o ]ZŽhw8ã@sdZddlZddlZgd¢Zejdddddd„ƒZejdddddd „ƒZejddddd d „ƒZej j   d ¡ejddddd ddœdd„ƒƒZ ej   d¡ej   d¡ejdddiidddœdd„ƒƒƒZej j   d ¡ejdddddd ddœdd„ƒƒZdS)z3Provides explicit constructions of expander graphs.éN)Úmargulis_gabber_galil_graphÚchordal_cycle_graphÚ paley_graphÚmaybe_regular_expanderÚis_regular_expanderÚrandom_regular_expander_graphT)ZgraphsZ returns_graphcCsÔtjd|tjd}| ¡s| ¡sd}t |¡‚tjt|ƒddD]=\}}|d|||f|d|d||f||d||f||d|d|ffD]\}}|  ||f||f¡qOq!d|›d|j d <|S) aÐReturns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes. The undirected MultiGraph is regular with degree `8`. Nodes are integer pairs. The second-largest eigenvalue of the adjacency matrix of the graph is at most `5 \sqrt{2}`, regardless of `n`. Parameters ---------- n : int Determines the number of nodes in the graph: `n^2`. create_using : NetworkX graph constructor, optional (default MultiGraph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If the graph is directed or not a multigraph. r©Údefaultú0`create_using` must be an undirected multigraph.é)Úrepeatézmargulis_gabber_galil_graph(ú)Úname) ÚnxÚ empty_graphÚ MultiGraphÚ is_directedÚ is_multigraphÚ NetworkXErrorÚ itertoolsÚproductÚrangeÚadd_edgeÚgraph)ÚnÚ create_usingÚGÚmsgÚxÚyÚuÚv©r#úL/var/www/auris/lib/python3.10/site-packages/networkx/generators/expanders.pyr1s  üúrc Cs¤tjd|tjd}| ¡s| ¡sd}t |¡‚t|ƒD]*}|d|}|d|}|dkr6t||d|ƒnd}|||fD]}| ||¡q=qd|›d|j d<|S) uReturns the chordal cycle graph on `p` nodes. The returned graph is a cycle graph on `p` nodes with chords joining each vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit) 3-regular expander [1]_. `p` *must* be a prime number. Parameters ---------- p : a prime number The number of vertices in the graph. This also indicates where the chordal edges in the cycle will be created. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If `create_using` indicates directed or not a multigraph. References ---------- .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and invariant measures", volume 125 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1994. rrr r r zchordal_cycle_graph(rr) rrrrrrrÚpowrr) ÚprrrrÚleftÚrightZchordr r#r#r$r\s'     ÿrcs‚tjd|tjd}| ¡rd}t |¡‚‡fdd„tdˆƒDƒ}tˆƒD]}|D] }| |||ˆ¡q(q$dˆ›d|jd <|S) a%Returns the Paley $\frac{(p-1)}{2}$ -regular graph on $p$ nodes. The returned graph is a graph on $\mathbb{Z}/p\mathbb{Z}$ with edges between $x$ and $y$ if and only if $x-y$ is a nonzero square in $\mathbb{Z}/p\mathbb{Z}$. If $p \equiv 1 \pmod 4$, $-1$ is a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore $x-y$ is a square if and only if $y-x$ is also a square, i.e the edges in the Paley graph are symmetric. If $p \equiv 3 \pmod 4$, $-1$ is not a square in $\mathbb{Z}/p\mathbb{Z}$ and therefore either $x-y$ or $y-x$ is a square in $\mathbb{Z}/p\mathbb{Z}$ but not both. Note that a more general definition of Paley graphs extends this construction to graphs over $q=p^n$ vertices, by using the finite field $F_q$ instead of $\mathbb{Z}/p\mathbb{Z}$. This construction requires to compute squares in general finite fields and is not what is implemented here (i.e `paley_graph(25)` does not return the true Paley graph associated with $5^2$). Parameters ---------- p : int, an odd prime number. create_using : NetworkX graph constructor, optional (default=nx.Graph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed directed graph. Raises ------ NetworkXError If the graph is a multigraph. References ---------- Chapter 13 in B. Bollobas, Random Graphs. Second edition. Cambridge Studies in Advanced Mathematics, 73. Cambridge University Press, Cambridge (2001). rrz&`create_using` cannot be a multigraph.cs(h|]}|dˆdkr|dˆ’qS)r rr#)Ú.0r©r&r#r$Ú Îs(zpaley_graph..r zpaley(rr)rrZDiGraphrrrrr)r&rrrZ square_setrZx2r#r*r$rœs*  ÿrÚseedéd©rÚ max_triesr,c sPddl}|dkr t d¡‚|dkst d¡‚|ddks!t d¡‚|d|ks5t d|d›d |›d ¡‚t ||¡}|dkrA|Sg}tƒ‰t|dƒD]T}|} tˆƒ|d|kr | d8} | |d¡ ¡} |   |d¡‡fd d „tj j | d dDƒ} t| ƒ|kr|  | ¡ˆ  | ¡| dkr–t d¡‚tˆƒ|d|ksZqL|  ˆ¡|S)u­Utility for creating a random regular expander. Returns a random $d$-regular graph on $n$ nodes which is an expander graph with very good probability. Parameters ---------- n : int The number of nodes. d : int The degree of each node. create_using : Graph Instance or Constructor Indicator of type of graph to return. If a Graph-type instance, then clear and use it. If a constructor, call it to create an empty graph. Use the Graph constructor by default. max_tries : int. (default: 100) The number of allowed loops when generating each independent cycle seed : (default: None) Seed used to set random number generation state. See :ref`Randomness`. Notes ----- The nodes are numbered from $0$ to $n - 1$. The graph is generated by taking $d / 2$ random independent cycles. Joel Friedman proved that in this model the resulting graph is an expander with probability $1 - O(n^{-\tau})$ where $\tau = \lceil (\sqrt{d - 1}) / 2 \rceil - 1$. [1]_ Examples -------- >>> G = nx.maybe_regular_expander(n=200, d=6, seed=8020) Returns ------- G : graph The constructed undirected graph. Raises ------ NetworkXError If $d % 2 != 0$ as the degree must be even. If $n - 1$ is less than $ 2d $ as the graph is complete at most. If max_tries is reached See Also -------- is_regular_expander random_regular_expander_graph References ---------- .. [1] Joel Friedman, A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems, 2004 https://arxiv.org/abs/cs/0405020 rNr zn must be a positive integerr z$d must be greater than or equal to 2zd must be evenzNeed n-1>= d to have room for z independent cycles with z nodescs0h|]\}}||fˆvr||fˆvr||f’qSr#r#)r)r!r"©Úedgesr#r$r+9s þz)maybe_regular_expander..T)Zcyclicz-Too many iterations in maybe_regular_expander)ÚnumpyrrrÚsetrÚlenZ permutationÚtolistÚappendÚutilsÚpairwiseÚupdateZadd_edges_from) rÚdrr/r,ÚnprZcyclesÚiÚ iterationsÚcycleZ new_edgesr#r0r$r×sD?     ÿ  þ    í€ rZdirectedZ multigraphrÚweightr )Zpreserve_edge_attrs©Úepsilonc CsŽddl}ddlm}|dkrt d¡‚t |¡sdStj |j¡\}}tj |t d}||dddd }t |ƒ}t t |ƒd| |d ¡|kƒS) aDetermines whether the graph G is a regular expander. [1]_ An expander graph is a sparse graph with strong connectivity properties. More precisely, this helper checks whether the graph is a regular $(n, d, \lambda)$-expander with $\lambda$ close to the Alon-Boppana bound and given by $\lambda = 2 \sqrt{d - 1} + \epsilon$. [2]_ In the case where $\epsilon = 0$ then if the graph successfully passes the test it is a Ramanujan graph. [3]_ A Ramanujan graph has spectral gap almost as large as possible, which makes them excellent expanders. Parameters ---------- G : NetworkX graph epsilon : int, float, default=0 Returns ------- bool Whether the given graph is a regular $(n, d, \lambda)$-expander where $\lambda = 2 \sqrt{d - 1} + \epsilon$. Examples -------- >>> G = nx.random_regular_expander_graph(20, 4) >>> nx.is_regular_expander(G) True See Also -------- maybe_regular_expander random_regular_expander_graph References ---------- .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph rN)Úeigshzepsilon must be non negativeF)ZdtypeZLMr )ÚwhichÚkZreturn_eigenvectorsr )r2Zscipy.sparse.linalgrBrrZ is_regularr7Zarbitrary_elementZdegreeZadjacency_matrixÚfloatÚminÚboolÚabsÚsqrt) rrAr;rBÚ_r:ÚAZlamsZlambda2r#r#r$rLs1   "r)rArr/r,cCs^t|||||d}|}t||ds-|d8}t|||||d}|dkr't d¡‚t||dr|S)aReturns a random regular expander graph on $n$ nodes with degree $d$. An expander graph is a sparse graph with strong connectivity properties. [1]_ More precisely the returned graph is a $(n, d, \lambda)$-expander with $\lambda = 2 \sqrt{d - 1} + \epsilon$, close to the Alon-Boppana bound. [2]_ In the case where $\epsilon = 0$ it returns a Ramanujan graph. A Ramanujan graph has spectral gap almost as large as possible, which makes them excellent expanders. [3]_ Parameters ---------- n : int The number of nodes. d : int The degree of each node. epsilon : int, float, default=0 max_tries : int, (default: 100) The number of allowed loops, also used in the maybe_regular_expander utility seed : (default: None) Seed used to set random number generation state. See :ref`Randomness`. Raises ------ NetworkXError If max_tries is reached Examples -------- >>> G = nx.random_regular_expander_graph(20, 4) >>> nx.is_regular_expander(G) True Notes ----- This loops over `maybe_regular_expander` and can be slow when $n$ is too big or $\epsilon$ too small. See Also -------- maybe_regular_expander is_regular_expander References ---------- .. [1] Expander graph, https://en.wikipedia.org/wiki/Expander_graph .. [2] Alon-Boppana bound, https://en.wikipedia.org/wiki/Alon%E2%80%93Boppana_bound .. [3] Ramanujan graphs, https://en.wikipedia.org/wiki/Ramanujan_graph r.r@r )rr:rr/r,rz4Too many iterations in random_regular_expander_graph)rrrr)rr:rArr/r,rr=r#r#r$r’s8 ÿ  ÿÿ ù r)N)Ú__doc__rZnetworkxrÚ__all__Z _dispatchablerrrr7Z decoratorsZnp_random_staterZnot_implemented_forrrr#r#r#r$Ús* *  *  ?  :  s  C ÿ