=True)</span>
<span class="sd"> >>> list(M.edges())</span>
<span class="sd"> [(0, 1), (1, 2)]</span>
<span class="sd"> Partitions can be represented in various ways:</span>
<span class="sd"> 0. a list/tuple/set of block lists/tuples/sets</span>
<span class="sd"> 1. a dict with block labels as keys and blocks lists/tuples/sets as values</span>
<span class="sd"> 2. a dict with block lists/tuples/sets as keys and block labels as values</span>
<span class="sd"> 3. a function from nodes in the original iterable to block labels</span>
<span class="sd"> 4. an equivalence relation function on the target iterable</span>
<span class="sd"> As `quotient_graph` is designed to accept partitions represented as (0), (1) or</span>
<span class="sd"> (4) only, the `equivalence_classes` function can be used to get the partitions</span>
<span class="sd"> in the right form, in order to call `quotient_graph`.</span>
<span class="sd"> .. _Strongly connected component: https://en.wikipedia.org/wiki/Strongly_connected_component</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Patrick Doreian, Vladimir Batagelj, and Anuska Ferligoj.</span>
<span class="sd"> *Generalized Blockmodeling*.</span>
<span class="sd"> Cambridge University Press, 2004.</span>
<span class="sd"> """</span>
<span class="c1"># If the user provided an equivalence relation as a function to compute</span>
<span class="c1"># the blocks of the partition on the nodes of G induced by the</span>
<span class="c1"># equivalence relation.</span>
<span class="k">if</span> <span class="n">callable</span><span class="p">(</span><span class="n">partition</span><span class="p">):</span>
<span class="c1"># equivalence_classes always return partition of whole G.</span>
<span class="n">partition</span> <span class="o">=</span> <span class="n">equivalence_classes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">partition</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">community</span><span class="o">.</span><span class="n">is_partition</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">partition</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXException</span><span class="p">(</span>
<span class="s2">"Input `partition` is not an equivalence relation for nodes of G"</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">_quotient_graph</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">partition</span><span class="p">,</span> <span class="n">edge_relation</span><span class="p">,</span> <span class="n">node_data</span><span class="p">,</span> <span class="n">edge_data</span><span class="p">,</span> <span class="n">relabel</span><span class="p">,</span> <span class="n">create_using</span>
<span class="p">)</span>
<span class="c1"># If the partition is a dict, it is assumed to be one where the keys are</span>
<span class="c1"># user-defined block labels, and values are block lists, tuples or sets.</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">partition</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
<span class="n">partition</span> <span class="o">=</span> <span class="p">[</span><span class="n">block</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="n">partition</span><span class="o">.</span><span class="n">values</span><span class="p">()]</span>
<span class="c1"># If the user provided partition as a collection of sets. Then we</span>
<span class="c1"># need to check if partition covers all of G nodes. If the answer</span>
<span class="c1"># is 'No' then we need to prepare suitable subgraph view.</span>
<span class="n">partition_nodes</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="o">*</span><span class="n">partition</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">partition_nodes</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">partition_nodes</span><span class="p">)</span>
<span class="c1"># Each node in the graph/subgraph must be in exactly one block.</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">community</span><span class="o">.</span><span class="n">is_partition</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">partition</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">NetworkXException</span><span class="p">(</span><span class="s2">"each node must be in exactly one part of `partition`"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_quotient_graph</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">partition</span><span class="p">,</span> <span class="n">edge_relation</span><span class="p">,</span> <span class="n">node_data</span><span class="p">,</span> <span class="n">edge_data</span><span class="p">,</span> <span class="n">relabel</span><span class="p">,</span> <span class="n">create_using</span>
<span class="p">)</span></div>
<span class="k">def</span> <span class="nf">_quotient_graph</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">partition</span><span class="p">,</span>
