Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.9/dist-packages/networkx/generators/degree_seq.py: 13%
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« prev ^ index » next coverage.py v7.3.2, created at 2023-10-20 07:00 +0000
1"""Generate graphs with a given degree sequence or expected degree sequence.
2"""
4import heapq
5import math
6from itertools import chain, combinations, zip_longest
7from operator import itemgetter
9import networkx as nx
10from networkx.utils import py_random_state, random_weighted_sample
12__all__ = [
13 "configuration_model",
14 "directed_configuration_model",
15 "expected_degree_graph",
16 "havel_hakimi_graph",
17 "directed_havel_hakimi_graph",
18 "degree_sequence_tree",
19 "random_degree_sequence_graph",
20]
22chaini = chain.from_iterable
25def _to_stublist(degree_sequence):
26 """Returns a list of degree-repeated node numbers.
28 ``degree_sequence`` is a list of nonnegative integers representing
29 the degrees of nodes in a graph.
31 This function returns a list of node numbers with multiplicities
32 according to the given degree sequence. For example, if the first
33 element of ``degree_sequence`` is ``3``, then the first node number,
34 ``0``, will appear at the head of the returned list three times. The
35 node numbers are assumed to be the numbers zero through
36 ``len(degree_sequence) - 1``.
38 Examples
39 --------
41 >>> degree_sequence = [1, 2, 3]
42 >>> _to_stublist(degree_sequence)
43 [0, 1, 1, 2, 2, 2]
45 If a zero appears in the sequence, that means the node exists but
46 has degree zero, so that number will be skipped in the returned
47 list::
49 >>> degree_sequence = [2, 0, 1]
50 >>> _to_stublist(degree_sequence)
51 [0, 0, 2]
53 """
54 return list(chaini([n] * d for n, d in enumerate(degree_sequence)))
57def _configuration_model(
58 deg_sequence, create_using, directed=False, in_deg_sequence=None, seed=None
59):
60 """Helper function for generating either undirected or directed
61 configuration model graphs.
63 ``deg_sequence`` is a list of nonnegative integers representing the
64 degree of the node whose label is the index of the list element.
66 ``create_using`` see :func:`~networkx.empty_graph`.
68 ``directed`` and ``in_deg_sequence`` are required if you want the
69 returned graph to be generated using the directed configuration
70 model algorithm. If ``directed`` is ``False``, then ``deg_sequence``
71 is interpreted as the degree sequence of an undirected graph and
72 ``in_deg_sequence`` is ignored. Otherwise, if ``directed`` is
73 ``True``, then ``deg_sequence`` is interpreted as the out-degree
74 sequence and ``in_deg_sequence`` as the in-degree sequence of a
75 directed graph.
77 .. note::
79 ``deg_sequence`` and ``in_deg_sequence`` need not be the same
80 length.
82 ``seed`` is a random.Random or numpy.random.RandomState instance
84 This function returns a graph, directed if and only if ``directed``
85 is ``True``, generated according to the configuration model
86 algorithm. For more information on the algorithm, see the
87 :func:`configuration_model` or :func:`directed_configuration_model`
88 functions.
90 """
91 n = len(deg_sequence)
92 G = nx.empty_graph(n, create_using)
93 # If empty, return the null graph immediately.
94 if n == 0:
95 return G
96 # Build a list of available degree-repeated nodes. For example,
97 # for degree sequence [3, 2, 1, 1, 1], the "stub list" is
98 # initially [0, 0, 0, 1, 1, 2, 3, 4], that is, node 0 has degree
99 # 3 and thus is repeated 3 times, etc.
100 #
101 # Also, shuffle the stub list in order to get a random sequence of
102 # node pairs.
103 if directed:
104 pairs = zip_longest(deg_sequence, in_deg_sequence, fillvalue=0)
105 # Unzip the list of pairs into a pair of lists.
106 out_deg, in_deg = zip(*pairs)
108 out_stublist = _to_stublist(out_deg)
109 in_stublist = _to_stublist(in_deg)
111 seed.shuffle(out_stublist)
112 seed.shuffle(in_stublist)
113 else:
114 stublist = _to_stublist(deg_sequence)
115 # Choose a random balanced bipartition of the stublist, which
116 # gives a random pairing of nodes. In this implementation, we
117 # shuffle the list and then split it in half.
