Turing instability in Reaction-Diffusion models on complex networks

TL;DR

This paper investigates how the structure of complex networks influences Turing instability in reaction-diffusion systems, revealing that network architecture significantly affects stability regions through both numerical and theoretical analysis.

Contribution

It provides a comparative analysis of Turing instability across different complex network models, combining numerical and theoretical approaches to identify stability conditions.

Findings

Stable and unstable regions vary with network architecture.

Network type influences the parameter space of diffusion coefficients.

Theoretical results support numerical findings for large networks.

Abstract

In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied. Here, we focus on three types of models which generate complex networks, i.e. the Erd\H{o}s-R\'enyi, the Watts-Strogatz, and the threshold network models. From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion coefficients completely differ, depending on the network architecture. In addition, we theoretically discuss the stable and unstable regions in the cases of regular enhanced ring lattices which include regular circles, and networks generated by the threshold network model when the number of vertices is large enough.