Stability and Robustness of Tensor-Coupled Flow-Conservation Dynamical Systems on Hypergraphs

TL;DR

This paper introduces an entropy-based framework for analyzing the stability and robustness of nonlinear hypergraph dynamics with higher-order interactions, providing theoretical guarantees and numerical validation.

Contribution

It develops a novel entropy Lyapunov function under tensor generalized detailed-balance, ensuring global stability and quantifying robustness via spectral gap analysis.

Findings

Unique equilibrium exists under TGDB condition.

Spectral gap determines exponential convergence rate.

Larger spectral gaps imply greater robustness and faster recovery.

Abstract

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.