118 n = len(stublist)
119 half = n // 2
120 seed.shuffle(stublist)
121 out_stublist, in_stublist = stublist[:half], stublist[half:]
122 G.add_edges_from(zip(out_stublist, in_stublist))
123 return G
126@py_random_state(2)
127@nx._dispatch(graphs=None)
128def configuration_model(deg_sequence, create_using=None, seed=None):
129 """Returns a random graph with the given degree sequence.
131 The configuration model generates a random pseudograph (graph with
132 parallel edges and self loops) by randomly assigning edges to
133 match the given degree sequence.
135 Parameters
136 ----------
137 deg_sequence : list of nonnegative integers
138 Each list entry corresponds to the degree of a node.
139 create_using : NetworkX graph constructor, optional (default MultiGraph)
140 Graph type to create. If graph instance, then cleared before populated.
141 seed : integer, random_state, or None (default)
142 Indicator of random number generation state.
143 See :ref:`Randomness<randomness>`.
145 Returns
146 -------
147 G : MultiGraph
148 A graph with the specified degree sequence.
149 Nodes are labeled starting at 0 with an index
150 corresponding to the position in deg_sequence.
152 Raises
153 ------
154 NetworkXError
155 If the degree sequence does not have an even sum.
157 See Also
158 --------
159 is_graphical
161 Notes
162 -----
163 As described by Newman [1]_.
165 A non-graphical degree sequence (not realizable by some simple
166 graph) is allowed since this function returns graphs with self
167 loops and parallel edges. An exception is raised if the degree
168 sequence does not have an even sum.
170 This configuration model construction process can lead to
171 duplicate edges and loops. You can remove the self-loops and
172 parallel edges (see below) which will likely result in a graph
173 that doesn't have the exact degree sequence specified.
175 The density of self-loops and parallel edges tends to decrease as
176 the number of nodes increases. However, typically the number of
177 self-loops will approach a Poisson distribution with a nonzero mean,
178 and similarly for the number of parallel edges. Consider a node
179 with *k* stubs. The probability of being joined to another stub of
180 the same node is basically (*k* - *1*) / *N*, where *k* is the
181 degree and *N* is the number of nodes. So the probability of a
182 self-loop scales like *c* / *N* for some constant *c*. As *N* grows,
183 this means we expect *c* self-loops. Similarly for parallel edges.
185 References
186 ----------
187 .. [1] M.E.J. Newman, "The structure and function of complex networks",
188 SIAM REVIEW 45-2, pp 167-256, 2003.
190 Examples
191 --------
192 You can create a degree sequence following a particular distribution
193 by using the one of the distribution functions in
194 :mod:`~networkx.utils.random_sequence` (or one of your own). For
195 example, to create an undirected multigraph on one hundred nodes
196 with degree sequence chosen from the power law distribution:
198 >>> sequence = nx.random_powerlaw_tree_sequence(100, tries=5000)
199 >>> G = nx.configuration_model(sequence)
200 >>> len(G)
201 100
202 >>> actual_degrees = [d for v, d in G.degree()]
203 >>> actual_degrees == sequence
204 True
206 The returned graph is a multigraph, which may have parallel
207 edges. To remove any parallel edges from the returned graph:
209 >>> G = nx.Graph(G)
211 Similarly, to remove self-loops:
213 >>> G.remove_edges_from(nx.selfloop_edges(G))
215 """
216 if sum(deg_sequence) % 2 != 0:
217 msg = "Invalid degree sequence: sum of degrees must be even, not odd"
218 raise nx.NetworkXError(msg)
220 G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
221 if G.is_directed():
222 raise nx.NetworkXNotImplemented("not implemented for directed graphs")
224 G = _configuration_model(deg_sequence, G, seed=seed)
226 return G
229@py_random_state(3)
230@nx._dispatch(graphs=None)
231def directed_configuration_model(
232 in_degree_sequence, out_degree_sequence, create_using=None, seed=None
233):
234 """Returns a directed_random graph with the given degree sequences.
236 The configuration model generates a random directed pseudograph
237 (graph with parallel edges and self loops) by randomly assigning
238 edges to match the given degree sequences.
240 Parameters
241 ----------
242 in_degree_sequence : list of nonnegative integers
243 Each list entry corresponds to the in-degree of a node.
244 out_degree_sequence : list of nonnegative integers
245 Each list entry corresponds to the out-degree of a node.
246 create_using : NetworkX graph constructor, optional (default MultiDiGraph)
247 Graph type to create. If graph instance, then cleared before populated.
248 seed : integer, random_state, or None (default)
249 Indicator of random number generation state.
250 See :ref:`Randomness<randomness>`.
252 Returns
253 -------
254 G : MultiDiGraph
255 A graph with the specified degree sequences.
256 Nodes are labeled starting at 0 with an index
257 corresponding to the position in deg_sequence.
