Self-similarity of complex networks and hidden metric spaces

TL;DR

This paper shows that the self-similarity observed in certain scale-free networks can be explained by the existence of hidden metric spaces, with clustering reflecting geometric properties.

Contribution

It introduces a hidden metric space framework to interpret network self-similarity and validates it with models that replicate real network properties.

Findings

Hidden metric spaces explain network self-similarity.

Models with underlying metrics reproduce real network properties.

Clustering reflects the triangle inequality in hidden geometry.

Abstract

We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding 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.