+ j2Rt^RIHt^RIt^RIHt.ROtRtRt Rt R]R] R ] /t ]PRR l4t ] t]P!R R 7RR l4t]PR4t]!R4]PRRl44tRtR#)z+Functions for computing clustering of pairs) defaultdictN)not_implemented_forcP\W,4\W,4, #N)lennunvs&&f/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/bipartite/cluster.pycc_dotr s rw<#bg, &&cj\W,4\\V4\V44, #r)rmaxrs&&r cc_maxr" rw<#c"gs2w/ //r cj\W,4\\V4\V44, #r)rminrs&&r cc_minrrr dotrrc 8\PPPV4'g\P!R4h\ V,pTfTp/pTFpRpY,UU u0uFqT,FqkK K up pT0, p T F0p Ys!\Y ,4\Y,44, pK2 TR8dT\T 4,pYuT&K T# \ dp\P!R4ThRp?ii;iuup pi)uBCompute a bipartite clustering coefficient for nodes. The bipartite clustering coefficient is a measure of local density of connections defined as [1]_: .. math:: c_u = \frac{\sum_{v \in N(N(u))} c_{uv} }{|N(N(u))|} where `N(N(u))` are the second order neighbors of `u` in `G` excluding `u`, and `c_{uv}` is the pairwise clustering coefficient between nodes `u` and `v`. The mode selects the function for `c_{uv}` which can be: `dot`: .. math:: c_{uv}=\frac{|N(u)\cap N(v)|}{|N(u) \cup N(v)|} `min`: .. math:: c_{uv}=\frac{|N(u)\cap N(v)|}{min(|N(u)|,|N(v)|)} `max`: .. math:: c_{uv}=\frac{|N(u)\cap N(v)|}{max(|N(u)|,|N(v)|)} Parameters ---------- G : graph A bipartite graph nodes : list or iterable (optional) Compute bipartite clustering for these nodes. The default is all nodes in G. mode : string The pairwise bipartite clustering method to be used in the computation. It must be "dot", "max", or "min". Returns ------- clustering : dictionary A dictionary keyed by node with the clustering coefficient value. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.path_graph(4) # path graphs are bipartite >>> c = bipartite.clustering(G) >>> c[0] 0.5 >>> c = bipartite.clustering(G, mode="min") >>> c[0] 1.0 See Also -------- robins_alexander_clustering average_clustering networkx.algorithms.cluster.square_clustering References ---------- .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). Basic notions for the analysis of large two-mode networks. Social Networks 30(1), 31--48. zGraph is not bipartitez6Mode for bipartite clustering must be: dot, min or maxNg) nx algorithms bipartite is_bipartite NetworkXErrormodesKeyErrorsetr) Gnodesmodecc_funcerrccsvccnbrunbrs2s &&& r latapy_clusteringr) s\ == " " / / 2 2788+  } C  d3dscFFqFd3qc9A '#ad)SY/ /B 8 #e* BA J!  D  4s C,.D, D7DDbipartite_average_clustering)namectaVfTp\WVR7o\V3RlV44\V4, #)uCompute the average bipartite clustering coefficient. A clustering coefficient for the whole graph is the average, .. math:: C = \frac{1}{n}\sum_{v \in G} c_v, where `n` is the number of nodes in `G`. Similar measures for the two bipartite sets can be defined [1]_ .. math:: C_X = \frac{1}{|X|}\sum_{v \in X} c_v, where `X` is a bipartite set of `G`. Parameters ---------- G : graph a bipartite graph nodes : list or iterable, optional A container of nodes to use in computing the average. The nodes should be either the entire graph (the default) or one of the bipartite sets. mode : string The pairwise bipartite clustering method. It must be "dot", "max", or "min" Returns ------- clustering : float The average bipartite clustering for the given set of nodes or the entire graph if no nodes are specified. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.star_graph(3) # star graphs are bipartite >>> bipartite.average_clustering(G) 0.75 >>> X, Y = bipartite.sets(G) >>> bipartite.average_clustering(G, X) 0.0 >>> bipartite.average_clustering(G, Y) 1.0 See Also -------- clustering Notes ----- The container of nodes passed to this function must contain all of the nodes in one of the bipartite sets ("top" or "bottom") in order to compute the correct average bipartite clustering coefficients. See :mod:`bipartite documentation ` for further details on how bipartite graphs are handled in NetworkX. References ---------- .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). Basic notions for the analysis of large two-mode networks. Social Networks 30(1), 31--48. )rr c36<"TFpSV,xK R#5ir).0r$r#s& r %average_clustering..s%u!s1vvus)r)sumr)rrr r#s&&&@r average_clusteringr3s6N } A 6C %u% %E 22r cVP4^8gVP4^8d^#\V4pV^8Xd^#\\ V4P 44V, #)uCompute the bipartite clustering of G. Robins and Alexander [1]_ defined bipartite clustering coefficient as four times the number of four cycles `C_4` divided by the number of three paths `L_3` in a bipartite graph: .. math:: CC_4 = \frac{4 * C_4}{L_3} The four cycles counted here are *butterflies* (also called *squares* or *four-cycles* in other contexts) — complete bipartite subgraphs K_{2,2} where alternating nodes belong to different partitions. See :func:`butterflies` for per-node butterfly counts. Parameters ---------- G : graph a bipartite graph Returns ------- clustering : float The Robins and Alexander bipartite clustering