Evolving Communities with Individual Preferences
Thomas Cass, Terry Lyons

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…
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