A limit theorem for diffusions on graphs with variable configuration

TL;DR

This paper establishes a limit theorem for diffusion processes on variable graphs, providing explicit formulas for asymmetry parameters when groups of vertices contract into points, advancing understanding of stochastic processes on complex networks.

Contribution

It introduces a limit theorem for diffusions on graphs with changing configurations and parameters, including explicit formulas for asymmetry in contracting vertex groups.

Findings

Proved a limit theorem for diffusions on variable graphs.

Derived explicit formulas for asymmetry parameters in the limit.

Analyzed the case of vertex groups contracting into points.

Abstract

A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and configuration of graphs, where the processes are set on. The explicit formulae for the parameters of asymmetry for the vertices of the limiting graph are given in the case, when, in the pre-limiting graphs, some groups of vertices form knots contracting into a points.