D^(hdZddlmZddlmZddlmZddlZddl m Z ddl m Z e Z dgZe d ejdd Zd Zd Zd ZdZy)z, Moody and White algorithm for k-components ) defaultdict) combinations) itemgetterN) edmonds_karp)not_implemented_for k_componentsdirectedc Btt}|t}tj|D]0}t |}t |dkDs|dj|2tj|Dcgc]}|j|}}|D]0}t |}t |dkDs|dj|2|D]&} t | dkrtj| |} | dkDr|| jt | ttj| | |} | t| | | fg} | s| d\} } t|}| j|}tj||}|| kDr"|dkDr||jt |ttj|||} | r| j|t|| |f| r)t!|Scc}w#t$r| jY1wxYw)a7 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 j/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/networkx/algorithms/connectivity/kcomponents.pyrrstt$L% ,,Q/) 9~ t9q= O " "4 ( ) ,.+D+DQ+GHaAJJqMHLH#+ [! v;? O " "6 * +  q6Q;   i 8 q5 O " "3q6 *B$$Q!yAB(D!456$)"I !Xy YJJu%--a9EH$! (//A7B,,Q&INOLL&*=av*N!OP< %\ 22II4!   s7G=BHHHc#TKtj}tt||j |j fdt dDtj|D]'}tj|Dcgc]}| c})ycc}ww)asMerge 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. c3\K|]#\}}t||zk\s||f%ywN)r).0uvrr,s r/ z_consolidate..s81aSqE!H9L5MQR5RAs, ,r N) rGraphdict enumerateadd_nodes_fromadd_edges_fromrrrunion)setsrr r!nr,s ` @r/ _consolidater?s  A 4 !EU'q1,,Q/8 iiI6q%(67786sB B( B# B(c#Kd}g}|jDchc] \}}||kDs |c}}|Dchc] }|D]}| c}}z }|j|} tj| D]e} t | } |D]%}|D]} ||| | s| j |  't | |jksU|j| gt||dzEd{ycc}}wcc}}w7w)Nc2tfd||DS)Nc3&K|]}|v ywr2)r3r>r+s r/r6zE_generate_partition..has_nbrs_in_partition..s3a1 >3)any)r noder+s `r/has_nbrs_in_partitionz2_generate_partition..has_nbrs_in_partitions31T7333r ) degreerrrraddrorderrr?) r r(rrG componentsr>dcutr,Hccr!rFs r/rrs4J88: /41aQQ /2Rc2R12R12R RE 5A%%a()G  (C ((D"5MM$' ( ( y>AGGI %   i ()JA... 02R/s8C? 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