Generalized Master Equations for Non-Poisson Dynamics on Networks

TL;DR

This paper introduces a generalized master equation framework to analyze non-Poisson temporal dynamics on networks, improving understanding of real-world temporal network processes beyond traditional Poisson assumptions.

Contribution

It develops a generalized master equation approach for non-Poisson edge dynamics, linking it to standard rate equations and providing tools for analyzing temporal networks with complex stochastic behavior.

Findings

The generalized master equation reduces to standard rate equations for Poisson processes.

Stationary solutions are obtained via an effective transition matrix with an easily computable eigenvector.

Implications for modeling and diagnosing non-Poisson temporal networks are discussed.

Abstract

The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Consequently, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that the equation reduces to the standard rate equations when the underlying process is Poisson and that the stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.