标题： 高阶连接拉普拉斯算子用于有向单纯复形 显示英文标题

标题： Higher-order Connection Laplacians for Directed Simplicial Complexes

摘要： 高阶网络编码了复杂系统中存在的多体相互作用，例如大脑、蛋白质复合物和社会互动。 单纯复形是高阶网络，它们允许对拓扑结构和动态之间的相互作用进行全面研究。 然而，单纯复形有一个局限性，即它们仅能捕捉无向的高阶相互作用，而在现实场景中，常常需要引入单纯形的方向，从而扩展边方向的流行概念。 在图和网络中，磁拉普拉斯算子是连接拉普拉斯算子的一个特例，已成为处理边方向性的流行算子。 在这里，我们通过考虑单纯形方向引起的配置来制定高阶连接拉普拉斯算子，以解决处理方向性单纯复形的挑战。 具体而言，我们定义了二维方向单纯复形的所有连接拉普拉斯算子，并通过考虑单纯复形的指导性合成例子来讨论由单纯形方向引起的高阶扩散动力学。 当我们要考虑由于相邻单纯形的方向冲突而表现出非平凡挫折效应的高阶扩散时，可以采用所提出的高阶扩散过程。 显示英文摘要

摘要： Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of Connection Laplacian, is becoming a popular operator to treat edge directionality. Here we tackle the challenge of treating directional simplicial complexes by formulating Higher-order Connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the Connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.