Characterizing Continuous Time Random Walks on Time Varying Graphs

TL;DR

This paper analyzes the behavior of continuous time random walks on dynamic graphs, establishing conditions for stationarity and ergodicity, and characterizing the stationary distribution under specific scenarios.

Contribution

It introduces new theoretical conditions for the stationarity and ergodicity of CTRWs on time-varying graphs and characterizes the stationary distribution in key cases.

Findings

Stationary distribution depends on walker rate and graph dynamics.

Characterization of stationary distribution under time-scale separation.

Analysis of stationary distribution when node degrees are uniform within components.

Abstract

In this paper we study the behavior of a continuous time random walk (CTRW) on a stationary and ergodic time varying dynamic graph. We establish conditions under which the CTRW is a stationary and ergodic process. In general, the stationary distribution of the walker depends on the walker rate and is difficult to characterize. However, we characterize the stationary distribution in the following cases: i) the walker rate is significantly larger or smaller than the rate in which the graph changes (time-scale separation), ii) the walker rate is proportional to the degree of the node that it resides on (coupled dynamics), and iii) the degrees of node belonging to the same connected component are identical (structural constraints). We provide examples that illustrate our theoretical findings.