Computing the Number of Loop-Free k-hop Paths of Networks

Abstract

Computing the number of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop paths is crucial for selecting services in social networks and analyzing graph data, for example, a service consumer require to evaluate the trustworthiness of a service provider along the social trust paths from a service consumer to the service provider, there are usually many social trust paths between two unconnected participants, people need to know the number of loop-free <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop trust propogation paths; other applications include the similarity computation for services recommendation, information diffusion, etc. Previously, the number of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop paths is roughly estimated by the elements in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> multiplications of the network adjacency matrix. This method calculates much more <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop paths than those actually exist, due to many paths with loops counted as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop paths, which may result in obvious errors in applications. Based on the idea of loops removing, accurate mathematical formulas for counting loop-free paths are obtained in this article for paths with five or less hops, an approximate method is provided for larger hops. Based on the proposed loop removing algorithm (LRA), the typical method for predicting trust between any two people in social networks is improved, the error rate is dramatically reduced; the traditional path based similarity indices are improved, which are much accurate than their antecedent counterparts; and a method for computing the spreading probability for information spreading between two unconnected vertices in the famous independent cascade (IC) model is also obtained. To reveal the effectiveness of the proposed LRA, this article also provide a traversal depth-first search algorithm (DFSA) for finding the true number of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -hop loop-free paths.