259 Raises
260 ------
261 NetworkXError
262 If the degree sequences do not have the same sum.
264 See Also
265 --------
266 configuration_model
268 Notes
269 -----
270 Algorithm as described by Newman [1]_.
272 A non-graphical degree sequence (not realizable by some simple
273 graph) is allowed since this function returns graphs with self
274 loops and parallel edges. An exception is raised if the degree
275 sequences does not have the same sum.
277 This configuration model construction process can lead to
278 duplicate edges and loops. You can remove the self-loops and
279 parallel edges (see below) which will likely result in a graph
280 that doesn't have the exact degree sequence specified. This
281 "finite-size effect" decreases as the size of the graph increases.
283 References
284 ----------
285 .. [1] Newman, M. E. J. and Strogatz, S. H. and Watts, D. J.
286 Random graphs with arbitrary degree distributions and their applications
287 Phys. Rev. E, 64, 026118 (2001)
289 Examples
290 --------
291 One can modify the in- and out-degree sequences from an existing
292 directed graph in order to create a new directed graph. For example,
293 here we modify the directed path graph:
295 >>> D = nx.DiGraph([(0, 1), (1, 2), (2, 3)])
296 >>> din = list(d for n, d in D.in_degree())
297 >>> dout = list(d for n, d in D.out_degree())
298 >>> din.append(1)
299 >>> dout[0] = 2
300 >>> # We now expect an edge from node 0 to a new node, node 3.
301 ... D = nx.directed_configuration_model(din, dout)
303 The returned graph is a directed multigraph, which may have parallel
304 edges. To remove any parallel edges from the returned graph:
306 >>> D = nx.DiGraph(D)
308 Similarly, to remove self-loops:
310 >>> D.remove_edges_from(nx.selfloop_edges(D))
312 """
313 if sum(in_degree_sequence) != sum(out_degree_sequence):
314 msg = "Invalid degree sequences: sequences must have equal sums"
315 raise nx.NetworkXError(msg)
317 if create_using is None:
318 create_using = nx.MultiDiGraph
320 G = _configuration_model(
321 out_degree_sequence,
322 create_using,
323 directed=True,
324 in_deg_sequence=in_degree_sequence,
325 seed=seed,
326 )
328 name = "directed configuration_model {} nodes {} edges"
329 return G
332@py_random_state(1)
333@nx._dispatch(graphs=None)
334def expected_degree_graph(w, seed=None, selfloops=True):
335 r"""Returns a random graph with given expected degrees.
337 Given a sequence of expected degrees $W=(w_0,w_1,\ldots,w_{n-1})$
338 of length $n$ this algorithm assigns an edge between node $u$ and
339 node $v$ with probability
341 .. math::
343 p_{uv} = \frac{w_u w_v}{\sum_k w_k} .
345 Parameters
346 ----------
347 w : list
348 The list of expected degrees.
349 selfloops: bool (default=True)
350 Set to False to remove the possibility of self-loop edges.
351 seed : integer, random_state, or None (default)
352 Indicator of random number generation state.
353 See :ref:`Randomness<randomness>`.
355 Returns
356 -------
357 Graph
359 Examples
360 --------
361 >>> z = [10 for i in range(100)]
362 >>> G = nx.expected_degree_graph(z)
364 Notes
365 -----
366 The nodes have integer labels corresponding to index of expected degrees
367 input sequence.
369 The complexity of this algorithm is $\mathcal{O}(n+m)$ where $n$ is the
370 number of nodes and $m$ is the expected number of edges.
372 The model in [1]_ includes the possibility of self-loop edges.
373 Set selfloops=False to produce a graph without self loops.
375 For finite graphs this model doesn't produce exactly the given
376 expected degree sequence. Instead the expected degrees are as
377 follows.
379 For the case without self loops (selfloops=False),
381 .. math::
383 E[deg(u)] = \sum_{v \ne u} p_{uv}
384 = w_u \left( 1 - \frac{w_u}{\sum_k w_k} \right) .
