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1"""Provides explicit constructions of expander graphs."""
3import itertools
5import networkx as nx
7__all__ = [
8 "margulis_gabber_galil_graph",
9 "chordal_cycle_graph",
10 "paley_graph",
11 "maybe_regular_expander",
12 "maybe_regular_expander_graph",
13 "is_regular_expander",
14 "random_regular_expander_graph",
15]
18# Other discrete torus expanders can be constructed by using the following edge
19# sets. For more information, see Chapter 4, "Expander Graphs", in
20# "Pseudorandomness", by Salil Vadhan.
21#
22# For a directed expander, add edges from (x, y) to:
23#
24# (x, y),
25# ((x + 1) % n, y),
26# (x, (y + 1) % n),
27# (x, (x + y) % n),
28# (-y % n, x)
29#
30# For an undirected expander, add the reverse edges.
31#
32# Also appearing in the paper of Gabber and Galil:
33#
34# (x, y),
35# (x, (x + y) % n),
36# (x, (x + y + 1) % n),
37# ((x + y) % n, y),
38# ((x + y + 1) % n, y)
39#
40# and:
41#
42# (x, y),
43# ((x + 2*y) % n, y),
44# ((x + (2*y + 1)) % n, y),
45# ((x + (2*y + 2)) % n, y),
46# (x, (y + 2*x) % n),
47# (x, (y + (2*x + 1)) % n),
48# (x, (y + (2*x + 2)) % n),
49#
50@nx._dispatchable(graphs=None, returns_graph=True)
51def margulis_gabber_galil_graph(n, create_using=None):
52 r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes.
54 The undirected MultiGraph is regular with degree `8`. Nodes are integer
55 pairs. The second-largest eigenvalue of the adjacency matrix of the graph
56 is at most `5 \sqrt{2}`, regardless of `n`.
58 Parameters
59 ----------
60 n : int
61 Determines the number of nodes in the graph: `n^2`.
62 create_using : NetworkX graph constructor, optional (default MultiGraph)
63 Graph type to create. If graph instance, then cleared before populated.
65 Returns
66 -------
67 G : graph
68 The constructed undirected multigraph.
70 Raises
71 ------
72 NetworkXError
73 If the graph is directed or not a multigraph.
75 """
76 G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
77 if G.is_directed() or not G.is_multigraph():
78 msg = "`create_using` must be an undirected multigraph."
79 raise nx.NetworkXError(msg)
81 for x, y in itertools.product(range(n), repeat=2):
82 for u, v in (
83 ((x + 2 * y) % n, y),
84 ((x + (2 * y + 1)) % n, y),
85 (x, (y + 2 * x) % n),
86 (x, (y + (2 * x + 1)) % n),
87 ):
88 G.add_edge((x, y), (u, v))
89 G.graph["name"] = f"margulis_gabber_galil_graph({n})"
90 return G
93@nx._dispatchable(graphs=None, returns_graph=True)
94def chordal_cycle_graph(p, create_using=None):
95 """Returns the chordal cycle graph on `p` nodes.
97 The returned graph is a cycle graph on `p` nodes with chords joining each
98 vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit)
99 3-regular expander [1]_.
101 `p` *must* be a prime number.
103 Parameters
104 ----------
105 p : a prime number
107 The number of vertices in the graph. This also indicates where the
108 chordal edges in the cycle will be created.
110 create_using : NetworkX graph constructor, optional (default=nx.Graph)
111 Graph type to create. If graph instance, then cleared before populated.
113 Returns
114 -------
115 G : graph
116 The constructed undirected multigraph.
118 Raises
119 ------
120 NetworkXError
122 If `create_using` indicates directed or not a multigraph.
124 References
125 ----------
127 .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and
128 invariant measures", volume 125 of Progress in Mathematics.
129 Birkhäuser Verlag, Basel, 1994.
131 """
132 G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
133 if G.is_directed() or not G.is_multigraph():
134 msg = "`create_using` must be an undirected multigraph."
