Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the spectral properties of the simplicial complex model called "Network Geometry with Flavor" we provide evidence that the up and down higher-order Laplacians can have a finite spectral dimension whose value increases as the order of the Laplacian increases. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes.

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单纯复形：高阶谱维数和动力学

单纯复合体构成了交互复杂系统的基本拓扑结构，其中包括大脑和社会交互网络。它们是广义的网络结构，允许超越成对交互的框架，并捕获两个或多个节点之间强烈影响动态过程的多体交互。事实上，单纯复合拓扑不仅允许将动态变量分配给相互作用的复杂系统的节点，还允许分配给链接、三角形等。在这里，我们展示了证据，即使我们比较属于同一单纯复形的单纯形的动力学，在不同维度的单纯形上定义的动力学也可能显着不同。通过研究称为“