Multiscale network renormalization: Scale-invariance without geometry

Physical Review Research, vol. 5

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 the 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 they provide a geometry-free route to renormalization.

Authors 3

1. Elena Garuccio Aachen Instituut-Lorentz for Theoretical Physics

   Leiden University

   Affiliation as printed

   Instituut-Lorentz for Theoretical Physics, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, the Netherlands

2. Margherita Lalli

   IMT School for Advanced Studies Lucca

   Affiliation as printed

   IMT School for Advanced Studies, Piazza S. Francesco 19, 55100 Lucca, Italy

3. Diego Garlaschelli Aachen Instituut-Lorentz for Theoretical Physics

   Leiden University · IMT School for Advanced Studies Lucca · Istituto Nazionale di Alta Matematica Francesco Severi · Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni

   Affiliation as printed

   IMT School for Advanced Studies, Piazza S. Francesco 19, 55100 Lucca, Italy

   INdAM-GNAMPA Istituto Nazionale di Alta Matematica, Italy

   Instituut-Lorentz for Theoretical Physics, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, the Netherlands