P. Villegas

T. Gili

G. Caldarelli

A. Gabrielli

Applying diffusion-based graph operators to complex networks identifies the proper spatiotemporal scales by overcoming small-world effects.

The renormalisation group is a powerful tool for examining organisational scales in dynamical systems. But applying it to complex networks presents challenges because of correlations between intertwined scales. We develop a Laplacian renormalisation group approach that can identify proper spatiotemporal scales in complex networks, introducing so-called Kadanoff supernodes to resolve detrimental small-world effects.

The renormalization group is the cornerstone of the modern theory of universality and phase transitions and it is a powerful tool to scrutinize symmetries and organizational scales in dynamical systems. However, its application to complex networks has proven particularly challenging, owing to correlations between intertwined scales. To date, existing approaches have been based on hidden geometries hypotheses, which rely on the embedding of complex networks into underlying hidden metric spaces. Here we propose a Laplacian renormalization group difusion-based picture for complex networks, which is able to identify proper spatiotemporal scales in heterogeneous networks. In analogy with real-space renormalization group procedures, we frst introduce the concept of Kadanof supernodes as block nodes across multiple scales, which helps to overcome detrimental small-world efects that are responsible for cross-scale correlations. We then rigorously defne the momentum space procedure to progressively integrate out fast difusion modes and generate coarse-grained graphs. We validate the method through application to several real-world networks, demonstrating its ability to perform network reduction keeping crucial properties of the systems intact.

Network renormalization

Laplacian renormalization group for heterogeneous networks