Evolving Communities with Individual Preferences

TL;DR

This paper develops rigorous mathematical tools to model the evolution of communities with individual preferences, focusing on weakly interacting systems and their continuum limits without relying solely on PDEs.

Contribution

It introduces a measure-theoretic framework for community evolution, establishing existence, uniqueness, and continuity of solutions, and connects microscopic preferences with macroscopic population behavior.

Findings

Proves existence and uniqueness of solutions for weakly interacting systems.

Establishes continuity of system behavior with respect to measure changes.

Provides a framework applicable to physical and social sciences without requiring PDE modeling.

Abstract

The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space where the measure determines the abundance of individual preferences. The preferences of an individual are described by a measurable choice of a rough path. We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions. In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over-arching PDE. Although one can match the standard nonlinear parabolic PDEs of McKean-Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE. An important technical result is continuity of the behaviour of the system with respect to changes in the measure assigning weight to individuals. Replacing the deterministic measure with the empirical distribution of an i.i.d. sample from it leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos. The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE. This work builds on the fine probability literature considering the limit behaviour for systems where a large no of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi-martingale measure. We mention some of the key papers.