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.
Kim et al. (Thu,) studied this question.