<span class="n">edge_relation</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">node_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">edge_data</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">relabel</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">create_using</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="p">):</span>
<span class="w"> </span><span class="sd">"""Construct the quotient graph assuming input has been checked"""</span>
<span class="k">if</span> <span class="n">create_using</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="vm">__class__</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">empty_graph</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">create_using</span><span class="p">)</span>
<span class="c1"># By default set some basic information about the subgraph that each block</span>
<span class="c1"># represents on the nodes in the quotient graph.</span>
<span class="k">if</span> <span class="n">node_data</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">node_data</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">dict</span><span class="p">(</span>
<span class="n">graph</span><span class="o">=</span><span class="n">S</span><span class="p">,</span> <span class="n">nnodes</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">),</span> <span class="n">nedges</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">number_of_edges</span><span class="p">(),</span> <span class="n">density</span><span class="o">=</span><span class="n">density</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="p">)</span>
<span class="c1"># Each block of the partition becomes a node in the quotient graph.</span>
<span class="n">partition</span> <span class="o">=</span> <span class="p">[</span><span class="nb">frozenset</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">]</span>
<span class="n">H</span><span class="o">.</span><span class="n">add_nodes_from</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">node_data</span><span class="p">(</span><span class="n">b</span><span class="p">))</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">)</span>
<span class="c1"># By default, the edge relation is the relation defined as follows. B is</span>
<span class="c1"># adjacent to C if a node in B is adjacent to a node in C, according to the</span>
<span class="c1"># edge set of G.</span>
<span class="c1">#</span>
<span class="c1"># This is not a particularly efficient implementation of this relation:</span>
<span class="c1"># there are O(n^2) pairs to check and each check may require O(log n) time</span>
<span class="c1"># (to check set membership). This can certainly be parallelized.</span>
<span class="k">if</span> <span class="n">edge_relation</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">edge_relation</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">product</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
<span class="c1"># By default, sum the weights of the edges joining pairs of nodes across</span>
<span class="c1"># blocks to get the weight of the edge joining those two blocks.</span>
<span class="k">if</span> <span class="n">edge_data</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">edge_data</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
<span class="n">edgedata</span> <span class="o">=</span> <span class="p">(</span>
<span class="n">d</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">b</span> <span class="o">|</span> <span class="n">c</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">if</span> <span class="p">(</span><span class="n">u</span> <span class="ow">in</span> <span class="n">b</span> <span class="ow">and</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">c</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">u</span> <span class="ow">in</span> <span class="n">c</span> <span class="ow">and</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">b</span><span class="p">)</span>
<span class="p">)</span>
<span class="k">return</span> <span class="p">{</span><span class="s2">"weight"</span><span class="p">:</span> <span class="nb">sum</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s2">"weight"</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">edgedata</span><span class="p">)}</span>
<span class="n">block_pairs</span> <span class="o">=</span> <span class="n">permutations</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="k">if</span> <span class="n">H</span><span class="o">.</span><span class="n">is_directed</span><span class="p">()</span> <span class="k">else</span> <span class="n">combinations</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="c1"># In a multigraph, add one edge in the quotient graph for each edge</span>
<span class="c1"># in the original graph.</span>
<span class="k">if</span> <span class="n">H</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="n">edges</span> <span class="o">=</span> <span class="n">chaini</span><span class="p">(</span>
<span class="p">(</span>
