On sparsity and power-law properties of graphs based on exchangeable point processes

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: a dense regime, a sparse, almost dense regime, a sparse regime with power-law behavior, and an almost extremely sparse regime. Our results allow us to derive a consistent estimator for the scalar parameter tuning the sparsity of the graph. We also propose a class of models within this framework where one can separately control the local, latent structure and the global sparsity/power-law properties of the graph.