Pattern formation in reaction-diffusion information propagation model on multiplex simplicial complexes

Abstract Turing pattern for explaining the spatial distribution in nature mostly focused on continuous media and existing networks but there are few attempts at studying them on the systems with high-order interactions. Considering that high-order interactions have a particularly significant impact on rumor propagation, this article establishes a generalized reaction-diffusion rumor propagation model based on a multiplex network, where simplicial complexes are employed to describe the high-order structures. It aims to provide the spatial distribution patterns of the population participating in rumor propagation and identify structural factors that affect such patterns. We theoretically give the necessary conditions for Turing instability in the single-layer and multiplex networks with consideration of high-order interactions. In the numerical simulation, we demonstrate that Turing pattern is controlled by adjusting the diffusion coefficient, high-order structure intensity, and average degree of the network. The results conclude that: (i) in a single-layer network, Turing pattern only exists when high-order interactions appear, and the difference in diffusion rate plays a decisive role, (ii) in a multiplex network, Turing pattern can still be observed under the same diffusion rates, which are affected by the difference of higher-order intensity between two layers, and (iii) in the existing networks, the average degree of the network takes an important impact on Turing pattern. All these findings contribute to comprehending the impact of network structure on pattern formation, especially the high-order interactions on Turing pattern.