Equilibrium-preserving Laplacian renormalization group

TL;DR

This paper introduces a rigorous Laplacian renormalization group method that preserves equilibrium states, enabling better coarse-graining of networks and revealing scale-invariance properties in hypergraphs.

Contribution

It formulates an equilibrium-preserving LRG with a principled coarse-graining procedure based on spectral properties and equilibrium flows, advancing network renormalization techniques.

Findings

Vertex degree and hyperedge distributions flow toward Poissonian forms in hypertrees.

Hypergraphs without finite spectral dimension broaden toward power-law distributions.

The method reveals interrelations between informational, structural, and dynamical scale-invariances.

Abstract

Diffusion over networks has recently been used to define spatiotemporal scales and extend Kadanoff block spins of Euclidean space to supernodes of networks in the Laplacian renormalization group (LRG). Yet, its ad hoc coarse-graining procedure remains underdeveloped and unvalidated, limiting its broader applicability. Here we rigorously formulate an LRG preserving the equilibrium state, offering a principled coarse-graining procedure. We construct the renormalized Laplacian matrix preserving dominant spectral properties using a proper, quasi-complete basis transformation and the renormalized adjacency matrix preserving mean connectivity from equilibrium-state flows among supernodes. Applying recursively this equilibrium-preserving LRG to various hypergraphs, we find that in hypertrees with low spectral dimensions vertex degree and hyperedge cardinality distributions flow toward Poissonian forms, while in hypergraphs lacking a finite spectral dimension they broaden toward power-law forms when starting from Poissonian ones, revealing how informational, structural, and dynamical scale-invariances are interrelated.