U ª9%e! ã@s8dZddlmZddlmZdgZd dd„Zdd„ZdS) z%Node redundancy for bipartite graphs.é)Ú combinations)Ú NetworkXErrorÚnode_redundancyNcs<|dkr ˆ}t‡fdd„|Dƒƒr*tdƒ‚‡fdd„|DƒS)u©Computes the node redundancy coefficients for the nodes in the bipartite graph `G`. The redundancy coefficient of a node `v` is the fraction of pairs of neighbors of `v` that are both linked to other nodes. In a one-mode projection these nodes would be linked together even if `v` were not there. More formally, for any vertex `v`, the *redundancy coefficient of `v`* is defined by .. math:: rc(v) = \frac{|\{\{u, w\} \subseteq N(v), \: \exists v' \neq v,\: (v',u) \in E\: \mathrm{and}\: (v',w) \in E\}|}{ \frac{|N(v)|(|N(v)|-1)}{2}}, where `N(v)` is the set of neighbors of `v` in `G`. Parameters ---------- G : graph A bipartite graph nodes : list or iterable (optional) Compute redundancy for these nodes. The default is all nodes in G. Returns ------- redundancy : dictionary A dictionary keyed by node with the node redundancy value. Examples -------- Compute the redundancy coefficient of each node in a graph:: >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> rc[0] 1.0 Compute the average redundancy for the graph:: >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> sum(rc.values()) / len(G) 1.0 Compute the average redundancy for a set of nodes:: >>> from networkx.algorithms import bipartite >>> G = nx.cycle_graph(4) >>> rc = bipartite.node_redundancy(G) >>> nodes = [0, 2] >>> sum(rc[n] for n in nodes) / len(nodes) 1.0 Raises ------ NetworkXError If any of the nodes in the graph (or in `nodes`, if specified) has (out-)degree less than two (which would result in division by zero, according to the definition of the redundancy coefficient). 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. Nc3s|]}tˆ|ƒdkVqdS)éN)Úlen©Ú.0Úv©ÚG©úg/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/networkx/algorithms/bipartite/redundancy.pyÚ Usz"node_redundancy..zSCannot compute redundancy coefficient for a node that has fewer than two neighbors.csi|]}|tˆ|ƒ“qSr )Ú_node_redundancyrr r r Ú [sz#node_redundancy..)Úanyr)r Znodesr r r r sJÿcsBtˆˆƒ}t‡‡fdd„tˆˆdƒDƒƒ}d|||dS)a™Returns the redundancy of the node `v` in the bipartite graph `G`. If `G` is a graph with `n` nodes, the redundancy of a node is the ratio of the "overlap" of `v` to the maximum possible overlap of `v` according to its degree. The overlap of `v` is the number of pairs of neighbors that have mutual neighbors themselves, other than `v`. `v` must have at least two neighbors in `G`. c3s4|],\}}tˆ|ƒtˆ|ƒ@ˆhrdVqdS)éN)Úset)rÚuÚw©r r r r rjsz#_node_redundancy..rr)rÚsumr)r r ÚnÚoverlapr rr r^s  ÿr)N)Ú__doc__Ú itertoolsrZnetworkxrÚ__all__rrr r r r Ús    U