Non-linear consensus dynamics on temporal hypergraphs with random noisy higher-order interactions

Abstract

Complex networks encoding the topological architecture of real-world complex systems have recently been undergoing a fundamental transition beyond pairwise interactions described by dyadic connections among nodes. Higher-order structures such as hypergraphs and simplicial complexes have been utilized to model group interactions for varied networked systems from brain, society, to biological and physical systems. In this article, we investigate the consensus dynamics over temporal hypergraphs featuring non-linear modulating functions, time-dependent topology and random perturbations. Based upon analytical tools in matrix, hypergraph, stochastic process and real analysis, we establish the sufficient conditions for all nodes in the network to reach consensus in the sense of almost sure convergence and $\mathscr{L}^2$ convergence. The rate of consensus and the moments of the equilibrium have been determined. Our results offer a theoretical foundation for the recent series of numerical studies and physical observations in the multi-body non-linear dynamical systems.