\MhJ XdZddlmZddlZddlmZdgZejddZdZ dS) z%Node redundancy for bipartite graphs.) combinationsN) NetworkXErrornode_redundancyc||}tfd|Drtdfd|DS)uComputes 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. Nc3JK|]}t|dkVdS)N)len.0vGs h/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/algorithms/bipartite/redundancy.py z"node_redundancy..Xs1 ( (Q3qt99q= ( ( ( ( ( (zSCannot compute redundancy coefficient for a node that has fewer than two neighbors.c2i|]}|t|S)_node_redundancyr s r z#node_redundancy..^s& 5 5 5!A1%% 5 5 5r)anyr)r nodess` rrr shV } ( ( ( (% ( ( (((  2   6 5 5 5u 5 5 55rct}tfdtdD}d|z||dz zz S)aReturns 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`. c3K|]8\}}t|t|zhz 4dV9dS)N)set)r uwr r s rrz#_node_redundancy..ms`q!#ad))c!A$ii2GA31N rrr)r sumr)r r noverlaps`` rrrasv AaD A$QqT1--G KAQK ((r)N) __doc__ itertoolsrnetworkxnxr__all__ _dispatchablerrrrrr&s++""""""""""""  R6R6R6R6j)))))r