Mean-field Approximations for Stochastic Population Processes with Heterogeneous Interactions

TL;DR

This paper develops a general theory for when mean-field approximations accurately describe large stochastic population processes with heterogeneous interactions, extending previous results from complete to arbitrary networks.

Contribution

It establishes conditions under which different mean-field approximations are valid for arbitrary interaction networks, including spectral gap and Frobenius norm criteria.

Findings

Homogeneous mean-field approximation is accurate if the interaction matrix has a large spectral gap.

Interaction-aware mean-field approximation is accurate when the Frobenius norm of the interaction matrix is small.

Simulations show that assuming homogeneous interactions can lead to significant inaccuracies in predictions.

Abstract

This paper studies a general class of stochastic population processes in which agents interact with one another over a network. Agents update their behaviors in a random and decentralized manner according to a policy that depends only on the agent's current state and an estimate of the macroscopic population state, given by a weighted average of the neighboring states. When the number of agents is large and the network is a complete graph (has all-to-all information access), the macroscopic behavior of the population can be well-approximated by a set of deterministic differential equations called a {\it mean-field approximation}. For incomplete networks such characterizations remained previously unclear, i.e., in general whether a suitable mean-field approximation exists for the macroscopic behavior of the population. The paper addresses this gap by establishing a generic theory describing when various mean-field approximations are accurate for \emph{arbitrary} interaction structures. Our results are threefold. Letting $W$ be the matrix describing agent interactions, we first show that a simple mean-field approximation that incorrectly assumes a homogeneous interaction structure is accurate provided $W$ has a large spectral gap. Second, we show that a more complex mean-field approximation which takes into account agent interactions is accurate as long as the Frobenius norm of $W$ is small. Finally, we compare the predictions of the two mean-field approximations through simulations, highlighting cases where using mean-field approximations that assume a homogeneous interaction structure can lead to inaccurate qualitative and quantitative predictions.