A geometry-induced topological phase transition in random graphs

TL;DR

This paper demonstrates a topological phase transition in random geometric graphs, linking clustering behavior to a fundamental change in the network's geometric structure, with implications for understanding real-world networks.

Contribution

It proves the geometric-to-nongeometric phase transition in clustering as topological, revealing anomalous features and implications for modeling real networks.

Findings

Clustering undergoes a continuous phase transition in random geometric graphs.

The transition is topological with diverging entropy and unique finite size scaling.

High clustering in real networks may indicate a nongeometric phase.

Abstract

Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric-to-nongeometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finite size scaling behavior of clustering. Moreover, a slow decay of clustering in the nongeometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.