# Turing patterns mediated by network topology in homogeneous active systems

## TL;DR

This paper explores how Turing patterns can form in reaction-diffusion systems on various network topologies, revealing the influence of network structure and diffusion parameters on pattern stability and emergence.

## Contribution

It provides a general mathematical framework for Turing pattern formation on networks, linking network topology, diffusion, and eigenvector localization to pattern stability.

## Key findings

- Turing instability can occur in any network topology with appropriate diffusion.
- Network degree fluctuations influence pattern stability and stochasticity.
- Patterns are shaped by the interplay of diffusion coefficients and eigenvector properties.

## Abstract

Mechanisms of pattern formation---of which the Turing instability is an archetype---constitute an important class of dynamical processes occurring in biological, ecological and chemical systems. Recently, it has been shown that the Turing instability can induce pattern formation in discrete media such as complex networks, opening up the intriguing possibility of exploring it as a generative mechanism in a plethora of socioeconomic contexts. Yet, much remains to be understood in terms of the precise connection between network topology and its role in inducing the patterns. Here, we present a general mathematical description of a two-species reaction-diffusion process occurring on different flavors of network topology. The dynamical equations are of the predator-prey class, that while traditionally used to model species population, has also been used to model competition between antagonistic ideas in social systems. We demonstrate that the Turing instability can be induced in any network topology, by tuning the diffusion of the competing species, or by altering network connectivity. The extent to which the emergent patterns reflect topological properties is determined by a complex interplay between the diffusion coefficients and the localization properties of the eigenvectors of the graph Laplacian. We find that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic.