Mean-Field Analysis of Latent Variable Process Models on Dynamically Evolving Graphs with Feedback Effects

Abstract

We study the mean-field limit of a generic class of dynamic co-evolving latent space networks motivated by the social and opinion dynamics literature. Such models include $n$ agents, whose opinions are given by latent stochastic processes, and a dynamic network process describing agent interactions. Models in this class incorporate (a) bi-directional feedback between the latent processes and the network process, (b) persistence effects, meaning that the network structure at the current time depends on the value of the latent processes at the current time but also on the network structure at the previous time instance and (c) localized interactions, meaning that individual agents do not have global information. We characterize the distributional limit of a random sample taken from the latent space network as the number of nodes in the network diverges. We describe the rich conditional probabilistic structure of the resulting limiting model which we use to establish the limiting behavior of the following quantities: (i) the empirical measure of the latent process, (ii) a conditional empirical measure relating the latent process to the network process and (iii) the network process graphon. In proving our main results, we derive a general conditional propagation of chaos result, which is of independent interest. Our novel approach to studying the limiting behavior of random samples proves to be a very useful methodology for fully grasping the asymptotic behavior of co-evolving particle systems. Numerical results are included to illustrate the theoretical findings.