\Mh|8dZddlZddlZgdZejddddZejddddZejddddZej j d ejdddd dd d Z ej d ej dejdddiidddZej j d ejddddd dddZdS)z3Provides explicit constructions of expander graphs.N)margulis_gabber_galil_graphchordal_cycle_graph paley_graphmaybe_regular_expanderis_regular_expanderrandom_regular_expander_graphT)graphs returns_graphctjd|tj}|s|sd}tj|t jt|dD]]\}}|d|zz|z|f|d|zdzz|z|f||d|zz|zf||d|zdzz|zffD]\}}| ||f||f ^d|d|j d <|S) aReturns 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. rdefault0`create_using` must be an undirected multigraph.)repeatzmargulis_gabber_galil_graph()name) nx empty_graph MultiGraph is_directed is_multigraph NetworkXError itertoolsproductrangeadd_edgegraph)n create_usingGmsgxyuvs ]/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/generators/expanders.pyrr1s.4 q, >>>A}}$aoo//$@s###!%((1555''1!a%i1_a 1q519o "A & QUa a!eaiA% &   ' 'DAq JJ1v1v & & & &  ':Q999AGFO Hctjd|tj}|s|sd}tj|t |D]L}|dz |z}|dz|z}|dkrt||dz |nd}|||fD]}|||Md|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. rr rrrzchordal_cycle_graph(rr) rrrrrrrpowrr) pr r!r"r#leftrightchordr$s r'rr\sN q, >>>A}}$aoo//$@s### 1XXA{Q! %&EEAq1ua   qu%  A JJq!     1Q111AGFO Hr(cXtjd|tj}|rd}tj|fdt dD}t D]#}|D]}||||zz$dd|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). rr z&`create_using` cannot be a multigraph.c8h|]}|dzzdk|dzzS)rr).0r#r+s r' zpaley_graph..s.EEEadaZ1__1a41*___r(rzpaley(rr)rrDiGraphrrrrr)r+r r!r" square_setr#x2s` r'rrsX q, ;;;A$6s### FEEEeAqkkEEEJ 1XX(( ( (B JJq1r6Q, ' ' ' ' (#qmmmAGFO Hr(seeddr max_triesr7c ddl}|dkrtjd|dkstjd|dzdkstjd|dz |kstjd|dzd |d tj||}|dkr|Sg}t t |dzD]}|} t |dz|zkr| dz} ||dz } | |dz  fd tj | d D} t | |kr*| |  | | dkrtjdt |dz|zk| |S)aUtility 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 rNrzn must be a positive integerrz$d must be greater than or equal to 2zd must be evenzNeed n-1>= d to have room for z independent cycles with z nodesc6h|]\}}||fv ||fv||fSr1r1)r2r%r&edgess r'r3z)maybe_regular_expander..;sHAqq6&&Aq6+>+>A+>+>+>r(T)cyclicz-Too many iterations in maybe_regular_expander)numpyrrrsetrlen permutationtolistappendutilspairwiseupdateadd_edges_from) rdr r:r7npr!cyclesi iterationscycle new_edgesr=s @r'rrs~1uu=>>> FFEFFF EQJJ/000 EQJJ WQ!V W Wa W W W    q,''A1uu F EEE16]]XX %jjQUaK'' !OJ$$QU++2244E LLQ   H--eD-AAI9~~"" e$$$ Y'''Q&'VWWW'%jjQUaK''*U Hr(directed multigraphr!weightr)preserve_edge_attrsepsiloncddl}ddl}|dkrtjdtj|sdStj|j\}}tj|t}|j j |ddd}t|}tt|d||d z z|zkS) 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 rNzepsilon must be non negativeF)dtypeLMr)whichkreturn_eigenvectorsr)r?scipyrr is_regularrEarbitrary_elementdegreeadjacency_matrixfloatsparselinalgeigshminboolabssqrt) r!rUrJsp_rIAlamslambda2s r'rrNsb{{=>>> =  u 8 % %ah / /DAq AU+++A 9  ! !!41% ! P PD$iiG G q2771q5>>1G;; < <`. 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 r9rTr)rrIr r:r7rz4Too many iterations in random_regular_expander_graph)rrrr)rrIrUr r:r7r!rMs r'rrsp  1<94   AJ!!W555 a "1<94    ??"F "!W555  Hr()N)__doc__rnetworkxr__all__ _dispatchablerrrrE decoratorsnp_random_staternot_implemented_forrrr1r(r'rvs"99   TT222' ' ' 32' TT222< < < 32< ~T2229 9 9 329 x$$V,,T222154p p p p 32-,p fj))l++sXqM&:;;;&'@=@=@=@=<;,+*)@=F$$V,,T222TStF F F F 32-,F F F r(