<span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">G</span><span class="o">.</span><span class="n">get_edge_data</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="p">{}))</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">product</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">block_pairs</span>
<span class="k">if</span> <span class="n">edge_relation</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
<span class="p">)</span>
<span class="c1"># In a simple graph, apply the edge data function to each pair of</span>
<span class="c1"># blocks to determine the edge data attributes to apply to each edge</span>
<span class="c1"># in the quotient graph.</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">edges</span> <span class="o">=</span> <span class="p">(</span>
<span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">edge_data</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="n">block_pairs</span> <span class="k">if</span> <span class="n">edge_relation</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
<span class="p">)</span>
<span class="n">H</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span>
<span class="c1"># If requested by the user, relabel the nodes to be integers,</span>
<span class="c1"># numbered in increasing order from zero in the same order as the</span>
<span class="c1"># iteration order of `partition`.</span>
<span class="k">if</span> <span class="n">relabel</span><span class="p">:</span>
<span class="c1"># Can't use nx.convert_node_labels_to_integers() here since we</span>
<span class="c1"># want the order of iteration to be the same for backward</span>
<span class="c1"># compatibility with the nx.blockmodel() function.</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">{</span><span class="n">b</span><span class="p">:</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">partition</span><span class="p">)}</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">relabel_nodes</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">labels</span><span class="p">)</span>
<span class="k">return</span> <span class="n">H</span>
<div class="viewcode-block" id="contracted_nodes"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.minors.contracted_nodes.html#networkx.algorithms.minors.contracted_nodes">[docs]</a><span class="k">def</span> <span class="nf">contracted_nodes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">self_loops</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the graph that results from contracting `u` and `v`.</span>
<span class="sd"> Node contraction identifies the two nodes as a single node incident to any</span>
<span class="sd"> edge that was incident to the original two nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> The graph whose nodes will be contracted.</span>
<span class="sd"> u, v : nodes</span>
<span class="sd"> Must be nodes in `G`.</span>
<span class="sd"> self_loops : Boolean</span>
<span class="sd"> If this is True, any edges joining `u` and `v` in `G` become</span>
<span class="sd"> self-loops on the new node in the returned graph.</span>
<span class="sd"> copy : Boolean</span>
<span class="sd"> If this is True (default True), make a copy of</span>
<span class="sd"> `G` and return that instead of directly changing `G`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> Networkx graph</span>
<span class="sd"> If Copy is True,</span>
<span class="sd"> A new graph object of the same type as `G` (leaving `G` unmodified)</span>
<span class="sd"> with `u` and `v` identified in a single node. The right node `v`</span>
<span class="sd"> will be merged into the node `u`, so only `u` will appear in the</span>
<span class="sd"> returned graph.</span>
<span class="sd"> If copy is False,</span>
<span class="sd"> Modifies `G` with `u` and `v` identified in a single node.</span>
<span class="sd"> The right node `v` will be merged into the node `u`, so</span>
<span class="sd"> only `u` will appear in the returned graph.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> For multigraphs, the edge keys for the realigned edges may</span>
<span class="sd"> not be the same as the edge keys for the old edges. This is</span>
<span class="sd"> natural because edge keys are unique only within each pair of nodes.</span>
<span class="sd"> For non-multigraphs where `u` and `v` are adjacent to a third node</span>
<span class="sd"> `w`, the edge (`v`, `w`) will be contracted into the edge (`u`,</span>
<span class="sd"> `w`) with its attributes stored into a "contraction" attribute.</span>
<span class="sd"> This function is also available as `identified_nodes`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> Contracting two nonadjacent nodes of the cycle graph on four nodes `C_4`</span>
<span class="sd"> yields the path graph (ignoring parallel edges):</span>
<span class="sd"> >>> G = nx.cycle_graph(4)</span>
<span class="sd"> >>> M = nx.contracted_nodes(G, 1, 3)</span>
<span class="sd"> >>> P3 = nx.path_graph(3)</span>
<span class="sd"> >>> nx.is_isomorphic(M, P3)</span>
<span class="sd"> True</span>
<span class="sd"> >>> G = nx.MultiGraph(P3)</span>
