Pattern Formation on Networks: from Localised Activity to Turing Patterns

TL;DR

This paper investigates how patterns of activity, including localized and system-wide patterns, emerge in complex networks with reaction-diffusion dynamics, revealing the role of network structure and bifurcation phenomena.

Contribution

It demonstrates the connection between localized node activity and global patterns via snaking bifurcations, highlighting the influence of nodes with optimal degree in pattern formation.

Findings

Patterns are connected through snaking bifurcations.

Nodes with optimal degree significantly influence system behavior.

Multistable patterns are prevalent in complex network systems.

Abstract

Systems of dynamical interactions between competing species can be used to model many complex systems, and can be mathematically described by {\em random} networks. Understanding how patterns of activity arise in such systems is important for understanding many natural phenomena. The emergence of patterns of activity on complex networks with reaction-diffusion dynamics on the nodes is studied here. The connection between solutions with a single activated node, which can bifurcate from an undifferentiated state, and the fully developed system-scale patterns are investigated computationally. The different coexisting patterns of activity the network can exhibit are shown to be connected via a snaking type bifurcation structure, similar to those responsible for organising localised pattern formation in regular lattices. These results reveal the origin of the multistable patterns found in systems organised on complex networks. A key role is found to be played by nodes with so called {\em optimal degree}, on which the interaction between the reaction kinetics and the network structure organise the behaviour of the system. A statistical representation of the density of solutions over the parameter space is used as a means to answer important questions about the number of accessible states that can be exhibited in systems with such a high degree of complexity.