Abstract
In this article we report on a novel way to incorporate complex network structure into the analysis of interacting particle systems. More precisely, it is well-known that in well-mixed/homogeneous/all-to-all-coupled systems, one may derive mean-field limit equations such as Vlasov–Fokker–Planck equations (VFPEs). A mesoscopic VFPE describes the probability of finding a single vertex/particle in a certain state, forming a bridge between microscopic statistical physics and macroscopic fluid-type approximations. One major obstacle in this framework is to incorporate complex network structures into limiting equations. In many cases, only heuristic approximations exist, or the limits rely on particular classes of integral operators. In this paper, we notice that there is a much more elegant, and profoundly more general, way available due to recent progress in the theory of graph limits. In particular, we show how one may easily enter complex network dynamics via graphops (graph operators) into VFPEs.
Highlights
Interacting particle systems, or more generally, dynamical systems on graphs/networks, form one of the major building blocks of modern science [36, 2]
We shall refer to this class of equations as Vlasov-Fokker-Planck equation (VFPEs)
We are going to show that exactly this viewpoint is the missing ingredient to start a general theory of Vlasov-Fokker-Planck equations (VFPEs), and related classes of evolution equations, as limits of finite-dimensional network dynamics
Summary
Interacting particle systems, or more generally, dynamical systems on graphs/networks, form one of the major building blocks of modern science [36, 2]. A new approach to unify and extend graph limit theory was proposed by Backhausz and Szegedy [1] Their idea relies on viewing the adjacency matrix A(n) from an operator-theoretic perspective. We are going to show (formally) that exactly this viewpoint is the missing ingredient to start a general theory of VFPEs, and related classes of evolution equations, as limits of finite-dimensional network dynamics. How these kinetic models fit elegantly together with graphops This approach really seems to break a barrier that has held back the application of tools from dynamical systems, functional analysis, and evolution equations to broad classes of large-scale network dynamics problems.
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