Clusters determine local fluctuations of random walks on graphs

TL;DR

This paper investigates how the local fluctuations of random walks on graphs are influenced by the global structure, revealing that fluctuations are lower across edges connecting different clusters, and proposing a new centrality measure based on these fluctuations.

Contribution

It establishes a link between local fluctuation statistics of random walks and the global graph structure, introducing a fluctuation-based centrality score.

Findings

Fluctuations are reduced when edges connect different clusters.

Fluctuations relate to the algebraic connectivity and Fiedler vector.

A new centrality score based on fluctuations is proposed.

Abstract

The evolution of many stochastic systems is accurately described by random walks on graphs. We here explore the close connection between local steady-state fluctuations of random walks and the global structure of the underlying graph. Fluctuations are quantified by the number of traversals of the random walk across edges during a fixed time window, more precisely, by the corresponding counting statistics. The variance-to-mean ratio of the counting statistics is typically lowered if two end vertices of an edge belong to different clusters as defined by spectral clustering. In particular, we relate the fluctuations to the algebraic connectivity and the Fiedler vector of the graph. Building on these results we suggest a centrality score based on fluctuations of random walks. Our findings imply that local fluctuations of continuous-time Markov processes on discrete state space depend strongly on the global topology of the underlying graph in addition to the specific transition rates.