135 raise nx.NetworkXError(msg)
137 for x in range(p):
138 left = (x - 1) % p
139 right = (x + 1) % p
140 # Here we apply Fermat's Little Theorem to compute the multiplicative
141 # inverse of x in Z/pZ. By Fermat's Little Theorem,
142 #
143 # x^p = x (mod p)
144 #
145 # Therefore,
146 #
147 # x * x^(p - 2) = 1 (mod p)
148 #
149 # The number 0 is a special case: we just let its inverse be itself.
150 chord = pow(x, p - 2, p) if x > 0 else 0
151 for y in (left, right, chord):
152 G.add_edge(x, y)
153 G.graph["name"] = f"chordal_cycle_graph({p})"
154 return G
157@nx._dispatchable(graphs=None, returns_graph=True)
158def paley_graph(p, create_using=None):
159 r"""Returns the Paley $\frac{(p-1)}{2}$ -regular graph on $p$ nodes.
161 The returned graph is a graph on $\mathbb{Z}/p\mathbb{Z}$ with edges between $x$ and $y$
162 if and only if $x-y$ is a nonzero square in $\mathbb{Z}/p\mathbb{Z}$.
164 If $p \equiv 1 \pmod 4$, $-1$ is a square in
165 $\mathbb{Z}/p\mathbb{Z}$ and therefore $x-y$ is a square if and
166 only if $y-x$ is also a square, i.e the edges in the Paley graph are symmetric.
168 If $p \equiv 3 \pmod 4$, $-1$ is not a square in $\mathbb{Z}/p\mathbb{Z}$
169 and therefore either $x-y$ or $y-x$ is a square in $\mathbb{Z}/p\mathbb{Z}$ but not both.
171 Note that a more general definition of Paley graphs extends this construction
172 to graphs over $q=p^n$ vertices, by using the finite field $F_q$ instead of
173 $\mathbb{Z}/p\mathbb{Z}$.
174 This construction requires to compute squares in general finite fields and is
175 not what is implemented here (i.e `paley_graph(25)` does not return the true
176 Paley graph associated with $5^2$).
178 Parameters
179 ----------
180 p : int, an odd prime number.
182 create_using : NetworkX graph constructor, optional (default=nx.Graph)
183 Graph type to create. If graph instance, then cleared before populated.
185 Returns
186 -------
187 G : graph
188 The constructed directed graph.
190 Raises
191 ------
192 NetworkXError
193 If the graph is a multigraph.
195 References
196 ----------
197 Chapter 13 in B. Bollobas, Random Graphs. Second edition.
198 Cambridge Studies in Advanced Mathematics, 73.
199 Cambridge University Press, Cambridge (2001).
200 """
201 G = nx.empty_graph(0, create_using, default=nx.DiGraph)
202 if G.is_multigraph():
203 msg = "`create_using` cannot be a multigraph."
204 raise nx.NetworkXError(msg)
206 # Compute the squares in Z/pZ.
207 # Make it a set to uniquify (there are exactly (p-1)/2 squares in Z/pZ
208 # when is prime).
209 square_set = {(x**2) % p for x in range(1, p) if (x**2) % p != 0}
211 for x in range(p):
212 for x2 in square_set:
213 G.add_edge(x, (x + x2) % p)
214 G.graph["name"] = f"paley({p})"
215 return G
218@nx.utils.decorators.np_random_state("seed")
219@nx._dispatchable(graphs=None, returns_graph=True)
220def maybe_regular_expander_graph(n, d, *, create_using=None, max_tries=100, seed=None):
221 r"""Utility for creating a random regular expander.
223 Returns a random $d$-regular graph on $n$ nodes which is an expander
224 graph with very good probability.