for the input graph. Examples -------- >>> from networkx.algorithms import bipartite >>> G = nx.davis_southern_women_graph() >>> print(round(bipartite.robins_alexander_clustering(G), 3)) 0.468 See Also -------- butterflies : per-node butterfly (four-cycle) counts latapy_clustering networkx.algorithms.cluster.square_clustering References ---------- .. [1] Robins, G. and M. Alexander (2004). Small worlds among interlocking directors: Network structure and distance in bipartite graphs. Computational & Mathematical Organization Theory 10(1), 69–94. )ordersize _threepathsr2 butterfliesvalues)rL_3s& r robins_alexander_clusteringr;sQ^ wwy1}1  a.C ax {1~$$& '# --r directedc VP4^8XdR\PVP4^4pVfV#VP V4Uu/uF q3W#,bK up#\ VP 44UUUu/uF wpwrVWVV3bK ppppVUu/uF)q3\VPV4VPR7bK+ pp\PV^4p VEFp Wz,p \\4p \\4p W,F\pWs,V 8dMNW,F>pW~,V 8dK-W;;,^, uu&W,PV4K@ K^ V P4FvwrV^8dKW^, ,^,pW;;,V, uu&W;;,V, uu&W,FpW;;,V^, , uu&K Kx EK VfV #VP V4Uu/uF q3W,bK up#uupiuupppiuupiuupi)uCount the number of butterflies for each node in a bipartite graph. A *butterfly* is a complete bipartite subgraph K_{2,2} — four nodes (two from each partition) with all four cross-edges present. It is the bipartite analogue of a triangle in unipartite graphs. .. code-block:: none A1 A2 | \ / | | X | | / \ | B1 B2 Equivalently, a butterfly is a 4-cycle (C_4) in which alternating nodes belong to different partitions of the bipartite graph. This structure is also called a *square* in the physics and complex-networks literature [3]_, and a *four-cycle* in the sociology literature [4]_. The name *butterfly* is standard in the data-mining and bipartite-network literature [1]_ [2]_. Parameters ---------- G : NetworkX graph An undirected bipartite graph. nodes : iterable of nodes, optional (default: all nodes) Return butterfly counts only for these nodes. The computation always uses the full graph; ``nodes`` only filters the returned dictionary (same convention as :func:`~networkx.triangles`). When ``None`` (default), counts for all nodes are returned. Nodes not present in `G` are silently ignored. Returns ------- butterflies : dict A dictionary keyed by node to the number of butterflies that node participates in. Each butterfly is counted once per participating node, so:: sum(butterflies(G).values()) == 4 * total_butterfly_count Examples -------- A single K_{2,2} contains exactly one butterfly, and each of its four nodes participates in that butterfly: >>> G = nx.complete_bipartite_graph(2, 2) >>> nx.bipartite.butterflies(G) {0: 1, 1: 1, 2: 1, 3: 1} The total number of butterflies is the sum divided by 4: >>> bt = nx.bipartite.butterflies(G) >>> sum(bt.values()) // 4 1 K_{3,3} contains nine butterflies; every node participates in six: >>> G2 = nx.complete_bipartite_graph(3, 3) >>> bt2 = nx.bipartite.butterflies(G2) >>> sum(bt2.values()) // 4 9 Nodes not in any butterfly receive count zero: >>> G = nx.complete_bipartite_graph(2, 2) >>> G.add_edge(0, 5) >>> nx.bipartite.butterflies(G) {0: 1, 1: 1, 2: 1, 3: 1, 5: 0} Notes ----- This algorithm uses wedge enumerate to count butterflies. The wedge enumeration method uses the vertex-priority of BFC-VP [2]_ : nodes are ranked by degree, neighbor lists are pre-sorted by ascending rank, and for each start node ``u`` only neighbors with lower rank are processed as middle and end nodes, with early termination on the sorted lists. This gives time complexity :math:`O\!\bigl(\sum_{(u,v)\in E} \min(d(u), d(v))\bigr) = O(\alpha m)` where :math:`\alpha` is the arboricity, i.e. the minimum number of forests into which $E$ can be partitioned, and :math:`m` the number of edges. The per-node attribution (distributing butterfly credits to start, middle, and end nodes) is an extension of BFC-VP not present in the original paper, which computes only a global count. See Also -------- robins_alexander_clustering : graph-level bipartite clustering coefficient whose numerator is ``4 * total butterfly count``. networkx.algorithms.cluster.square_clustering : per-node square clustering coefficient for general graphs. References ---------- .. [1] Sanei-Mehri, S. V., Sariyuce, A. E., & Tirthapura, S. (2018). Butterfly counting in bipartite networks. *Proceedings of the 24th ACM SIGKDD*, 2150–2159. https://doi.org/10.1145/3219819.3220097 .. [2] Wang, K., Lin, X., Qin, L., Zhang, W., & Zhang, Y. (2023). Accelerated butterfly counting with vertex priority on bipartite graphs. *The VLDB Journal*, 32, 257–281. https://doi.org/10.1007/s00778-022-00746-0 .. [3] Lind, P. G., Gonzalez, M. C., & Herrmann, H. J. (2005). Cycles and clustering in bipartite networks. *Physical Review E*, 72, 056127. https://doi.org/10.1103/PhysRevE.72.056127 .. [4] Robins, G. and M. Alexander (2004). 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