<span class="sd"> >>> M = nx.contracted_nodes(G, 0, 2)</span>
<span class="sd"> >>> M.edges</span>
<span class="sd"> MultiEdgeView([(0, 1, 0), (0, 1, 1)])</span>
<span class="sd"> >>> G = nx.Graph([(1, 2), (2, 2)])</span>
<span class="sd"> >>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)</span>
<span class="sd"> >>> list(H.nodes())</span>
<span class="sd"> [1]</span>
<span class="sd"> >>> list(H.edges())</span>
<span class="sd"> [(1, 1)]</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> contracted_edge</span>
<span class="sd"> quotient_graph</span>
<span class="sd"> """</span>
<span class="c1"># Copying has significant overhead and can be disabled if needed</span>
<span class="k">if</span> <span class="n">copy</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span> <span class="o">=</span> <span class="n">G</span>
<span class="c1"># edge code uses G.edges(v) instead of G.adj[v] to handle multiedges</span>
<span class="k">if</span> <span class="n">H</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">edges_to_remap</span> <span class="o">=</span> <span class="n">chain</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">in_edges</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span> <span class="n">G</span><span class="o">.</span><span class="n">out_edges</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">edges_to_remap</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">edges</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># If the H=G, the generators change as H changes</span>
<span class="c1"># This makes the edges_to_remap independent of H</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">copy</span><span class="p">:</span>
<span class="n">edges_to_remap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">edges_to_remap</span><span class="p">)</span>
<span class="n">v_data</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="n">H</span><span class="o">.</span><span class="n">remove_node</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">for</span> <span class="p">(</span><span class="n">prev_w</span><span class="p">,</span> <span class="n">prev_x</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="ow">in</span> <span class="n">edges_to_remap</span><span class="p">:</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">prev_w</span> <span class="k">if</span> <span class="n">prev_w</span> <span class="o">!=</span> <span class="n">v</span> <span class="k">else</span> <span class="n">u</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">prev_x</span> <span class="k">if</span> <span class="n">prev_x</span> <span class="o">!=</span> <span class="n">v</span> <span class="k">else</span> <span class="n">u</span>
<span class="k">if</span> <span class="p">({</span><span class="n">prev_w</span><span class="p">,</span> <span class="n">prev_x</span><span class="p">}</span> <span class="o">==</span> <span class="p">{</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">})</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">self_loops</span><span class="p">:</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">H</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="ow">or</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="n">H</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">d</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="s2">"contraction"</span> <span class="ow">in</span> <span class="n">H</span><span class="o">.</span><span class="n">edges</span><span class="p">[(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)]:</span>
<span class="n">H</span><span class="o">.</span><span class="n">edges</span><span class="p">[(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)][</span><span class="s2">"contraction"</span><span class="p">][(</span><span class="n">prev_w</span><span class="p">,</span> <span class="n">prev_x</span><span class="p">)]</span> <span class="o">=</span> <span class="n">d</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">edges</span><span class="p">[(</span><span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">)][</span><span class="s2">"contraction"</span><span class="p">]</span> <span class="o">=</span> <span class="p">{(</span><span class="n">prev_w</span><span class="p">,</span> <span class="n">prev_x</span><span class="p">):</span> <span class="n">d</span><span class="p">}</span>
<span class="k">if</span> <span class="s2">"contraction"</span> <span class="ow">in</span> <span class="n">H</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="n">H</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="s2">"contraction"</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">v_data</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">H</span><span class="o">.</span><span class="n">nodes</span><span class="p">[</span><span class="n">u</span><span class="p">][</span><span class="s2">"contraction"</span><span class="p">]</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">v_data</span><span class="p">}</span>
