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arXiv · 2502.08276

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

Shaoxuan Cui·Chencheng Zhang·Bin Jiang·Hildeberto Jardón Kojakhmetov·Ming Cao
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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.

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Shaoxuan Cui, Chencheng Zhang, Bin Jiang, Hildeberto Jardón Kojakhmetov, Ming Cao. 2025-02-12. Higher-order Laplacian dynamics on hypergraphs with cooperative and antagonistic interactions. https://arxiv.org/abs/2502.08276

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