2021/01/24 by Anirudh Sridhar, Soummya Kar, Sridhar, Anirudh +1
Physics and Astronomy · Psychology · #Advanced Thermodynamics and Statistical Mechanics #FOS: Computer and information sciences #FOS: Mathematics #Mental Health Research Topics #Multiagent Systems (cs.MA) #Opinion Dynamics and Social Influence #Probability (math.PR) #Social and Information Networks (cs.SI)
paper · pdf · doi:10.48550/arxiv.2101.09644
openalex publication_date 2021/01/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 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.