226 Parameters
227 ----------
228 n : int
229 The number of nodes.
230 d : int
231 The degree of each node.
232 create_using : Graph Instance or Constructor
233 Indicator of type of graph to return.
234 If a Graph-type instance, then clear and use it.
235 If a constructor, call it to create an empty graph.
236 Use the Graph constructor by default.
237 max_tries : int. (default: 100)
238 The number of allowed loops when generating each independent cycle
239 seed : (default: None)
240 Seed used to set random number generation state. See :ref`Randomness<randomness>`.
242 Notes
243 -----
244 The nodes are numbered from $0$ to $n - 1$.
246 The graph is generated by taking $d / 2$ random independent cycles.
248 Joel Friedman proved that in this model the resulting
249 graph is an expander with probability
250 $1 - O(n^{-\tau})$ where $\tau = \lceil (\sqrt{d - 1}) / 2 \rceil - 1$. [1]_
252 Examples
253 --------
254 >>> G = nx.maybe_regular_expander_graph(n=200, d=6, seed=8020)
256 Returns
257 -------
258 G : graph
259 The constructed undirected graph.
261 Raises
262 ------
263 NetworkXError
264 If $d % 2 != 0$ as the degree must be even.
265 If $n - 1$ is less than $ 2d $ as the graph is complete at most.
266 If max_tries is reached
268 See Also
269 --------
270 is_regular_expander
271 random_regular_expander_graph
273 References
274 ----------
275 .. [1] Joel Friedman,
276 A Proof of Alon's Second Eigenvalue Conjecture and Related Problems, 2004
277 https://arxiv.org/abs/cs/0405020
279 """
281 import numpy as np
283 if n < 1:
284 raise nx.NetworkXError("n must be a positive integer")
286 if not (d >= 2):
287 raise nx.NetworkXError("d must be greater than or equal to 2")
289 if not (d % 2 == 0):
290 raise nx.NetworkXError("d must be even")
292 if not (n - 1 >= d):
293 raise nx.NetworkXError(
294 f"Need n-1>= d to have room for {d // 2} independent cycles with {n} nodes"
295 )
297 G = nx.empty_graph(n, create_using)
299 if n < 2:
300 return G
302 cycles = []
303 edges = set()
305 # Create d / 2 cycles
306 for i in range(d // 2):
307 iterations = max_tries
308 # Make sure the cycles are independent to have a regular graph
309 while len(edges) != (i + 1) * n:
310 iterations -= 1
311 # Faster than random.permutation(n) since there are only
312 # (n-1)! distinct cycles against n! permutations of size n
313 cycle = seed.permutation(n - 1).tolist()
314 cycle.append(n - 1)
316 new_edges = {
317 (u, v)
318 for u, v in nx.utils.pairwise(cycle, cyclic=True)
319 if (u, v) not in edges and (v, u) not in edges
320 }
321 # If the new cycle has no edges in common with previous cycles
322 # then add it to the list otherwise try again
323 if len(new_edges) == n:
324 cycles.append(cycle)
325 edges.update(new_edges)
327 if iterations == 0:
328 msg = "Too many iterations in maybe_regular_expander_graph"
329 raise nx.NetworkXError(msg)
331 G.add_edges_from(edges)
333 return G
336def maybe_regular_expander(n, d, *, create_using=None, max_tries=100, seed=None):
337 """
338 .. deprecated:: 3.6
339 `maybe_regular_expander` is a deprecated alias
340 for `maybe_regular_expander_graph`.
341 Use `maybe_regular_expander_graph` instead.
342 """
343 import warnings
345 warnings.warn(
346 "maybe_regular_expander is deprecated, "
347 "use `maybe_regular_expander_graph` instead.",
348 category=DeprecationWarning,
349 stacklevel=2,
350 )
351 return maybe_regular_expander_graph(
352 n, d, create_using=create_using, max_tries=max_tries, seed=seed
353 )
356@nx.utils.not_implemented_for("directed")
357@nx.utils.not_implemented_for("multigraph")
358@nx._dispatchable(preserve_edge_attrs={"G": {"weight": 1}})
359def is_regular_expander(G, *, epsilon=0):
360 r"""Determines whether the graph G is a regular expander. [1]_
362 An expander graph is a sparse graph with strong connectivity properties.
364 More precisely, this helper checks whether the graph is a
365 regular $(n, d, \lambda)$-expander with $\lambda$ close to
366 the Alon-Boppana bound and given by
367 $\lambda = 2 \sqrt{d - 1} + \epsilon$. [2]_
369 In the case where $\epsilon = 0$ then if the graph successfully passes the test
370 it is a Ramanujan graph. [3]_
372 A Ramanujan graph has spectral gap almost as large as possible, which makes them
373 excellent expanders.
