Clustering implies geometry in networks

TL;DR

This paper demonstrates that networks with fixed degree and clustering properties are inherently geometric, with strong clustering implying an underlying geometric structure, especially on the real line.

Contribution

It establishes a link between fixed clustering and geometricity in networks, providing conditions under which networks are essentially geometric graphs.

Findings

Networks with fixed degree and clustering are equivalent to geometric graphs on the real line.

Strong clustering across nodes indicates an underlying geometric structure.

The methods are broadly applicable to various network ensembles and quantum gravity problems.

Abstract

Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in an ensemble of random geometric graphs. Here we identify structural properties of networks that guarantee that random graphs having these properties are geometric. Specifically we show that random graphs in which expected degree and clustering of every node are fixed to some constants are equivalent to random geometric graphs on the real line, if clustering is sufficiently strong. Large numbers of triangles, homogeneously distributed across all nodes as in real networks, are thus a consequence of network geometricity. The methods we use to prove this are quite general and applicable to other network ensembles, geometric or not, and to certain problems in quantum gravity.