[Paper Review] Sparse exchangeable graphs and their limits via graphon processes

Christian Borgs, Jennifer Chayes+2 more|arXiv (Cornell University)|Jan 1, 2017

Limits and Structures in Graph TheoryMathematics31 references46 citations

TL;DR

This paper introduces graphon processes—random families of sparse, exchangeable graphs built via Poisson point processes on σ-finite measure spaces—where edge probabilities are governed by an integrable graphon function W. The key result is that such processes exhibit convergent subgraph frequencies and converge in a generalized cut metric to their generating graphon, with the graphon identifiable only up to cut-distance equivalence.

ABSTRACT

In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in R+. Here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over σ-finite measure spaces, we can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases. Explicitly, modelling the underlying space of features by a σ-finite measure space (S, S, µ) and the connection probabilities by an integrable function W : S × S → [0, 1], we construct a random family (Gt)t≥0 of growing graphs such that the vertices of Gt are given by a Poisson point process on S with intensity tµ, with two points x, y of the point process connected with probability W(x, y). We call such a random family a graphon process. We prove that a graphon process has convergent subgraph frequencies (with possibly infinite limits) and that, in the natural extension of the cut metric to our setting, the sequence converges to the generating graphon. We also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero. More generally, we study metric convergence for arbitrary (not necessarily random) sequences of graphs, and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails. Finally, we prove that every graphon is equivalent to a graphon on R+ equipped with Lebesgue measure.

Motivation & Objective

• To extend classical graphon theory to sparse graphs by generalizing the underlying measure space from probability to σ-finite measures.
• To model exchangeable sparse graphs, including the Caron-Fox model, using Poisson point processes on σ-finite measure spaces.
• To establish convergence of subgraph frequencies and metric convergence for graph sequences in this generalized setting.
• To characterize when a sequence of graphs admits a convergent subsequence via the notion of uniform regularity of tails.
• To show every graphon is equivalent to one defined on R+ with Lebesgue measure, enabling canonical representation.

Proposed method

• Define a graphon process as a random family (Gt)t≥0 where vertices are generated by a Poisson point process on a σ-finite measure space (S, S, µ) with intensity tµ.
• Specify edge formation between vertices x and y with probability W(x, y), where W: S × S → [0, 1] is an integrable function.
• Generalize the cut metric to σ-finite measure spaces to define convergence of graph sequences.
• Prove that subgraph frequencies in the graphon process converge, possibly to infinite limits, under this generalized metric.
• Introduce the concept of uniform regularity of tails as a necessary and sufficient condition for subsequence convergence in arbitrary graph sequences.
• Demonstrate that every graphon is equivalent to a graphon on R+ with Lebesgue measure under cut distance.

Experimental results

Research questions

• RQ1 Can exchangeable sparse graphs be modeled using a generalization of graphons beyond probability spaces?
• RQ2 How can convergence of subgraph frequencies be established in sparse, exchangeable graph models?
• RQ3 What conditions ensure metric convergence of arbitrary graph sequences in the generalized setting?
• RQ4 Is the generating graphon of a graphon process identifiable, and if so, up to what equivalence?
• RQ5 Can every graphon be represented equivalently on R+ with Lebesgue measure under the cut distance?

Key findings

• Graphon processes with integrable graphons W on σ-finite measure spaces exhibit convergent subgraph frequencies, even when limits are infinite.
• The sequence of graphs generated by a graphon process converges in the generalized cut metric to the generating graphon W.
• The graphon W is identifiable only up to equivalence under cut distance zero, meaning graphs with zero cut distance are indistinguishable in the limit.
• A sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying uniform regularity of tails.
• Every graphon is equivalent to a graphon defined on R+ equipped with Lebesgue measure, providing a canonical representation.