  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.
