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Evolving communities with individual preferences

Authors: T. Cass; T. Lyons;
APC: 2,903.65 EUR

Evolving communities with individual preferences

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.

Country
United Kingdom
Keywords

McKean-Vlasov model, propagation of chaos, rough stochastic differential equation, Probability (math.PR), Models of societies, social and urban evolution, FOS: Mathematics, Interacting random processes; statistical mechanics type models; percolation theory, Mathematics - Probability, Stochastic ordinary differential equations (aspects of stochastic analysis)

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
17
Top 10%
Top 10%
Average
Green
hybrid