[Paper Review] Graph limits and exchangeable random graphs

Persi Diaconis, Svante Janson|ArXiv.org|Dec 17, 2007

Limits and Structures in Graph TheoryMathematics17 references279 citations

TL;DR

This paper establishes a rigorous connection between de Finetti's theorem on exchangeable random arrays and the theory of graph limits, showing that exchangeable random graphs arise as mixtures of graphon-based models. The key contribution is a representation theorem proving that every exchangeable random infinite directed graph corresponds to a graph limit via a random graphon, unifying classical probability with modern graph limit theory.

ABSTRACT

We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.

Motivation & Objective

• To unify de Finetti’s theory of exchangeable arrays with the emerging theory of graph limits.
• To clarify the probabilistic foundations of graph limit theory using exchangeability and representation theorems.
• To extend the Aldous–Hoover representation theorem to directed graphs and graph limits.
• To show that graph limits correspond precisely to distributions of exchangeable random graphs with random graphons.
• To provide a probabilistic interpretation of graph limits through exchangeable random arrays and measure-preserving transformations.

Proposed method

• Uses the Aldous–Hoover representation theorem for jointly exchangeable arrays to decompose edge indicators into functions of i.i.d. uniform random variables.
• Defines a graphon quintuple $\mathbf{W} \in \mathcal{W}_5$ encoding edge probabilities between vertices with random labels, including loop indicators.
• Constructs random infinite directed graphs $G(\infty, \mathbf{W})$ and $G(\infty, \mathbf{W}, p)$ using independent uniform variables and measurable functions.
• Applies the extreme point characterization from de Finetti’s theorem to show that graph limits arise as distributions of such exchangeable graphs.
• Uses measure-preserving maps to reparameterize loop probabilities and reduce the representation to a quadruple $\mathbf{W}$ and a marginal $p$.
• Establishes that convergence of finite graphs to a limit is almost sure, with the limit characterized by the graphon $\mathbf{W}$.

Experimental results

Research questions

• RQ1 How can de Finetti’s theorem on exchangeable arrays be extended to model random graphs and their limits?
• RQ2 What is the precise probabilistic structure underlying graph limits in the context of exchangeable random graphs?
• RQ3 How do the Aldous–Hoover and de Finetti representations unify in the setting of directed graphs?
• RQ4 What conditions ensure that a graph limit corresponds to an exchangeable random graph distribution?
• RQ5 Can the representation of exchangeable arrays be adapted to include directed edges and loops in a consistent way?

Key findings

• Every exchangeable random infinite directed graph arises as a mixture of graphs generated by a random graphon $\mathbf{W}$, establishing a one-to-one correspondence with graph limits.
• The graph limit $\Gamma_{\mathbf{W}}$ is characterized by the homomorphism density $t(F, \Gamma_{\mathbf{W}}) = \mathbb{P}(F \subseteq G(k, \mathbf{W}))$ for any finite directed graph $F$.
• Almost sure convergence of finite graphs $G(n, \mathbf{W})$ to $\Gamma_{\mathbf{W}}$ holds as $n \to \infty$, confirming the limit's stability.
• The loop indicators $X_{ii}$ form an exchangeable sequence, and their distribution is a mixture of i.i.d. Bernoulli($p$) variables, recovering de Finetti’s theorem.
• The representation of exchangeable arrays via $f_1(\xi_i)$ and $f_2(\xi_i, \xi_j, \xi_{ij})$ allows construction of graph limits through measurable functions on $[0,1]^4$.
• Graph limits in $\mathcal{D}_\infty$ are precisely the distributions $\Gamma_{\mathbf{W}}$ for $\mathbf{W} \in \mathcal{W}_5$, or equivalently $\Gamma_{\mathbf{W},p}$ for $\mathbf{W} \in \mathcal{W}_4$ and $p \in [0,1]$.