Geometric renormalization of weighted networks

TL;DR

This paper extends geometric renormalization to weighted networks, revealing that weights exhibit multiscale self-similarity and enabling the creation of scaled-down network replicas for analysis.

Contribution

It introduces a new framework for applying geometric renormalization to weighted networks, highlighting the significance of maximum weights in multiscale self-similarity.

Findings

Weights show multiscale self-similarity under renormalization.

Maximum weight selection is validated as a meaningful procedure.

Scaled-down network replicas can be generated for analysis.

Abstract

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. In this letter, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that weights in real networks exhibit multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications.