Multiscale network renormalization: scale-invariance without geometry

TL;DR

This paper introduces a novel graph renormalization method applicable to any network, revealing a class of scale-invariant networks driven by hidden variables, and demonstrates its effectiveness on real-world networks like the International Trade Network.

Contribution

It presents a general, geometry-free renormalization scheme for heterogeneous networks, enabling multiscale analysis without relying on assumptions like community structure or hyperbolicity.

Findings

Identifies a class of scale-invariant networks with hidden variables.

Produces realistic scale-free networks with clustering and assortativity.

Develops a multiscale model of the International Trade Network.

Abstract

Systems with lattice geometry can be renormalized exploiting their coordinates in metric space, which naturally define the coarse-grained nodes. By contrast, complex networks defy the usual techniques, due to their small-world character and lack of explicit geometric embedding. Current network renormalization approaches require strong assumptions (e.g. community structure, hyperbolicity, scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Here we introduce a graph renormalization scheme valid for any hierarchy of heterogeneous coarse-grainings, thereby allowing for the definition of 'block-nodes' across multiple scales. This approach identifies a class of scale-invariant networks characterized by a necessary and specific dependence on additive hidden variables attached to nodes, plus optional dyadic factors. If the hidden variables are annealed, they lead to realistic scale-free networks with assortativity and finite local clustering, even in the sparse regime and in absence of geometry. If they are quenched, they can guide the renormalization of real-world networks with node attributes and distance-dependence or communities. As an application, we derive an accurate multiscale model of the International Trade Network applicable across arbitrary geographic partitions. These results highlight a deep conceptual distinction between scale-free and scale-invariant networks, and provide a geometry-free route to renormalization.