Multi-agent systems with multiple-wise interaction: Propagation of chaos and macroscopic limit
Thierry Paul, Stefano Rossi, Emmanuel Trélat
TL;DR
The paper studies multi-agent systems with $m$-wise interactions and develops a three-step limit framework to derive continuum descriptions. It first proves well-posedness and chaos propagation for a mesoscopic Vlasov-type equation, establishing quantitative convergence of the particle system to the mean-field limit. It then passes to a macroscopic description via a monokinetic ansatz, proving existence and uniqueness of a nonlocal evolution for the label-dependent opinion function $y(t,x)$, and analyzes the limit as the interaction order grows to infinity, yielding a limiting nonlocal operator $G^\infty$ and a convergent macroscopic solution $y^\infty$. The results connect microscopic dynamics to macroscopic behavior and provide a rigorous route to understand how increasingly complex multi-agent interactions shape collective dynamics.
Abstract
We consider interacting multi-agent systems where the interaction is not only pairwise but involves simultaneous interactions among multiple agents (multiple-wise interaction). By passing through the mesoscopic and macroscopic limits with a fixed multiple-wise interaction of order $m$, we derive a macroscopic equation in the limit $m \rightarrow \infty$, capturing the dominant effects in large-size multiple-wise order.