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

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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.