375 Parameters
376 ----------
377 G : NetworkX graph
378 epsilon : int, float, default=0
380 Returns
381 -------
382 bool
383 Whether the given graph is a regular $(n, d, \lambda)$-expander
384 where $\lambda = 2 \sqrt{d - 1} + \epsilon$.
386 Examples
387 --------
388 >>> G = nx.random_regular_expander_graph(20, 4)
389 >>> nx.is_regular_expander(G)
390 True
392 See Also
393 --------
394 maybe_regular_expander_graph
395 random_regular_expander_graph
397 References
398 ----------
399 .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
400 .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
401 .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph
403 """
405 import numpy as np
406 import scipy as sp
408 if epsilon < 0:
409 raise nx.NetworkXError("epsilon must be non negative")
411 if not nx.is_regular(G):
412 return False
414 _, d = nx.utils.arbitrary_element(G.degree)
416 A = nx.adjacency_matrix(G, dtype=float)
417 lams = sp.sparse.linalg.eigsh(A, which="LM", k=2, return_eigenvectors=False)
419 # lambda2 is the second biggest eigenvalue
420 lambda2 = min(lams)
422 # Use bool() to convert numpy scalar to Python Boolean
423 return bool(abs(lambda2) < 2 * np.sqrt(d - 1) + epsilon)
426@nx.utils.decorators.np_random_state("seed")
427@nx._dispatchable(graphs=None, returns_graph=True)
428def random_regular_expander_graph(
429 n, d, *, epsilon=0, create_using=None, max_tries=100, seed=None
430):
431 r"""Returns a random regular expander graph on $n$ nodes with degree $d$.
433 An expander graph is a sparse graph with strong connectivity properties. [1]_
435 More precisely the returned graph is a $(n, d, \lambda)$-expander with
436 $\lambda = 2 \sqrt{d - 1} + \epsilon$, close to the Alon-Boppana bound. [2]_
438 In the case where $\epsilon = 0$ it returns a Ramanujan graph.
439 A Ramanujan graph has spectral gap almost as large as possible,
440 which makes them excellent expanders. [3]_
442 Parameters
443 ----------
444 n : int
445 The number of nodes.
446 d : int
447 The degree of each node.
448 epsilon : int, float, default=0
449 max_tries : int, (default: 100)
450 The number of allowed loops,
451 also used in the `maybe_regular_expander_graph` utility
452 seed : (default: None)
453 Seed used to set random number generation state. See :ref`Randomness<randomness>`.
455 Raises
456 ------
457 NetworkXError
458 If max_tries is reached
460 Examples
461 --------
462 >>> G = nx.random_regular_expander_graph(20, 4)
463 >>> nx.is_regular_expander(G)
464 True
466 Notes
467 -----
468 This loops over `maybe_regular_expander_graph` and can be slow when
469 $n$ is too big or $\epsilon$ too small.
471 See Also
472 --------
473 maybe_regular_expander_graph
474 is_regular_expander
476 References
477 ----------
478 .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph
479 .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound
480 .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph
482 """
483 G = maybe_regular_expander_graph(
484 n, d, create_using=create_using, max_tries=max_tries, seed=seed
485 )
486 iterations = max_tries
488 while not is_regular_expander(G, epsilon=epsilon):
489 iterations -= 1
490 G = maybe_regular_expander_graph(
491 n=n, d=d, create_using=create_using, max_tries=max_tries, seed=seed
492 )
494 if iterations == 0:
495 raise nx.NetworkXError(
496 "Too many iterations in random_regular_expander_graph"
497 )
499 return G