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The notion of return probability -- explored most famously by GeorgeP\'{o}lya on d-dimensional lattices -- has potential as a measure for theanalysis of networks. We present an efficient method for finding returnprobability distributions for connected undirected graphs. We argue that returnprobability has the same discriminatory power as existing k-step measures -- inparticular, beta centrality (with negative beta), the graph-theoretical powerindex (GPI), and subgraph centrality. We compare the running time of ouralgorithm to beta centrality and subgraph centrality and find that it issignificantly faster. When return probability is used to measure the samephenomena as beta centrality, it runs in linear time -- O(n+m), where n and mare the number of nodes and edges, respectively -- which takes much less timethan either the matrix inversion or the sequence of matrix multiplicationsrequired for calculating the exact or approximate forms of beta centrality,respectively. We call this form of return probability the P\'{o}lya power index(PPI). Computing subgraph centrality requires an expensive eigendecompositionof the adjacency matrix; return probability runs in half the time of theeigendecomposition on a 2000-node network. These performance improvements areimportant because computationally efficient measures are necessary in order toanalyze large networks.

Return probability and k-step measures