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

We hereby develop the theory of Turing instability for reaction–diffusion systems defined on $m$-directed hypergraphs, the latter being a 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, emerge 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.