Nonlocal diffusion of variable order on complex networks

TL;DR

This paper explores nonlocal diffusion processes with variable fractional order on complex networks, establishing mathematical properties and demonstrating dynamic behaviors through examples.

Contribution

It introduces a novel extension of nonlocal dynamics on networks using variable-order fractional Laplacians, proving key theoretical properties.

Findings

Existence and uniqueness of solutions are established.

Solutions exhibit uniform asymptotic stability.

Examples illustrate diverse dynamic behaviors.

Abstract

Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. This is a new extension to non-stationary behavior of a class of non-local phenomena on complex networks for which both directed and undirected graphs are considered. Under appropriate assumptions, the existence, uniqueness, and uniform asymptotic stability of the solutions of the underlying initial value problem are proved. Some examples giving a sample of the behavior of the dynamics are also included.