Inference for dynamic Erd\H{o}s-R\'enyi random graphs under regime switching

TL;DR

This paper introduces a method to estimate the underlying on-off dynamics and regime switching behavior of two parallel evolving Erdős-Rényi graphs from aggregate network data, applicable in various real-world systems.

Contribution

It develops a novel method of moments approach for estimating on-off time distributions and regime switching parameters from aggregate network observations.

Findings

Effective parameter recovery demonstrated in experiments

Method handles unobserved regime states

Applicable to real-world dynamic networks

Abstract

This paper examines a model involving two dynamic Erd\H{o}s-R\'enyi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key feature of our setup is regime switching: the graph that is observed at any given moment depends on the state of an underlying background process, which is modeled as an alternating renewal process. This modeling framework captures a common situation in various real-world applications, where the observed network is influenced by a (typically unobservable) background process. Such scenarios arise, for example, in economics, communication networks, and biological systems. In our setup we only have access to aggregate quantities such as the number of active edges or the counts of specific subgraphs (such as stars or complete graphs) in the observed graph; importantly, we do not observe the mode. The objective is to estimate the on- and off-time distributions of the edges in each of the two dynamic Erd\H{o}s-R\'enyi random graphs, as well as the distribution of time spent in each of the two modes. By employing parametric models for the on- and off-times and the background process, we develop a method of moments approach to estimate the relevant parameters. Experimental evaluations are conducted to demonstrate the effectiveness of the proposed method in recovering these parameters.