On sparsity, power-law and clustering properties of graphex processes

TL;DR

This paper analyzes the structural properties of graphex-based graphs, revealing regimes of density and sparsity, and provides asymptotic and clustering results, along with models controlling graph features.

Contribution

It introduces a framework for understanding graph regimes and properties, including new asymptotic expressions and models for controlling graph structure.

Findings

Identifies four regimes: dense, almost dense, power-law sparse, and extremely sparse.

Shows convergence of clustering coefficients to constants under mild conditions.

Derives a central limit theorem for the number of nodes.

Abstract

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.