Estimating Graph Dynamics from Population Observations

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.