Higher-order Laplacian dynamics on hypergraphs with cooperative and antagonistic interactions

TL;DR

This paper extends Laplacian dynamics to hypergraphs with cooperative and antagonistic interactions, analyzing collective behaviors and convergence properties in complex group-wise interaction networks.

Contribution

It introduces higher-order Laplacian dynamics on signless and signed hypergraphs, extending classical models to capture complex group interactions and analyzing their convergence behaviors.

Findings

Hypergraph Laplacian dynamics exhibit diverse convergence behaviors.

Extension of Altafini model to hypergraph structures.

Numerical examples validate theoretical results.

Abstract

Laplacian dynamics on a signless graph characterize a class of linear interactions, where pairwise cooperative interactions between all agents lead to the convergence to a common state. On a structurally balanced signed graph, the agents converge to values of the same magnitude but opposite signs (bipartite consensus), as illustrated by the well-known Altafini model. These interactions have been modeled using traditional graphs, where the relationships between agents are always pairwise. In comparison, higher-order networks (such as hypergraphs), offer the possibility to capture more complex, group-wise interactions among agents. This raises a natural question: can collective behavior be analyzed by using hypergraphs? The answer is affirmative. In this paper, higher-order Laplacian dynamics on signless hypergraphs are first introduced and various collective convergence behaviors are investigated, in the framework of homogeneous and non-homogeneous polynomial systems. Furthermore, by employing gauge transformations and leveraging tensor similarities, we extend these dynamics to signed hypergraphs, drawing parallels to the Altafini model. Moreover, we explore non-polynomial interaction functions within this framework. The theoretical results are demonstrated through several numerical examples.