Markov Chains and Random Walks with Memory on Hypergraphs: A Tensor-Based Approach
This provides new tools for analyzing higher-order networks with time-dependent effects, though it appears incremental as an extension of existing tensor methods to memory-based scenarios.
The authors tackled the problem of modeling complex systems with group interactions and memory effects by developing a tensor framework for higher-order Markov chains, showing it can approximate such chains with low-dimensional nonlinear systems and applying it to random walks on hypergraphs.
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.