标题： 基于通用缩放关系的谱空间RG理论 显示英文标题

标题： Spectral-Space RG Theory Based on Universal Scaling Relations

摘要： 无标度网络——从互联网到生物系统——表现出分层组织结构，这种结构抵抗传统的重整化群（RG）分析。 它们的尺度不变性和小世界连通性挑战了依赖于明确长度尺度的标准RG方法。 我们通过制定一种谱空间重整化群框架来解决这一挑战，该框架能够捕捉复杂网络中的结构和动态尺度。 利用拉普拉斯特征值谱，我们实现了不受几何约束的粗粒化变换。 这产生了连接分形维数、谱维数和度指数的普遍尺度关系，建立了网络重整化的第一个系统基础。 一种新颖的元图重构算法使得可以直接从谱数据中提取重整化的拓扑结构。 我们在多种现实网络中验证了我们的预测，并发现了新现象：演化的网络表现出多尺度行为，表明结构转变的发生，而谱非递归性揭示了静态拓扑中不可见的隐藏动态相关性。 应用于欧洲电力网络，我们的方法识别出远距离区域之间的潜在连接，这与观察到的故障传播一致。 我们的结果将谱空间重整化置于分析尺度不变网络的统一框架中，对网络科学、基础设施韧性以及统计物理具有广泛的影响。 显示英文摘要

摘要： Scale-free networks -- from the Internet to biological systems -- exhibit hierarchical organization that resists conventional renormalization group (RG) analysis. Their combination of scale invariance and small-world connectivity challenges standard RG methods, which rely on well-defined length scales. We resolve this challenge by formulating a spectral-space RG framework that captures both structural and dynamical scaling in complex networks. Leveraging the Laplacian eigenspectrum, we implement coarse-graining transformations unconstrained by geometry. This yields universal scaling relations connecting fractal dimensions, spectral dimensions, and degree exponents, establishing the first systematic foundation for network renormalization. A novel meta-graph reconstruction algorithm enables direct extraction of renormalized topologies from spectral data. We validate our predictions across diverse real-world networks and uncover new phenomena: evolving networks display multi-scaling behavior indicative of structural transitions, and spectral non-recursiveness reveals hidden dynamical correlations invisible in static topology. Applied to the European power grid, our method identifies latent connections between distant regions, consistent with observed fault propagation. Our results position spectral-space renormalization as a unified framework for analyzing scale-invariant networks, with broad implications for network science, infrastructure resilience, and statistical physics.