387 NetworkX uses the standard convention that a self-loop edge counts 2
388 in the degree of a node, so with self loops (selfloops=True),
390 .. math::
392 E[deg(u)] = \sum_{v \ne u} p_{uv} + 2 p_{uu}
393 = w_u \left( 1 + \frac{w_u}{\sum_k w_k} \right) .
395 References
396 ----------
397 .. [1] Fan Chung and L. Lu, Connected components in random graphs with
398 given expected degree sequences, Ann. Combinatorics, 6,
399 pp. 125-145, 2002.
400 .. [2] Joel Miller and Aric Hagberg,
401 Efficient generation of networks with given expected degrees,
402 in Algorithms and Models for the Web-Graph (WAW 2011),
403 Alan Frieze, Paul Horn, and Paweł Prałat (Eds), LNCS 6732,
404 pp. 115-126, 2011.
405 """
406 n = len(w)
407 G = nx.empty_graph(n)
409 # If there are no nodes are no edges in the graph, return the empty graph.
410 if n == 0 or max(w) == 0:
411 return G
413 rho = 1 / sum(w)
414 # Sort the weights in decreasing order. The original order of the
415 # weights dictates the order of the (integer) node labels, so we
416 # need to remember the permutation applied in the sorting.
417 order = sorted(enumerate(w), key=itemgetter(1), reverse=True)
418 mapping = {c: u for c, (u, v) in enumerate(order)}
419 seq = [v for u, v in order]
420 last = n
421 if not selfloops:
422 last -= 1
423 for u in range(last):
424 v = u
425 if not selfloops:
426 v += 1
427 factor = seq[u] * rho
428 p = min(seq[v] * factor, 1)
429 while v < n and p > 0:
430 if p != 1:
431 r = seed.random()
432 v += math.floor(math.log(r, 1 - p))
433 if v < n:
434 q = min(seq[v] * factor, 1)
435 if seed.random() < q / p:
436 G.add_edge(mapping[u], mapping[v])
437 v += 1
438 p = q
439 return G
442@nx._dispatch(graphs=None)
443def havel_hakimi_graph(deg_sequence, create_using=None):
444 """Returns a simple graph with given degree sequence constructed
445 using the Havel-Hakimi algorithm.
447 Parameters
448 ----------
449 deg_sequence: list of integers
450 Each integer corresponds to the degree of a node (need not be sorted).
451 create_using : NetworkX graph constructor, optional (default=nx.Graph)
452 Graph type to create. If graph instance, then cleared before populated.
453 Directed graphs are not allowed.
455 Raises
456 ------
457 NetworkXException
458 For a non-graphical degree sequence (i.e. one
459 not realizable by some simple graph).
461 Notes
462 -----
463 The Havel-Hakimi algorithm constructs a simple graph by
464 successively connecting the node of highest degree to other nodes
465 of highest degree, resorting remaining nodes by degree, and
466 repeating the process. The resulting graph has a high
467 degree-associativity. Nodes are labeled 1,.., len(deg_sequence),
468 corresponding to their position in deg_sequence.
470 The basic algorithm is from Hakimi [1]_ and was generalized by
471 Kleitman and Wang [2]_.
473 References
474 ----------
475 .. [1] Hakimi S., On Realizability of a Set of Integers as
476 Degrees of the Vertices of a Linear Graph. I,
477 Journal of SIAM, 10(3), pp. 496-506 (1962)
478 .. [2] Kleitman D.J. and Wang D.L.
