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

François Caron, Francesca Panero, Judith Rousseau

TL;DR

This work analyzes graphex processes, linking sparsity, power-law degree behavior, and transitivity to the tail of the marginal graphon via regular variation. It derives precise asymptotics for the number of nodes $N_lpha$, edges $N^{(e)}_lpha$, and degree counts $N_{lpha,j}$, delineating four regimes determined by the tail index $c3$ (with $bc^{-1}(x) hicksim   x^{-c2}$ as $x o0$), including dense, sparse almost-dense, sparse with power-law ($c2 o0$, $0<c2<1$), and almost extremely sparse cases; in the sparse-power-law regime, the degree distribution exhibits a power-law with exponent in $(1,2)$. The paper also proves convergence results for global and local clustering coefficients and establishes central limit theorems for subgraph counts and $N_lpha$, under technical Assumptions that control small and large-degree behaviour and interactions. Finally, it offers a flexible parametrisation to separate latent structure from global sparsity and power-law properties, showing how to sparsify dense graphon models while preserving clustering characteristics, and demonstrates applicability to a broad class of graphex models including Caron2017. These results yield practical guidance for inference and model design in large, sparse networks.

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.

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

TL;DR

This work analyzes graphex processes, linking sparsity, power-law degree behavior, and transitivity to the tail of the marginal graphon via regular variation. It derives precise asymptotics for the number of nodes , edges , and degree counts , delineating four regimes determined by the tail index (with as ), including dense, sparse almost-dense, sparse with power-law (, ), and almost extremely sparse cases; in the sparse-power-law regime, the degree distribution exhibits a power-law with exponent in . The paper also proves convergence results for global and local clustering coefficients and establishes central limit theorems for subgraph counts and , under technical Assumptions that control small and large-degree behaviour and interactions. Finally, it offers a flexible parametrisation to separate latent structure from global sparsity and power-law properties, showing how to sparsify dense graphon models while preserving clustering characteristics, and demonstrates applicability to a broad class of graphex models including Caron2017. These results yield practical guidance for inference and model design in large, sparse networks.

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.

Paper Structure

This paper contains 3 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Assumptions
  3. Notations

Figures (3)

  • Figure 1: Illustration of the graph model based on exchangeable point processes. (left) A unit-rate Poisson process $(\theta_{i},\vartheta_{i})$, $i\in\mathbb{N}$ on $(0,\alpha]\times \mathbb{R}_{+}$. (right) For each pair $i\leq j$, set $Z_{ij}=Z_{ji}=1$ with probability $W(\vartheta_{i},\vartheta_{j})$. Here, $W$ is indicated by the red shading (darker shading indicates higher value). Similar to Figure 5 in Caron2017.
  • Figure 2: Illustration of some of the asymptotic results developed in this paper, applied to the generalised graphon model defined by Equations \ref{['eq:graphonCF']} and \ref{['eq:GGP']} with $\sigma_0=0.2$ and $\tau_0=2$. (a) Empirical degree distribution for a graph of size $\alpha=1000$ (red) and asymptotic degree distribution (dashed blue, see Corollary \ref{['th:sparsity1']}). (b) Average local (blue) and global (red) clustering coefficients for 10 graphs of growing sizes. Limit values are represented by dashed lines (see Propositions \ref{['prop:globalclustering']} and \ref{['prop:localclustering']}). (c) Local clustering coefficient for nodes of a given degree $j$, for a graph of size $\alpha=1000$. The limit value is represented by a dashed line (see Proposition \ref{['prop:localclustering']}).
  • Figure :