hSrSSKJr SSKJr SSKJr SSKrSSK J r SSK J r \ r S/r\ "S 5\RSS j55rS rS rS rSrg)z, Moody and White algorithm for k-components ) defaultdict) combinations) itemgetterN) edmonds_karp)not_implemented_for k_componentsdirectedc t[[5nUc[n[R"U5H3n[ U5n[ U5S:dMUSRU5 M5 [R"U5Vs/sHoPRU5PM nnUH3n[ U5n[ U5S:dMUSRU5 M5 UGH0n [ U 5S::aM[R"XS9n U S:aX*R[ U 55 [[R"XUS95n U [XU 54/n U (dMU Sup[U5nU RU5n[R"UUS9nUU :a#US:aUUR[ U55 [[R"UUUS95n U (aU RU[UU U545 U (aMGM3 [!U5$s snf![a U R5 N:f=f)a/ Returns the k-component structure of a graph G. A `k`-component is a maximal subgraph of a graph G that has, at least, node connectivity `k`: we need to remove at least `k` nodes to break it into more components. `k`-components have an inherent hierarchical structure because they are nested in terms of connectivity: a connected graph can contain several 2-components, each of which can contain one or more 3-components, and so forth. Parameters ---------- G : NetworkX graph flow_func : function Function to perform the underlying flow computations. Default value :meth:`edmonds_karp`. This function performs better in sparse graphs with right tailed degree distributions. :meth:`shortest_augmenting_path` will perform better in denser graphs. Returns ------- k_components : dict Dictionary with all connectivity levels `k` in the input Graph as keys and a list of sets of nodes that form a k-component of level `k` as values. Raises ------ NetworkXNotImplemented If the input graph is directed. Examples -------- >>> # Petersen graph has 10 nodes and it is triconnected, thus all >>> # nodes are in a single component on all three connectivity levels >>> G = nx.petersen_graph() >>> k_components = nx.k_components(G) Notes ----- Moody and White [1]_ (appendix A) provide an algorithm for identifying k-components in a graph, which is based on Kanevsky's algorithm [2]_ for finding all minimum-size node cut-sets of a graph (implemented in :meth:`all_node_cuts` function): 1. Compute node connectivity, k, of the input graph G. 2. Identify all k-cutsets at the current level of connectivity using Kanevsky's algorithm. 3. Generate new graph components based on the removal of these cutsets. Nodes in a cutset belong to both sides of the induced cut. 4. If the graph is neither complete nor trivial, return to 1; else end. This implementation also uses some heuristics (see [3]_ for details) to speed up the computation. See also -------- node_connectivity all_node_cuts biconnected_components : special case of this function when k=2 k_edge_components : similar to this function, but uses edge-connectivity instead of node-connectivity References ---------- .. [1] Moody, J. and D. White (2003). Social cohesion and embeddedness: A hierarchical conception of social groups. American Sociological Review 68(1), 103--28. http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf .. [2] Kanevsky, A. (1993). Finding all minimum-size separating vertex sets in a graph. Networks 23(6), 533--541. http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract .. [3] Torrents, J. and F. Ferraro (2015). Structural Cohesion: Visualization and Heuristics for Fast Computation. https://arxiv.org/pdf/1503.04476v1 ) flow_func)kr )rlistdefault_flow_funcnxconnected_componentssetlenappendbiconnected_componentssubgraphnode_connectivity all_node_cuts_generate_partitionnext StopIterationpop_reconstruct_k_components)Gr r componentcompc bicomponents bicomponentbicompBrcutsstackparent_k partitionnodesCthis_ks ^/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/connectivity/kcomponents.pyrrstt$L% ,,Q/ 9~ t9q= O " "4 ( 0 ,.+D+DQ+GH+GaJJqM+GLH# [! v;? O " "6 * $  q6Q;    8 q5 O " "3q6 *B$$QyAB(!456e$)"I !X YJJu%--a9EH$! (//A7B,,Q&INOLL&*=av*N!OPe< %\ 22II4!   s>HBHH76H7c#f^^# [R"5n[[U55mUR T5 UR UU4Sj[ TS555 [R"U5H,n[R"UVs/sHnTUPM sn6v M. gs snf7f)aWMerge sets that share k or more elements. See: http://rosettacode.org/wiki/Set_consolidation The iterative python implementation posted there is faster than this because of the overhead of building a Graph and calling nx.connected_components, but it's not clear for us if we can use it in NetworkX because there is no licence for the code. c3b># UH$up[TUTU-5T:dMX4v M& g7fN)r).0uvrr,s r/ _consolidate..s5141SqE!H9L5MQR5R1s/ /r N) rGraphdict enumerateadd_nodes_fromadd_edges_fromrrrunion)setsrr r!nr,s ` @r/ _consolidater@s  A 4 !EU'q1,,Q/ iiI6Iq%(I67706sBB1B, !B1c#*# Sn/nUR5VVs1sHupVXb:dM UiM snnUVVs1sH owHoUiM M snn- nURU5n [R"U 5Hqn [ U 5n UH-nUH$n U"X U 5(dMU R U 5 M& M/ [ U 5UR5:dM`URU 5 Ms [XBS-5ShvN gs snnfs snnfN7f)Nc2^[U4SjX55$)Nc3,># UH oT;v M g7fr2)r3r?r+s r/r6E_generate_partition..has_nbrs_in_partition..s37a >7)any)r noder+s `r/has_nbrs_in_partition2_generate_partition..has_nbrs_in_partitions317333r ) degreerrrraddrorderrr@) r r(rrI componentsr?dcutr,Hccr!rHs r/rrs4J88: /:41Q: /2Rc1c12R RE 5A%%a(G C("55MM$' y>AGGI %   i ()JA... 02R/s8D DDDD  AD!4D&D?DDc^0n[U5n[[SUS-55HnX2:Xa[[ XU55X'M#X0;a[[ XS-U55X'MF[ R "X6mXS-Vs/sH!n[U4SjU55(dMUPM# nnU(a[[ XU-U55X'M[[ XU55X'M U$s snf)Nr c3,># UH oT;v M g7fr2rD)r3r? nodes_at_ks r/r6,_reconstruct_k_components..s5USTaz6ISTrF)maxreversedrangerr@rr=rG)k_compsresultmax_krr#to_addrVs @r/rrs F LE eAuqy) * :\'*a89FI  \&Q-;z!}=D t > MrKr2)__doc__ collectionsr itertoolsroperatorrnetworkxrnetworkx.algorithms.flowrnetworkx.utilsrr__all__ _dispatchablerr@rrrerDrKr/rosl$"2.   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