hw8SrSSKrSSKr/SQr\R "SSS9SSj5r\R "SSS9SSj5r\R "SSS9SSj5r\RRRS 5\R "SSS9SS SS .S j55r \RRS 5\RRS5\R "SSS00S9SS.Sj555r\RRRS 5\R "SSS9SSS SS.Sj55rg)z3Provides explicit constructions of expander graphs.N)margulis_gabber_galil_graphchordal_cycle_graph paley_graphmaybe_regular_expanderis_regular_expanderrandom_regular_expander_graphT)graphs returns_graphc[R"SU[RS9nUR5(dUR 5(dSn[R "U5e[ R"[U5SS9HXupEUSU--U-U4USU-S--U-U4XESU--U-4XESU-S--U-44HupgURXE4Xg45 M MZ SUS3URS 'U$) aReturns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes. The undirected MultiGraph is regular with degree `8`. Nodes are integer pairs. The second-largest eigenvalue of the adjacency matrix of the graph is at most `5 \sqrt{2}`, regardless of `n`. Parameters ---------- n : int Determines the number of nodes in the graph: `n^2`. create_using : NetworkX graph constructor, optional (default MultiGraph) Graph type to create. If graph instance, then cleared before populated. Returns ------- G : graph The constructed undirected multigraph. Raises ------ NetworkXError If the graph is directed or not a multigraph. rdefault0`create_using` must be an undirected multigraph.)repeatzmargulis_gabber_galil_graph()name) nx empty_graph MultiGraph is_directed is_multigraph NetworkXError itertoolsproductrangeadd_edgegraph)n create_usingGmsgxyuvs O/var/www/html/env/lib/python3.13/site-packages/networkx/generators/expanders.pyrr1s4 q, >A}}aoo//@s##!!%(15!a%i1_a 1q519o "A & QUa a!eaiA% &  DA JJvv &  65QCq9AGGFO Hc[R"SU[RS9nUR5(dUR 5(dSn[R "U5e[ U5HFnUS- U-nUS-U-nUS:a[X@S- U5OSnXVU4HnURXH5 M MH SUS3URS'U$) 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$s r'rr\sN q, >A}}aoo//@s## 1XA{Q! %&EA1ua qu%A JJq & -QCq1AGGFO Hr(c[R"SU[RS9nUR5(aSn[R"U5e[ SU5Vs1sHoDS-U-S:wdMUS-U-iM nn[ U5H#nUHnUR XDU-U-5 M M% SUS3URS'U$s snf) aReturns 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.rrzpaley(rr)rrDiGraphrrrrr)r+r r!r"r# square_setx2s r'rrsT q, ;A6s## ',AqkEkdaZ1_*1a41*kJE 1XB JJqr6Q, 'qcmAGGFO H Fs C3 Cseeddr max_triesr3cSSKnUS:a[R"S5eUS:d[R"S5eUS-S:Xd[R"S5eUS- U:d [R"SUS-S US 35e[R"X5nUS:aU$/n[ 5n[ US-5GHn Un [ U5U S-U-:wdMU S-n URUS- 5R5n U RUS- 5 [RRU S S 9V V s1sHupX4U;dMX4U;dMX4iM nn n [ U5U:Xa"URU 5 URU5 U S:Xa[R"S 5e[ U5U S-U-:waMGM URU5 U$s sn n f)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 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 nodesT)cyclicz-Too many iterations in maybe_regular_expander)numpyrrrsetrlen permutationtolistappendutilspairwiseupdateadd_edges_from)rdr r6r3npr!cyclesedgesi iterationscycler%r& new_edgess r'rrs~1u=>> FEFF EQJ/00 EQJ,QTF2KA3f U   q'A1u F EE16] %jQUaK' !OJ$$QU+224E LLQ HH--eD-AADA6&,-6+>A 9~" e$ Y'Q&&'VWW'%jQUaK''0U H!s6 GGGdirected multigraphr!weightr)preserve_edge_attrsepsiloncSSKnSSKJn US:a[R"S5e[R "U5(dg[R RUR5upE[R"U[S9nU"USSSS 9n[U5n[[U5SURUS - 5-U-:5$) 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)eigshzepsilon must be non negativeF)dtypeLMr)whichkreturn_eigenvectorsr)r9scipy.sparse.linalgrRrr is_regularr?arbitrary_elementdegreeadjacency_matrixfloatminboolabssqrt) r!rPrDrR_rCAlamslambda2s r'rrLsb){=>> ==   88 % %ahh /DA AU+A $! ?D$iG G qBGGAEN2W<< ==r()rPr r6r3c[XX4US9nUn[XbS9(d<US-n[XX4US9nUS:Xa[R"S5e[XbS9(dM<U$)apReturns 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 r5rOr)rrCr r6r3rz4Too many iterations in random_regular_expander_graph)rrrr)rrCrPr r6r3r!rHs r'rrsrp  <4 AJ!!5a "<4  ?""F "!55 Hr()N)__doc__rnetworkxr__all__ _dispatchablerrrr? decoratorsnp_random_staternot_implemented_forrrr(r'rosi9 TT2' 3' TT2< 3< ~T27 37 t$$V,T2154p 3-p fj)l+sXqM&:;&'@><,*@>F$$V,T2TStF 3-F r(