Self-similarity of complex networks and hidden metric spaces

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.