Cliques in geometric inhomogeneous random graph

TL;DR

This paper analyzes the formation and quantity of cliques in a geometric inhomogeneous random graph model, revealing phase transitions influenced by degree distribution and geometry, with implications for understanding real-world network clustering.

Contribution

It formalizes the trade-off between geometry and degree in clique formation and characterizes the typical clique type and its phase transition behavior.

Findings

Number of cliques exhibits a phase transition depending on degree exponent tau.

For small tau, geometry has negligible effect on clique count.

Identifies the typical structure of cliques in the model.

Abstract

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model containing geometry. The interplay between scale-freeness and geometry ensures that connections are likely to form between either high-degree vertices, or between close by vertices. At the same time it is rare for a vertex to have a high degree, and most vertices are not close to one another. This trade-off makes cliques more likely to appear between specific vertices. In this paper, we formalize this trade-off and prove that there exists a typical type of clique in terms of the degrees and the positions of the vertices that span the clique. Moreover, we show that the asymptotic number of cliques as well as the typical clique type undergoes a phase transition, in which only k and the degree-exponent tau are involved. Interestingly, this phase transition shows that for small values of tau, the underlying geometry of the model is irrelevant: the number of cliques scales the same as in a non-geometric network