Diffusion-driven instability of topological signals coupled by the Dirac operator

TL;DR

This paper explores how reaction-diffusion systems involving topological signals on networks, coupled via the Dirac operator, can exhibit Turing patterns that involve multiple topological dimensions, extending classical node-based models.

Contribution

It introduces a novel framework for reaction-diffusion processes on topological signals coupled through the Dirac operator, revealing conditions for Turing pattern formation involving multiple dimensions.

Findings

Turing patterns involve both nodes and links, never localized on a single dimension.

Projection of topological signals also exhibits Turing patterns.

Validated on network models and lattices with periodic boundary conditions.

Abstract

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover when the topological signals display Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.