h4SrSSKrSSKJrJr /SQr\"S5\"S5\"S5\R"SS 9SS j5555r\"S5\"S5\"S 5\R"SS 9SS j5555r \"S5\"S5\"S5\RSS j5555r \"S5\"S5\"S5\RSSj5555r g)aGFunctions for estimating the small-world-ness of graphs. A small world network is characterized by a small average shortest path length, and a large clustering coefficient. Small-worldness is commonly measured with the coefficient sigma or omega. Both coefficients compare the average clustering coefficient and shortest path length of a given graph against the same quantities for an equivalent random or lattice graph. For more information, see the Wikipedia article on small-world network [1]_. .. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network N)not_implemented_forpy_random_state)random_referencelattice_referencesigmaomegadirected multigraphT) returns_graphc[U5S:a[R"S5e[UR5S:a[R"S5eSSKJnJn [RRnUR5n[UR56upxU"U5n [U5n [R"U5n X-n[X-XS- -S- - 5n Sn [U5GH\nSnX:dM U"SXS9unnUU:XaMUUnUUnUR[!UR#U555nUR[!UR#U555nUUUU4;d UUUU4;aMUUU;aUUU;aUR%UU5 UR%UU5 UR'UU5 UR'UU5 U(aWU"UUU5S:XaIUR'UU5 UR'UU5 UR%UU5 UR%UU5 OU S- n GMOUS- nX:aGMOGM_ U$) aqCompute a random graph by swapping edges of a given graph. Parameters ---------- G : graph An undirected graph with 4 or more nodes. niter : integer (optional, default=1) An edge is rewired approximately `niter` times. connectivity : boolean (optional, default=True) When True, ensure connectivity for the randomized graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- G : graph The randomized graph. Raises ------ NetworkXError If there are fewer than 4 nodes or 2 edges in `G` Notes ----- The implementation is adapted from the algorithm by Maslov and Sneppen (2002) [1]_. References ---------- .. [1] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913.  Graph has fewer than four nodes.Graph has fewer that 2 edgesrcumulative_distributiondiscrete_sequence cdistributionseed)lennx NetworkXErroredgesnetworkx.utilsrr connectivitylocal_edge_connectivitycopyzipdegreenumber_of_edgesintrangechoicelist neighborsadd_edge remove_edge)Gniterrrrr local_connkeysdegreescdfnnodesnedgesntries swapcountinaiciacbds P/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/smallworld.pyrrsV 1vzABB 177|a=>>I88J A$MD !' *C VF    "F NE Fqj$9A$=> ?FI 5\ j)#IHRRxRARA DQ01A DQ01AQ1I~q!Qi1 AQqTM 1a  1a  a# a# Jq!Q$71$<MM!Q'MM!Q'JJq!$JJq!$NI FA?jjD HrcSSKnSSKJnJn [R R n[U5S:a[R"S5e[UR5S:a[R"S5eUR5n[UR56upU"U 5n [U5n [R"U5n UcURX45nURSU 5nURU S- SS 5nUR!S UR#X:X55n[%['UR)U S- 555HHnUR!UUS-SUSUS-5X,U- S- SS24'X,U- S- SS24SSS 2UUSS24'MJ X-n['X-XS- -S- - 5n[%U5GHnSnUU:dMU"SXS 9unnUU:XaMU UnU UnUR+[-UR/U555nUR+[-UR/U555nU R1U5nU R1U5nUUUU4;d UUUU4;aMUUU;aUUU;aUUU4UUU4-UUU4UUU4-:aUR3UU5 UR3UU5 UR5UU5 UR5UU5 U(aWU"UUU5S:XaIUR5UU5 UR5UU5 UR3UU5 UR3UU5 OGMUS- nUU:aGMGM U$) aXLatticize the given graph by swapping edges. Parameters ---------- G : graph An undirected graph. niter : integer (optional, default=1) An edge is rewired approximately niter times. D : numpy.array (optional, default=None) Distance to the diagonal matrix. connectivity : boolean (optional, default=True) Ensure connectivity for the latticized graph when set to True. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- G : graph The latticized graph. Raises ------ NetworkXError If there are fewer than 4 nodes or 2 edges in `G` Notes ----- The implementation is adapted from the algorithm by Sporns et al. [1]_. which is inspired from the original work by Maslov and Sneppen(2002) [2]_. References ---------- .. [1] Sporns, Olaf, and Jonathan D. Zwi. "The small world of the cerebral cortex." Neuroinformatics 2.2 (2004): 145-162. .. [2] Maslov, Sergei, and Kim Sneppen. "Specificity and stability in topology of protein networks." Science 296.5569 (2002): 910-913. rNrrrrrr)rr)numpyrrrrrrrrrr r!r"r#zerosarangeappendwherer%r$ceilr&r'r(indexr)r*)r+r,Drrnprrr-r.r/r0r1r2unumuv max_attempts_r6r7r8r9r:r;r<bidis r=rrzs:bI88J 1vzABB 177|a=>> A$MD !' *C VF    "Fy HHf% & YYq& ! YYvz1b ) IIdBHHRWb5 6s2776A:./0A#%99Qq1uwZ7QU#DAqj1na  Q)*4R40AadG1 NEv&QJ*?!*CDEL 5\ ,)#IHRRxRARA DQ01A DQ01AABABQ1I~q!Qi1 AQqTMRV9qRy(Ab"fI"b& ,AAJJq!$JJq!$MM!Q'MM!Q'$ 1a(;q(@ a+ a+ 1a( 1a( FAG,N Hr>cSSKn//S.n[U5H]n[XUS9nUSR[R "U55 USR[R "U55 M_ [R "U5n[R "U5n URUS5n URUS5n X- X- - n [U 5$)aReturns the small-world coefficient (sigma) of the given graph. The small-world coefficient is defined as: sigma = C/Cr / L/Lr where C and L are respectively the average clustering coefficient and average shortest path length of G. Cr and Lr are respectively the average clustering coefficient and average shortest path length of an equivalent random graph. A graph is commonly classified as small-world if sigma>1. Parameters ---------- G : NetworkX graph An undirected graph. niter : integer (optional, default=100) Approximate number of rewiring per edge to compute the equivalent random graph. nrand : integer (optional, default=10) Number of random graphs generated to compute the average clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- sigma : float The small-world coefficient of G. Notes ----- The implementation is adapted from Humphries et al. [1]_ [2]_. References ---------- .. [1] The brainstem reticular formation is a small-world, not scale-free, network M. D. Humphries, K. Gurney and T. J. Prescott, Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354. .. [2] Humphries and Gurney (2008). "Network 'Small-World-Ness': A Quantitative Method for Determining Canonical Network Equivalence". PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051. rNCLr,rrTrU) rAr%rrDr transitivityaverage_shortest_path_lengthmeanfloat) r+r,nrandrrI randMetricsr5GrrTrUCrLrrs r=rrsb$K 5\ a4 8C 34C ? ? CD A ''*A S! "B S! "B V E <r>cSSKn//S.n[R"U5nUnUS-n[U5H^n [ XUS9n USR [R "U 55 [XUS9n [R"U 5n X:dM\U nM` [R"U5n [R "U5nURUS5nX- X- - n[U5$)aReturns the small-world coefficient (omega) of a graph The small-world coefficient of a graph G is: omega = Lr/L - C/Cl where C and L are respectively the average clustering coefficient and average shortest path length of G. Lr is the average shortest path length of an equivalent random graph and Cl is the average clustering coefficient of an equivalent lattice graph. The small-world coefficient (omega) measures how much G is like a lattice or a random graph. Negative values mean G is similar to a lattice whereas positive values mean G is a random graph. Values close to 0 mean that G has small-world characteristics. Parameters ---------- G : NetworkX graph An undirected graph. niter: integer (optional, default=5) Approximate number of rewiring per edge to compute the equivalent random graph. nrand: integer (optional, default=10) Number of random graphs generated to compute the maximal clustering coefficient (Cr) and average shortest path length (Lr). seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- omega : float The small-world coefficient (omega) Notes ----- The implementation is adapted from the algorithm by Telesford et al. [1]_. References ---------- .. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). "The Ubiquity of Small-World Networks". Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451. doi:10.1089/brain.2011.0038. rNrSrrVrU) rAraverage_clusteringr%rrDrXrrYrZ)r+r,r[rrIr\Clniter_lattice_referenceniter_random_referencerOr]GlCl_temprTrUr_rs r=rr;sn$K  q !B#"QY 5\ aD IC ? ? CDqd K''+ <B a A ''*A S! "B V E <r>)rTN)NTN)d N)rgriN) __doc__networkxrrrr__all__ _dispatchablerrrrr>r=ros."? EZ \"%[ &#![ |Z \"%u &#!u pZ \">#!>BZ \"U#!Ur>