From Spatial to Spectral: Network Renormalization via Dynamical Correlations

Abstract

Network renormalization has traditionally relied on spatial adjacency-grouping nearby nodes together, but this approach fails to capture the dynamical correlations that govern system-wide behavior in scale-free networks. We present a spectral-space renormalization framework that enables coarse-graining based on dynamical coherence rather than geometric proximity. Within this framework, diffusion processes naturally constitute renormalization transformations in spectral space, yielding scaling relations that connect network dimensions with critical exponents. Building on this foundation, we develop a meta-graph reconstruction algorithm that systematically maps spectral information back into explicit topology while preserving dynamical correlations. The resulting renormalized networks uncover organizational structures that remain invisible to adjacency-based methods, including long-range correlations between structurally distant nodes that reflect coherent dynamical responses. Applications to Internet topologies, yeast regulatory networks, and European power grids demonstrate the broad applicability of this framework. The algorithm consistently extracts fractal, spectral, and random-walk dimensions with theoretical consistency across diverse systems. In power grids, it further reveals hidden failure pathways, exposing transcontinental correlations that match documented cascade patterns. In Internet networks, it reveals multiscaling behavior as the topology evolves over time. By shifting network renormalization from spatial geometry to dynamical flow, this work provides a unified foundation for understanding how information, energy, and failures propagate through complex systems, with direct implications for infrastructure resilience and network vulnerability assessment.