Impact of directionality on the emergence of Turing patterns on m-directed higher-order structures

TL;DR

This paper extends the theory of Turing instability to reaction-diffusion systems on m-directed hypergraphs, showing that directionality broadens the conditions for pattern formation, with analytical and simulation validation.

Contribution

It introduces a novel framework for Turing patterns on m-directed hypergraphs, generalizing directed networks and demonstrating the role of directionality in pattern emergence.

Findings

Directionality promotes Turing instability.

Two Laplace matrices enable analytical proof of pattern emergence.

Simulations confirm theoretical predictions on hypergraphs.

Abstract

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on m-directed hypergraphs, the latter being generalization of hypergraphs where nodes forming hyperedges can be shared into two disjoint sets, the head nodes and the tail nodes. This framework encodes thus for a privileged direction for the reaction to occur: the joint action of tail nodes is a driver for the reaction involving head nodes. It thus results a natural generalization of directed networks. Based on a linear stability analysis we have shown the existence of two Laplace matrices, allowing to analytically prove that Turing patterns, stationary or wave-like, emerges for a much broader set of parameters in the m-directed setting. In particular directionality promotes Turing instability, otherwise absent in the symmetric case. Analytical results are compared to simulations performed by using the Brusselator model defined on a m-directed d-hyperring as well as on a m-directed random hypergraph.