Harmonic morphisms and dynamical invariants in network renormalization

TL;DR

This paper establishes that discrete harmonic morphisms are the minimal condition for preserving random walk dynamics during network coarse-graining, providing a new framework for evaluating multiscale network models.

Contribution

It formalizes the concept of harmonic morphisms in network renormalization, introduces the harmonic degree as a diagnostic, and demonstrates their application across various real-world network renormalization methods.

Findings

Laplacian renormalization often yields exact harmonic morphisms in networks.

Harmonic morphisms preserve random walk transition structures at specific scales.

Different renormalization methods produce distinct dynamical fingerprints.

Abstract

Renormalization of complex networks requires principled criteria for assessing whether a coarse-graining preserves dynamical content. We prove that discrete harmonic morphisms -- surjective maps preserving harmonic functions -- provide the minimal condition under which random walks on a fine-grained network project exactly onto random walks on its coarse-grained image, through an appropriate random time change. We formalize this via the harmonic degree, a diagnostic quantifying how closely any network coarse-graining approximates a harmonic morphism. Applying this framework to geometric, Laplacian, and GNN-based renormalization across real-world networks, we find that each method produces a distinct dynamical fingerprint encoding its underlying physical assumptions. Most strikingly, Laplacian renormalization spontaneously yields exact harmonic morphisms in several networks, achieving exact preservation of first-exit random-walk transition structure at specific scales, a property that entropic susceptibility fails to detect. Our results identify a discrete analog of diffusion-preserving conformal maps for irregular network topologies and provide quantitative tools for designing and evaluating multi-scale network descriptions.