479 Algorithms for Constructing Graphs and Digraphs with Given Valences
480 and Factors Discrete Mathematics, 6(1), pp. 79-88 (1973)
481 """
482 if not nx.is_graphical(deg_sequence):
483 raise nx.NetworkXError("Invalid degree sequence")
485 p = len(deg_sequence)
486 G = nx.empty_graph(p, create_using)
487 if G.is_directed():
488 raise nx.NetworkXError("Directed graphs are not supported")
489 num_degs = [[] for i in range(p)]
490 dmax, dsum, n = 0, 0, 0
491 for d in deg_sequence:
492 # Process only the non-zero integers
493 if d > 0:
494 num_degs[d].append(n)
495 dmax, dsum, n = max(dmax, d), dsum + d, n + 1
496 # Return graph if no edges
497 if n == 0:
498 return G
500 modstubs = [(0, 0)] * (dmax + 1)
501 # Successively reduce degree sequence by removing the maximum degree
502 while n > 0:
503 # Retrieve the maximum degree in the sequence
504 while len(num_degs[dmax]) == 0:
505 dmax -= 1
506 # If there are not enough stubs to connect to, then the sequence is
507 # not graphical
508 if dmax > n - 1:
509 raise nx.NetworkXError("Non-graphical integer sequence")
511 # Remove largest stub in list
512 source = num_degs[dmax].pop()
513 n -= 1
514 # Reduce the next dmax largest stubs
515 mslen = 0
516 k = dmax
517 for i in range(dmax):
518 while len(num_degs[k]) == 0:
519 k -= 1
520 target = num_degs[k].pop()
521 G.add_edge(source, target)
522 n -= 1
523 if k > 1:
524 modstubs[mslen] = (k - 1, target)
525 mslen += 1
526 # Add back to the list any nonzero stubs that were removed
527 for i in range(mslen):
528 (stubval, stubtarget) = modstubs[i]
529 num_degs[stubval].append(stubtarget)
530 n += 1
532 return G
535@nx._dispatch(graphs=None)
536def directed_havel_hakimi_graph(in_deg_sequence, out_deg_sequence, create_using=None):
537 """Returns a directed graph with the given degree sequences.
539 Parameters
540 ----------
541 in_deg_sequence : list of integers
542 Each list entry corresponds to the in-degree of a node.
543 out_deg_sequence : list of integers
544 Each list entry corresponds to the out-degree of a node.
545 create_using : NetworkX graph constructor, optional (default DiGraph)
546 Graph type to create. If graph instance, then cleared before populated.
548 Returns
549 -------
550 G : DiGraph
551 A graph with the specified degree sequences.
552 Nodes are labeled starting at 0 with an index
553 corresponding to the position in deg_sequence
555 Raises
556 ------
557 NetworkXError
558 If the degree sequences are not digraphical.
560 See Also
561 --------
562 configuration_model
564 Notes
565 -----
566 Algorithm as described by Kleitman and Wang [1]_.
568 References
569 ----------
570 .. [1] D.J. Kleitman and D.L. Wang
571 Algorithms for Constructing Graphs and Digraphs with Given Valences
572 and Factors Discrete Mathematics, 6(1), pp. 79-88 (1973)
573 """
574 in_deg_sequence = nx.utils.make_list_of_ints(in_deg_sequence)
575 out_deg_sequence = nx.utils.make_list_of_ints(out_deg_sequence)
577 # Process the sequences and form two heaps to store degree pairs with
578 # either zero or nonzero out degrees
579 sumin, sumout = 0, 0
580 nin, nout = len(in_deg_sequence), len(out_deg_sequence)
581 maxn = max(nin, nout)
582 G = nx.empty_graph(maxn, create_using, default=nx.DiGraph)
583 if maxn == 0:
584 return G
585 maxin = 0
586 stubheap, zeroheap = [], []
587 for n in range(maxn):
588 in_deg, out_deg = 0, 0
589 if n < nout:
590 out_deg = out_deg_sequence[n]
591 if n < nin:
592 in_deg = in_deg_sequence[n]
593 if in_deg < 0 or out_deg < 0:
594 raise nx.NetworkXError(
595 "Invalid degree sequences. Sequence values must be positive."
596 )
597 sumin, sumout, maxin = sumin + in_deg, sumout + out_deg, max(maxin, in_deg)
598 if in_deg > 0:
599 stubheap.append((-1 * out_deg, -1 * in_deg, n))
600 elif out_deg > 0:
601 zeroheap.append((-1 * out_deg, n))
602 if sumin != sumout:
603 raise nx.NetworkXError(
604 "Invalid degree sequences. Sequences must have equal sums."
