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

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