Random Walk Fundamental Tensor and Its Applications to Network Analysis

We first present a comprehensive review of various random walk metrics used in the literature and express them in a consistent framework. We then introduce fundamental tensor -- a generalization of the well-known fundamental matrix -- and show that classical random walk metrics can be derived from it in a unified manner. We provide a collection of useful relations for random walk metrics that are useful and insightful for network studies. To demonstrate the usefulness and efficacy of the proposed fundamental tensor in network analysis, we present four important applications: 1) unification of network centrality measures, 2) characterization of (generalized) network articulation points, 3) identification of network most influential nodes, and 4) fast computation of network reachability after failures.