605 )
606 heapq.heapify(stubheap)
607 heapq.heapify(zeroheap)
609 modstubs = [(0, 0, 0)] * (maxin + 1)
610 # Successively reduce degree sequence by removing the maximum
611 while stubheap:
612 # Remove first value in the sequence with a non-zero in degree
613 (freeout, freein, target) = heapq.heappop(stubheap)
614 freein *= -1
615 if freein > len(stubheap) + len(zeroheap):
616 raise nx.NetworkXError("Non-digraphical integer sequence")
618 # Attach arcs from the nodes with the most stubs
619 mslen = 0
620 for i in range(freein):
621 if zeroheap and (not stubheap or stubheap[0][0] > zeroheap[0][0]):
622 (stubout, stubsource) = heapq.heappop(zeroheap)
623 stubin = 0
624 else:
625 (stubout, stubin, stubsource) = heapq.heappop(stubheap)
626 if stubout == 0:
627 raise nx.NetworkXError("Non-digraphical integer sequence")
628 G.add_edge(stubsource, target)
629 # Check if source is now totally connected
630 if stubout + 1 < 0 or stubin < 0:
631 modstubs[mslen] = (stubout + 1, stubin, stubsource)
632 mslen += 1
634 # Add the nodes back to the heaps that still have available stubs
635 for i in range(mslen):
636 stub = modstubs[i]
637 if stub[1] < 0:
638 heapq.heappush(stubheap, stub)
639 else:
640 heapq.heappush(zeroheap, (stub[0], stub[2]))
641 if freeout < 0:
642 heapq.heappush(zeroheap, (freeout, target))
644 return G
647@nx._dispatch(graphs=None)
648def degree_sequence_tree(deg_sequence, create_using=None):
649 """Make a tree for the given degree sequence.
651 A tree has #nodes-#edges=1 so
652 the degree sequence must have
653 len(deg_sequence)-sum(deg_sequence)/2=1
654 """
655 # The sum of the degree sequence must be even (for any undirected graph).
656 degree_sum = sum(deg_sequence)
657 if degree_sum % 2 != 0:
658 msg = "Invalid degree sequence: sum of degrees must be even, not odd"
659 raise nx.NetworkXError(msg)
660 if len(deg_sequence) - degree_sum // 2 != 1:
661 msg = (
662 "Invalid degree sequence: tree must have number of nodes equal"
663 " to one less than the number of edges"
664 )
665 raise nx.NetworkXError(msg)
666 G = nx.empty_graph(0, create_using)
667 if G.is_directed():
668 raise nx.NetworkXError("Directed Graph not supported")
670 # Sort all degrees greater than 1 in decreasing order.
671 #
672 # TODO Does this need to be sorted in reverse order?
673 deg = sorted((s for s in deg_sequence if s > 1), reverse=True)
675 # make path graph as backbone
676 n = len(deg) + 2
677 nx.add_path(G, range(n))
678 last = n
680 # add the leaves
681 for source in range(1, n - 1):
682 nedges = deg.pop() - 2
683 for target in range(last, last + nedges):
684 G.add_edge(source, target)
685 last += nedges
687 # in case we added one too many
688 if len(G) > len(deg_sequence):
689 G.remove_node(0)
690 return G
693@py_random_state(1)
694@nx._dispatch(graphs=None)
695def random_degree_sequence_graph(sequence, seed=None, tries=10):
696 r"""Returns a simple random graph with the given degree sequence.
698 If the maximum degree $d_m$ in the sequence is $O(m^{1/4})$ then the
699 algorithm produces almost uniform random graphs in $O(m d_m)$ time
700 where $m$ is the number of edges.
702 Parameters
703 ----------
704 sequence : list of integers
705 Sequence of degrees
706 seed : integer, random_state, or None (default)
707 Indicator of random number generation state.
708 See :ref:`Randomness<randomness>`.
709 tries : int, optional
710 Maximum number of tries to create a graph
712 Returns
713 -------
714 G : Graph
715 A graph with the specified degree sequence.
716 Nodes are labeled starting at 0 with an index
717 corresponding to the position in the sequence.
