[Paper Review] The Class of Random Graphs Arising from Exchangeable Random Measures

Victor Veitch, Daniel M. Roy|arXiv (Cornell University)|Dec 7, 2015

Point processes and geometric inequalitiesMathematics37 references72 citations

TL;DR

This paper introduces a general class of random graphs based on exchangeable random measures, characterized by a graphex (I, S, W) that generalizes both dense exchangeable graphs and sparse models. It establishes a representation theorem via Kallenberg's framework, enabling tractable modeling of real-world networks with sparse, heavy-tailed, and small-world structures.

ABSTRACT

We introduce a class of random graphs that we argue meets many of the desiderata one would demand of a model to serve as the foundation for a statistical analysis of real-world networks. The class of random graphs is defined by a probabilistic symmetry: invariance of the distribution of each graph to an arbitrary relabelings of its vertices. In particular, following Caron and Fox, we interpret a symmetric simple point process on $\mathbb{R}_+^2$ as the edge set of a random graph, and formalize the probabilistic symmetry as joint exchangeability of the point process. We give a representation theorem for the class of random graphs satisfying this symmetry via a straightforward specialization of Kallenberg's representation theorem for jointly exchangeable random measures on $\mathbb{R}_+^2$. The distribution of every such random graph is characterized by three (potentially random) components: a nonnegative real $I \in \mathbb{R}_+$, an integrable function $S: \mathbb{R}_+ o \mathbb{R}_+$, and a symmetric measurable function $W: \mathbb{R}_+^2 o [0,1]$ that satisfies several weak integrability conditions. We call the triple $(I,S,W)$ a graphex, in analogy to graphons, which characterize the (dense) exchangeable graphs on $\mathbb{N}$. Indeed, the model we introduce here contains the exchangeable graphs as a special case, as well as the "sparse exchangeable" model of Caron and Fox. We study the structure of these random graphs, and show that they can give rise to interesting structure, including sparse graph sequences. We give explicit equations for expectations of certain graph statistics, as well as the limiting degree distribution. We also show that certain families of graphexes give rise to random graphs that, asymptotically, contain an arbitrarily large fraction of the vertices in a single connected component.

Motivation & Objective

• To develop a statistically principled framework for modeling real-world networks that are sparse, heterogeneous, and exhibit complex structures like power-law degree distributions and small-world properties.
• To address the lack of a general, tractable, and flexible model for statistical network analysis by grounding it in a fundamental symmetry: invariance to vertex relabeling.
• To unify dense exchangeable graph models and sparse models (e.g., Caron-Fox) under a single theoretical framework using exchangeable random measures.
• To provide a representation theorem for such random graphs using Kallenberg’s theory of jointly exchangeable random measures.
• To enable statistical inference by characterizing the distribution of graphs through a minimal set of components: a graphex (I, S, W)

Proposed method

• Define random graphs via a symmetric simple point process on ℝ₊², interpreted as edge sets, with distribution invariant under arbitrary vertex relabelings (exchangeability).
• Apply Kallenberg’s representation theorem for jointly exchangeable random measures to derive a canonical form for the distribution of such graphs.
• Characterize the distribution of every graph in the class by a triple (I, S, W): a nonnegative constant I, a function S:ℝ₊→ℝ₊, and a symmetric measurable function W:ℝ₊²→[0,1] satisfying weak integrability conditions.
• Establish that the model includes dense exchangeable graphs (via graphons) and sparse models (e.g., Caron-Fox) as special cases.
• Derive explicit expressions for the expected number of edges and the limiting degree distribution using the graphex components.
• Propose a sampling scheme for subgraphs that yields KEGs, suggesting a notion of empirical graphex for estimation and model fitting.

Experimental results

Research questions

• RQ1 Can a general class of random graphs be defined that satisfies exchangeability (invariance to vertex relabeling) and captures both dense and sparse network structures?
• RQ2 What is the full probabilistic representation of such exchangeable random graphs, and how can their distributions be characterized in terms of tractable parameters?
• RQ3 How do the components (I, S, W) of the graphex relate to structural properties like edge density, degree distribution, and connectivity?
• RQ4 What sampling mechanisms yield KEGs from finite networks, and how can this inform empirical model fitting and estimation?
• RQ5 How does this framework relate to existing theories such as graph limit theory (e.g., L^p graphons) and the dense exchangeable graph framework?

Key findings

• The class of random graphs defined by exchangeable random measures admits a representation via a graphex (I, S, W), which fully characterizes the distribution of the graph.
• The model generalizes both dense exchangeable graphs (via graphons) and sparse models (e.g., Caron-Fox), unifying them under a single framework.
• The expected number of edges in a graphex-based model is given by an explicit integral expression involving I, S, and W, enabling statistical inference.
• The limiting degree distribution is derived in closed form, showing that the model can produce power-law-like degree distributions under appropriate choices of W.
• Certain families of graphexes can generate graphs where, asymptotically, a vanishing fraction of vertices are isolated and a positive fraction lies in a single connected component.
• A sampling scheme is proposed where subgraphs induced by Poisson-distributed, uniformly sampled vertices yield KEGs, suggesting a path to empirical graphex estimation.