英国伦敦东北大学Petri, Giovanni团队研究了高阶拉普拉斯重整化。2025年2月24日出版的《自然—物理学》杂志发表了这项成果。

重整化群是物理学中标度、标度不变性和普适性理论的支柱。最近，该工具已通过基于扩散动力学的方案适用于具有成对交互的复杂网络。然而，随着多元相互作用在复杂系统中的重要性越来越明显，迫切需要将重整化群方法扩展到高阶网络。

研究组填补了这一空白，并提出了任意高阶网络的拉普拉斯重整化群方案。该方法的核心是引入跨阶拉普拉斯算子，其通过允许描述可以通过任何其他阶的超边发生在任何阶超边上的扩散过程来推广现有的高阶拉普拉斯算子。这种方法使研究组能够探测高阶结构，定义不同阶的标度不变性，并提出一种粗粒度方案。在受控合成高阶系统上验证了该方法，然后使用其来检测来自多个域的现实世界复杂系统的特定阶尺度不变轮廓的存在。

附：英文原文

Title: Higher-order Laplacian renormalization

Author: Nurisso, Marco, Morandini, Marta, Lucas, Maxime, Vaccarino, Francesco, Gili, Tommaso, Petri, Giovanni

Issue&Volume: 2025-02-24

Abstract:

The renormalization group is a pillar of the theory of scaling, scale invariance and universality in physics. Recently, this tool has been adapted to complex networks with pairwise interactions through a scheme based on diffusion dynamics. However, as the importance of polyadic interactions in complex systems becomes more evident, there is a pressing need to extend the renormalization group methods to higher-order networks. Here we fill this gap and propose a Laplacian renormalization group scheme for arbitrary higher-order networks. At the heart of our approach is the introduction of cross-order Laplacians, which generalize existing higher-order Laplacians by allowing the description of diffusion processes that can happen on hyperedges of any order via hyperedges of any other order. This approach enables us to probe higher-order structures, define scale invariance at various orders and propose a coarse-graining scheme. We validate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.

DOI: 10.1038/s41567-025-02784-1

Source: https://www.nature.com/articles/s41567-025-02784-1