\MhdZddlmZddlmZddlmZddlZddl m Z ddl m Z e Z dgZe d ejdd Zd Zd Zd ZdZdS)z, Moody and White algorithm for k-components ) defaultdict) combinations) itemgetterN) edmonds_karp)not_implemented_for k_componentsdirectedc tt}|t}tjD]?}t |}t |dkr|d|@fdtjD}|D]?}t |}t |dkr|d|@|D]}t |dkrtj ||} | dkr(|| t |ttj || |} | t|| | fg} | r| d\} } t| }| |}tj ||}|| kr.|dkr(||t |ttj |||} | r&| |t|| |fn$#t$r| YnwxYw| t!|S)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 Nc:g|]}|S)subgraph).0cGs l/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/algorithms/connectivity/kcomponents.py z k_components..xs#HHHaAJJqMMHHH) flow_func)kr)rlistdefault_flow_funcnxconnected_componentssetlenappendbiconnected_componentsnode_connectivity all_node_cuts_generate_partitionnextr StopIterationpop_reconstruct_k_components)rrr componentcomp bicomponents bicomponentbicompBrcutsstackparent_k partitionnodesCthis_ks` rrrs{tt$$L% ,Q//)) 9~~ t99q== O " "4 ( ( (HHHH2+DQ+G+GHHHL#++ [!! v;;?? O " "6 * * *  q66Q;;   i 8 8 8 q55 O " "3q66 * * *B$Q!yAAABB(D!4456 $)"I !Xy YJJu%%-a9EEEH$$! (//A777B,Q&INNNOOQLL&*=av*N*N!OPPP       * %\ 2 22s8B:H33IIc#\Ktj}tt|||fdt dDtj|D]}tj fd|DVdS)asMerge sets that share k or more elements. 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