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We introduce a new method to efficiently approximate the number of infections resulting from a given initially-infected node in a network of susceptible individuals, based on counting the number of possible infection paths of various lengths to each other node in the network. We analytically study the properties of our method systematically, in particular demonstrating different forms for SIS and SIR disease spreading (e.g. under the SIR model our method counts self-avoiding walks). In comparison to existing methods to infer the spreading efficiency of different nodes in the network (based on degree, k-shell decomposition analysis and different centrality measures), our method directly considers the spreading process, and as such is unique in providing estimation of actual numbers of infections. Crucially, in simulating infections on various real-world networks with the SIR model, we show that our walks-based method improves the inference of effectiveness of nodes over a wide range of infection rates compared to existing methods. We also analyse the trade-off between estimate accuracy and computational cost of our method, showing that the better accuracy here can still be obtained at a comparable computational cost to other methods.
