Invariance of Competition Outcomes in Hypergraph Competitive Dynamics

TL;DR

This paper investigates how higher-order interactions in hypergraph networks influence competitive outcomes, demonstrating that the type of winner-take-all result is robust to complex multi-way interactions and depends mainly on key scalar parameters.

Contribution

It provides a rigorous mathematical analysis showing the invariance of competition outcomes in hypergraph dynamics, extending classical models to higher-order interactions.

Findings

Outcome types (WTA, WSA, VWTA) are insensitive to hyperedge order.

Steady states depend mainly on scalar parameters like self-inhibition ratios.

Numerical experiments confirm similar outcome taxonomy as in standard graphs.

Abstract

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.