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

Shaoxuan Cui, Chencheng Zhang, Bin Jiang, Hildeberto Jardón Kojakhmetov, Ming Cao

TL;DR

The paper addresses collective behavior on higher-order networks by developing higher-order Laplacian dynamics on signless and signed hypergraphs. It uses Metzler tensors and $H$-eigenvalue theory to establish global convergence properties, including a globally attractive line of equilibria for positive Metzler systems with zero Perron eigenvalue and consensus results for signless Laplacians on connected hypergraphs. For signed hypergraphs, the authors extend to bipartite consensus via gauge transformations and structural balance, and they also generalize to non-polynomial interaction functions with analogous convergence properties. Numerical examples validate the theoretical results and illustrate how higher-order interactions extend classical graph-based consensus (Altafini) to hypergraphs, with potential applications in opinion dynamics and group interactions.

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.

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

TL;DR

The paper addresses collective behavior on higher-order networks by developing higher-order Laplacian dynamics on signless and signed hypergraphs. It uses Metzler tensors and -eigenvalue theory to establish global convergence properties, including a globally attractive line of equilibria for positive Metzler systems with zero Perron eigenvalue and consensus results for signless Laplacians on connected hypergraphs. For signed hypergraphs, the authors extend to bipartite consensus via gauge transformations and structural balance, and they also generalize to non-polynomial interaction functions with analogous convergence properties. Numerical examples validate the theoretical results and illustrate how higher-order interactions extend classical graph-based consensus (Altafini) to hypergraphs, with potential applications in opinion dynamics and group interactions.

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.

Paper Structure

This paper contains 9 sections, 20 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on tensors and hypergraphs
  3. Positive Metzler systems with a zero Perron-eigenvalue
  4. Laplacian tensors, higher-order Laplacian dynamics, and consensus dynamics on a signless hypergraph
  5. Higher-order Laplacian dynamic and bipartite consensus on a signed hypergraph
  6. Interaction functions other than polynomials
  7. Numerical Examples
  8. Conclusions
  9. Extended invariance principle

Key Result

Lemma 1

Suppose that the two tensors ${A}$ and ${B}$ are diagonally similar, namely ${B}=D^{-(k-1)} {A} D$ for some invertible diagonal matrix $D$. Then $x$ is an eigenvector of ${B}$ corresponding to the eigenvalue $\lambda$ if and only if $y=D x$ is an eigenvector of ${A}$ corresponding to the same eigenv

Figures (2)

  • Figure 1: The horizontal axis is the time $t$. (a) The consensus example on a $4$-uniform hypergraph. (b) The consensus example on a non-uniform hypergraph containing both a $3$-uniform sub-hypergraph and a sub-graph. (c) The bipartite consensus example on a $4$-uniform hypergraph. (d) The bipartite consensus example on a non-uniform hypergraph containing both a $4$-uniform sub-hypergraph and a sub-graph.
  • Figure 2: The bipartite consensus example of a non-polynomial interaction on a non-uniform hypergraph containing both a $4$-uniform sub-hypergraph and a sub-graph. The horizontal axis is the time $t$.

Theorems & Definitions (42)

  • Lemma 1: shao2013general
  • Lemma 2: cui2024metzler
  • Lemma 3: cui2024metzler
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • ...and 32 more