+ j ^Rt^RIHt^RIt^RIHtR.t]PRRl4tRt R#)z%Node redundancy for bipartite graphs.) combinationsN) NetworkXErrornode_redundancycaVfSp\;QJdV3RlV4F 'gK RM RM!V3RlV44'd \R4hVUu/uFq"\SV4bK up#uupi)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` [1]_. 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. c3N<"TFp\SV,4^8xK R#5i)N)len).0vGs& i/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/bipartite/redundancy.py "node_redundancy..Xs (%Q3qt9q=%s"%TFzSCannot compute redundancy coefficient for a node that has fewer than two neighbors.)anyr_node_redundancy)r nodesr sf& r rr shV } s (% (sss (% ((( 2  05 5u!1% %u 55 5sA/caa\SS,4p\VV3Rl\SS,^444p^V,W"^, ,, #)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`. c3<"TFBwr\SV,4\SV,4,S0, 'gK>^xKD R#5i)N)set)r uwr r s& r r #_node_redundancy..ms:-fq#ad)c!A$i2GA31N1N-s ;A  A )rsumr)r r noverlapsff r rrasK AaD A$QqT1-G KAQK (()N) __doc__ itertoolsrnetworkxnxr__all__ _dispatchablerrrr r$s;+""  R6R6j)r