标题： 高阶拉普拉斯在具有合作与对抗相互作用的超图上的动力学 显示英文标题

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

摘要： 无符号图上的拉普拉斯动力学表征了一类线性相互作用，其中所有代理之间的成对合作相互作用导致收敛到一个共同状态。 在结构平衡的带符号图上，代理收敛到相同幅度但相反符号的值（双部分共识），如著名的Altafini模型所示。 这些相互作用已被传统图建模，其中代理之间的关系始终是成对的。 相比之下，高阶网络（如超图）提供了捕捉代理之间更复杂群体相互作用的可能性。 这引发了一个自然的问题：能否通过使用超图来分析集体行为？ 答案是肯定的。 在本文中，首先引入了无符号超图上的高阶拉普拉斯动力学，并在齐次和非齐次多项式系统的框架下研究了各种集体收敛行为。 此外，通过采用规范变换并利用张量相似性，我们将这些动力学扩展到带符号超图，与Altafini模型相呼应。 此外，我们在此框架内探索了非多项式相互作用函数。 理论结果通过几个数值例子进行了演示。 显示英文摘要

摘要： 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.