Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. We show that higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order m . We provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models...

中文翻译：

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单纯形配合物的高阶谱：重归一化组方法

网络拓扑是一个蓬勃发展的跨学科学科，与包括量子引力和大脑研究在内的不同学科有关。在网络拓扑中研究的离散拓扑对象是简单复形。简单复合体不仅通过考虑成对交互，而且还考虑了两个以上节点之间的多体交互来对网络进行泛化。高阶拉普拉斯算子是拓扑算子，它们描述简单复数上的高阶扩散，并构成捕获网络拓扑与动力学之间相互作用的自然数学对象。我们表明，高阶上下拉普拉斯算子可以具有有限的谱维，从而表征了依赖于阶m的简单复形上扩散过程的长时间行为。