Markov Chains and Random Walks with Memory on Hypergraphs: A Tensor-Based Approach

Abstract: Many complex systems exhibit interactions that depend not only on pairwise connections, but also group structures and memory effects. To capture such effects, we develop a unified tensor framework for modeling higher-order Markov chains with memory. Our formulation introduces an even-order paired tensor that links folded and unfolded dynamics and characterizes their steady states and convergence. We further show that a Markov chain with memory can be approximated by a low-dimensional nonlinear tensor-based system and then provide a full system analysis. As an application, we define random walks on hypergraphs where memory naturally arises from the hyperedge structure, providing new tools for analyzing higher-order networks with time-dependent effects.