Estimating Graph Dynamics from Population Observations

TL;DR

This paper introduces methods to estimate the edge probability of a dynamic Erdős–Rényi graph based on population observations, providing theoretical guarantees for estimator consistency and asymptotic normality.

Contribution

It proposes two estimators for the edge probability in a dynamic random graph and proves their statistical properties under population observation data.

Findings

Two consistent estimators for edge probability p.

Establishes asymptotic normality of the estimators.

Provides theoretical analysis for estimation in dynamic graph models.

Abstract

In this paper we consider a population process evolving on a dynamic random graph. The dynamic random graph is an Erd\H{o}s--R\'enyi graph that is resampled every time unit, independently of the previous ones, with `edge existence probability' $p$. The population process consists of $M$ individuals which reside at the vertices of the dynamic graph. At each point in time any of the $M$ individuals, supposing it resides at a vertex with $k$ neighbors, jumps to an adjacent vertex with probability $k/(k+1)$ (where this adjacent vertex is picked uniformly at random), and with probability $1/(k+1)$ it stays where it is. We suppose we observe the numbers of individuals at each of the vertices, but not the evolving random graph itself. We propose two estimators for $p$, and establish their consistency and asymptotic normality.