Random Walk Fundamental Tensor and Its Applications to Network Analysis

TL;DR

This paper introduces the fundamental tensor, a unified framework for random walk metrics, enabling comprehensive network analysis and applications such as centrality, articulation points, influential nodes, and reachability.

Contribution

The paper presents the fundamental tensor as a generalization of the fundamental matrix, unifying various random walk metrics and demonstrating its applications in network analysis.

Findings

Unified framework for random walk metrics.

Effective identification of influential nodes and articulation points.

Fast computation of network reachability after failures.

Abstract

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.