<span class="k">return</span> <span class="n">H</span></div>
<span class="n">identified_nodes</span> <span class="o">=</span> <span class="n">contracted_nodes</span>
<div class="viewcode-block" id="contracted_edge"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.minors.contracted_edge.html#networkx.algorithms.minors.contracted_edge">[docs]</a><span class="k">def</span> <span class="nf">contracted_edge</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">edge</span><span class="p">,</span> <span class="n">self_loops</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the graph that results from contracting the specified edge.</span>
<span class="sd"> Edge contraction identifies the two endpoints of the edge as a single node</span>
<span class="sd"> incident to any edge that was incident to the original two nodes. A graph</span>
<span class="sd"> that results from edge contraction is called a *minor* of the original</span>
<span class="sd"> graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> The graph whose edge will be contracted.</span>
<span class="sd"> edge : tuple</span>
<span class="sd"> Must be a pair of nodes in `G`.</span>
<span class="sd"> self_loops : Boolean</span>
<span class="sd"> If this is True, any edges (including `edge`) joining the</span>
<span class="sd"> endpoints of `edge` in `G` become self-loops on the new node in the</span>
<span class="sd"> returned graph.</span>
<span class="sd"> copy : Boolean (default True)</span>
<span class="sd"> If this is True, a the contraction will be performed on a copy of `G`,</span>
<span class="sd"> otherwise the contraction will happen in place.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> Networkx graph</span>
<span class="sd"> A new graph object of the same type as `G` (leaving `G` unmodified)</span>
<span class="sd"> with endpoints of `edge` identified in a single node. The right node</span>
<span class="sd"> of `edge` will be merged into the left one, so only the left one will</span>
<span class="sd"> appear in the returned graph.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ValueError</span>
<span class="sd"> If `edge` is not an edge in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> Attempting to contract two nonadjacent nodes yields an error:</span>
<span class="sd"> >>> G = nx.cycle_graph(4)</span>
<span class="sd"> >>> nx.contracted_edge(G, (1, 3))</span>
<span class="sd"> Traceback (most recent call last):</span>
<span class="sd"> ...</span>
<span class="sd"> ValueError: Edge (1, 3) does not exist in graph G; cannot contract it</span>
<span class="sd"> Contracting two adjacent nodes in the cycle graph on *n* nodes yields the</span>
<span class="sd"> cycle graph on *n - 1* nodes:</span>
<span class="sd"> >>> C5 = nx.cycle_graph(5)</span>
<span class="sd"> >>> C4 = nx.cycle_graph(4)</span>
<span class="sd"> >>> M = nx.contracted_edge(C5, (0, 1), self_loops=False)</span>
<span class="sd"> >>> nx.is_isomorphic(M, C4)</span>
<span class="sd"> True</span>
<span class="sd"> See also</span>
<span class="sd"> --------</span>
<span class="sd"> contracted_nodes</span>
<span class="sd"> quotient_graph</span>
<span class="sd"> """</span>
<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">edge</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">G</span><span class="o">.</span><span class="n">has_edge</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Edge </span><span class="si">{</span><span class="n">edge</span><span class="si">}</span><span class="s2"> does not exist in graph G; cannot contract it"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">contracted_nodes</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">self_loops</span><span class="o">=</span><span class="n">self_loops</span><span class="p">,</span> <span class="n">copy</span><span class="o">=</span><span class="n">copy</span><span class="p">)</span></div>
</pre></div>
</article>
</div>
<div class="bd-sidebar-secondary bd-toc">
<div class="toc-item">
<div id="searchbox"></div>
</div>
<div class="toc-item">
</div>
<div class="toc-item">
</div>
</div>
</div>
<footer class="bd-footer-content">
<div class="bd-footer-content__inner">
</div>
</footer>
</main>
</div>
</div>
<!-- Scripts loaded after <body> so the DOM is not blocked -->
<script src="../../../../_static/scripts/bootstrap.js?digest=796348d33e8b1d947c94"></script>
<script src="../../../../_static/scripts/pydata-sphinx-theme.js?digest=796348d33e8b1d947c94"></script>
<footer class="bd-footer"><div class="bd-footer__inner container">
<div class="footer-item">
<p class="copyright">
© Copyright 2004-2023, NetworkX Developers.<br>
</p>
</div>
<div class="footer-item">
<p class="theme-version">
Built with the
<a href="https://pydata-sphinx-theme.readthedocs.io/en/stable/index.html">
PyData Sphinx Theme
</a>
0.12.0.
</p>
</div>
<div class="footer-item">
<p class="sphinx-version">
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 5.2.3.<br>
</p>
</div>
</div>
</footer>
</body>
</html>
|