719 Raises
720 ------
721 NetworkXUnfeasible
722 If the degree sequence is not graphical.
723 NetworkXError
724 If a graph is not produced in specified number of tries
726 See Also
727 --------
728 is_graphical, configuration_model
730 Notes
731 -----
732 The generator algorithm [1]_ is not guaranteed to produce a graph.
734 References
735 ----------
736 .. [1] Moshen Bayati, Jeong Han Kim, and Amin Saberi,
737 A sequential algorithm for generating random graphs.
738 Algorithmica, Volume 58, Number 4, 860-910,
739 DOI: 10.1007/s00453-009-9340-1
741 Examples
742 --------
743 >>> sequence = [1, 2, 2, 3]
744 >>> G = nx.random_degree_sequence_graph(sequence, seed=42)
745 >>> sorted(d for n, d in G.degree())
746 [1, 2, 2, 3]
747 """
748 DSRG = DegreeSequenceRandomGraph(sequence, seed)
749 for try_n in range(tries):
750 try:
751 return DSRG.generate()
752 except nx.NetworkXUnfeasible:
753 pass
754 raise nx.NetworkXError(f"failed to generate graph in {tries} tries")
757class DegreeSequenceRandomGraph:
758 # class to generate random graphs with a given degree sequence
759 # use random_degree_sequence_graph()
760 def __init__(self, degree, rng):
761 if not nx.is_graphical(degree):
762 raise nx.NetworkXUnfeasible("degree sequence is not graphical")
763 self.rng = rng
764 self.degree = list(degree)
765 # node labels are integers 0,...,n-1
766 self.m = sum(self.degree) / 2.0 # number of edges
767 try:
768 self.dmax = max(self.degree) # maximum degree
769 except ValueError:
770 self.dmax = 0
772 def generate(self):
773 # remaining_degree is mapping from int->remaining degree
774 self.remaining_degree = dict(enumerate(self.degree))
775 # add all nodes to make sure we get isolated nodes
776 self.graph = nx.Graph()
777 self.graph.add_nodes_from(self.remaining_degree)
778 # remove zero degree nodes
779 for n, d in list(self.remaining_degree.items()):
780 if d == 0:
781 del self.remaining_degree[n]
782 if len(self.remaining_degree) > 0:
783 # build graph in three phases according to how many unmatched edges
784 self.phase1()
785 self.phase2()
786 self.phase3()
787 return self.graph
789 def update_remaining(self, u, v, aux_graph=None):
790 # decrement remaining nodes, modify auxiliary graph if in phase3
791 if aux_graph is not None:
792 # remove edges from auxiliary graph
793 aux_graph.remove_edge(u, v)
794 if self.remaining_degree[u] == 1:
795 del self.remaining_degree[u]
796 if aux_graph is not None:
797 aux_graph.remove_node(u)
798 else:
799 self.remaining_degree[u] -= 1
800 if self.remaining_degree[v] == 1:
801 del self.remaining_degree[v]
802 if aux_graph is not None:
803 aux_graph.remove_node(v)
804 else:
805 self.remaining_degree[v] -= 1
807 def p(self, u, v):
808 # degree probability
809 return 1 - self.degree[u] * self.degree[v] / (4.0 * self.m)
811 def q(self, u, v):
812 # remaining degree probability
813 norm = max(self.remaining_degree.values()) ** 2
814 return self.remaining_degree[u] * self.remaining_degree[v] / norm
816 def suitable_edge(self):
817 """Returns True if and only if an arbitrary remaining node can
818 potentially be joined with some other remaining node.
820 """
821 nodes = iter(self.remaining_degree)
822 u = next(nodes)
823 return any(v not in self.graph[u] for v in nodes)
825 def phase1(self):
826 # choose node pairs from (degree) weighted distribution
827 rem_deg = self.remaining_degree
828 while sum(rem_deg.values()) >= 2 * self.dmax**2:
829 u, v = sorted(random_weighted_sample(rem_deg, 2, self.rng))
830 if self.graph.has_edge(u, v):
831 continue
832 if self.rng.random() < self.p(u, v): # accept edge
833 self.graph.add_edge(u, v)
834 self.update_remaining(u, v)
836 def phase2(self):
837 # choose remaining nodes uniformly at random and use rejection sampling
838 remaining_deg = self.remaining_degree
839 rng = self.rng
840 while len(remaining_deg) >= 2 * self.dmax:
841 while True:
842 u, v = sorted(rng.sample(list(remaining_deg.keys()), 2))
843 if self.graph.has_edge(u, v):
844 continue
845 if rng.random() < self.q(u, v):
846 break
847 if rng.random() < self.p(u, v): # accept edge
848 self.graph.add_edge(u, v)
849 self.update_remaining(u, v)
851 def phase3(self):
852 # build potential remaining edges and choose with rejection sampling
853 potential_edges = combinations(self.remaining_degree, 2)
854 # build auxiliary graph of potential edges not already in graph
855 H = nx.Graph(
856 [(u, v) for (u, v) in potential_edges if not self.graph.has_edge(u, v)]
857 )
858 rng = self.rng
859 while self.remaining_degree:
860 if not self.suitable_edge():
861 raise nx.NetworkXUnfeasible("no suitable edges left")
862 while True:
863 u, v = sorted(rng.choice(list(H.edges())))
864 if rng.random() < self.q(u, v):
865 break
866 if rng.random() < self.p(u, v): # accept edge
867 self.graph.add_edge(u, v)
868 self.update_remaining(u, v, aux_graph=H)