Shaoxuan Cui

Shaoxuan Cui, Lingfei Wang, Hildeberto Jardon-Kojakhmetov et al.

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.

Chencheng Zhang, Hao Yang, Bin Jiang et al.

This paper develops an entropy-based stability and robustness framework for nonlinear hypergraph dynamics with conservation and flow balance. We consider generator-form systems on the simplex whose state-dependent transition rates capture higher-order (tensor) interactions among nodes. Under a tensor generalized detailed-balance (TGDB) condition, we show that the system admits a unique equilibrium and an entropy Lyapunov function ensuring global asymptotic stability. The Jacobian restricted to the tangent subspace of the simplex is Hurwitz, and its spectral gap determines the exponential convergence rate. Building on this structure, we derive first-order sensitivity bounds of the equilibrium under perturbations of the coupling tensor and establish a local input-to-state stability (ISS) estimate with respect to external inputs. The results reveal a quantitative link between the spectral gap and the system's robustness margin: larger spectral gaps imply smaller equilibrium shifts and faster recovery under structural or parametric perturbations. Numerical experiments on tensor-coupled flow models confirm the theoretical predictions and illustrate how the proposed entropy-dissipating framework unifies stability and robustness analysis for conservative higher-order network systems.

Shaoxuan Cui, Qi Zhao, Guanlin Li et al.

This paper studies stabilization and its corresponding closed-loop region-of-attraction (ROA) for homogeneous polynomial dynamical systems whose nonlinear term admits an orthogonally decomposable (ODECO) tensor representation. While recent tensor-based results provide explicit solutions and sharp global characterizations for open-loop ODECO systems, closed-loop synthesis and computable ROA estimates are still often dominated by local linearization or Lyapunov/SOS (sum of squares) methods, which can be conservative and computationally demanding. We propose a structure-preserving linear feedback design that shares the ODECO eigenbasis of the system's tensor, thereby enabling closed-form trajectory expressions, explicit convergence/escape thresholds, and sharp ROA characterizations. Under mild conditions, we further derive robustness/ISS-type bounds for bounded disturbances. Numerical examples validate the theoretical results.

Qi Zhao, Shaoxuan Cui, Baolin Zhang et al.

Winner-take-all (WTA)--type selection is a fundamental mechanism in networked competition, yet its dependence on higher-order interactions remains insufficiently understood. We study a Lotka--Volterra competitive dynamics on higher-order networks, where classical pairwise inhibition is augmented by multi-way interaction terms induced by hyperedges of uniform hypergraphs. The proposed model shows multiple competitive outcomes, including WTA, winner-share-all (WSA), and variant winner-take-all (VWTA). The existence, uniqueness and stability of equilibria are rigorously proved through mathematical analysis, which relies on classical stability theory and recent advances in tensor algebra. We show that the eventual selection outcome is relatively insensitive to the hyperedge order and the specific higher-order coupling structure, and is instead determined by a small set of interpretable scalar parameters, such as the ratio between self-inhibition and lateral-inhibition and the external inputs. Numerical experiments support the theory by showing that higher-order interactions affect convergence and steady states, yet yield the similar outcome taxonomy (WTA/WSA/VWTA) as in standard graphs. These results provide a network-scientific explanation of the robustness of WTA-type outcomes under complex group interactions and offer principled guidance for designing selection mechanisms on higher-order networks.