h{ :SrSSKrS/r\RS5rg)z(Function for computing walks in a graph.Nnumber_of_walkscrSSKnUS:a[SU35e[R"USS9nURR UR 5U5n[U5VVVVs0sH5upVU[U5VVs0sHupxXRXW5_M snn_M7 n nnnnU $s snnfs snnnnf)a7Returns the number of walks connecting each pair of nodes in `G` A *walk* is a sequence of nodes in which each adjacent pair of nodes in the sequence is adjacent in the graph. A walk can repeat the same edge and go in the opposite direction just as people can walk on a set of paths, but standing still is not counted as part of the walk. This function only counts the walks with `walk_length` edges. Note that the number of nodes in the walk sequence is one more than `walk_length`. The number of walks can grow very quickly on a larger graph and with a larger walk length. Parameters ---------- G : NetworkX graph walk_length : int A nonnegative integer representing the length of a walk. Returns ------- dict A dictionary of dictionaries in which outer keys are source nodes, inner keys are target nodes, and inner values are the number of walks of length `walk_length` connecting those nodes. Raises ------ ValueError If `walk_length` is negative Examples -------- >>> G = nx.Graph([(0, 1), (1, 2)]) >>> walks = nx.number_of_walks(G, 2) >>> walks {0: {0: 1, 1: 0, 2: 1}, 1: {0: 0, 1: 2, 2: 0}, 2: {0: 1, 1: 0, 2: 1}} >>> total_walks = sum(sum(tgts.values()) for _, tgts in walks.items()) You can also get the number of walks from a specific source node using the returned dictionary. For example, number of walks of length 1 from node 0 can be found as follows: >>> walks = nx.number_of_walks(G, 1) >>> walks[0] {0: 0, 1: 1, 2: 0} >>> sum(walks[0].values()) # walks from 0 of length 1 1 Similarly, a target node can also be specified: >>> walks[0][1] 1 rNz"`walk_length` cannot be negative: )weight) numpy ValueErrornxadjacency_matrixlinalg matrix_powertoarray enumerateitem) G walk_lengthnpApoweru_idxuv_idxvresults K/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/walks.pyrrstQ=k]KLL Ad+A II " "199; r s/.  FFr