Self-similarity of complex networks and hidden metric spaces
Published in Physical Review Letters in 2008.

M. Ángeles Serrano
Institute of Theoretical Physics, LBS, SB, EPFL,
1015 Lausanne, Switzerland

Dmitri Krioukov
Cooperative Association for Internet Data Analysis - CAIDA
San Diego Supercomputer Center,
University of California, San Diego

Marián Boguñá
Departament de Física Fonamental,
Universitat de Barcelona, Martí i Franquès 1,
08028 Barcelona, Spain

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.