Categorical approach to graph limits

We define and study a natural category of graph limits. The objects are pairs $(π,μ)$, where $π$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $μ$ (the distribution of edges) is an abstract finite measure on the square $(X,\mathcal{A})^2$. Morphisms are random maps between the underlying measurable spaces which preserve the distribution of vertices as well as the distribution of edges. We also define a convergence notion (inspired by s-convergence) for sequences of graph limits. We apply tools from category theory to prove the compactness of the space of all graph limits.

💡 Research Summary

The paper introduces a categorical framework for graph limits by representing a limit object as a pair $(\pi,\mu)$, called a □‑graphon, where $\pi$ is a probability measure describing the distribution of vertices on an abstract measurable space $(X,\mathcal A)$ and $\mu$ is a finite (symmetric) measure on the product space $(X,\mathcal A)^2$ describing the distribution of edges. This formulation generalizes the usual graphon (or s‑graphon) model, which fixes the vertex distribution to Lebesgue measure and only records an edge measure.

The central technical contribution is the definition of morphisms between □‑graphons. A morphism from $(\pi_X,\mu_X)$ on $(X,\mathcal A)$ to $(\pi_Y,\mu_Y)$ on $(Y,\mathcal B)$ is a Markov kernel $\kappa\colon X\to\mathcal B$ satisfying two preservation conditions: (i) the push‑forward of the vertex measure equals the target vertex measure, $\pi_Y=\kappa_\pi_X$, and (ii) the push‑forward of the edge measure via the product kernel $\kappa\otimes^2$ equals the target edge measure, $\mu_Y=\kappa\otimes^2_\mu_X$. Condition (i) guarantees that the random map respects the distribution of vertices, while (ii) ensures that the induced random mapping on pairs of vertices respects the edge distribution. This double‑preservation requirement is the novel ingredient that distinguishes the category from the classical category of measurable spaces with Markov kernels.

With these morphisms, the authors construct the category $\mathbf{Graphon}_\square$: objects are all □‑graphons (allowing arbitrary underlying measurable spaces), morphisms are the measure‑preserving Markov kernels described above, composition is the usual composition of kernels, and identities are Dirac kernels. An isomorphism in this category is a pair of kernels that are inverses of each other; equivalently, two □‑graphons are isomorphic precisely when there exists a random bijection that simultaneously preserves vertex and edge distributions.

The paper then defines a convergence notion for sequences of □‑graphons, inspired by the s‑convergence introduced by Borgs, Chayes, and Lovász. For each integer $k$, the $k$‑shape of a □‑graphon is the set of weighted graphs on $k$ vertices obtained by sampling $k$ points according to $\pi$ and recording the induced edge weights according to $\mu$. A sequence $(\pi_n,\mu_n)$ converges if, for every $k$, the corresponding $k$‑shapes converge in the Vietoris topology on the space of compact subsets of weighted graphs. This definition extends s‑convergence by also tracking vertex weights. The authors prove that this convergence coincides with the original s‑convergence when the vertex distribution is fixed to Lebesgue measure, showing that the new framework does not alter the underlying topology.

The main theorem (Theorem 4.5) establishes compactness of the space of all □‑graphons with respect to the above convergence. The proof proceeds by constructing inverse systems for the vertex measures and for the edge measures separately. For a convergent sequence, the limiting $k$‑shapes determine consistent finite‑dimensional marginals of the vertex and edge measures. Using these marginals, the authors define inverse systems indexed by finite subsets of the natural numbers, with bonding maps given by canonical projections. By invoking a specialized version of the Kolmogorov Extension Theorem (proved as Lemma 2.12), they obtain unique inverse limits $\pi$ and $\mu$, which live on an infinite product of finite measurable spaces. The pair $(\pi,\mu)$ is then shown to be a □‑graphon that realizes the prescribed limit $k$‑shapes, thereby serving as the limit of the original sequence. This argument demonstrates that the compactness does not rely on fixing a particular underlying space (such as the unit interval) but works in full generality.

Section 5 compares the new convergence with s‑convergence, re‑deriving the compactness of the space of s‑graphons as a corollary. Section 6 characterizes isomorphisms in $\mathbf{Graphon}_\square$ and begins an investigation of the equivalence relation “having the same $k$‑shapes for all $k$”. The authors prove that this relation coincides with categorical isomorphism under mild regularity conditions, and they outline further work needed to obtain a complete classification.

Overall, the paper blends measure theory, probability (through Markov kernels), and category theory to provide a unified and more flexible language for graph limits. By allowing arbitrary vertex distributions and by treating both vertex and edge measures as first‑class citizens, the authors open the door to studying limits of heterogeneous random graph models, to formulating new notions of homomorphism between limits, and to applying categorical tools such as limits, colimits, and adjunctions in the analysis of large networks. The compactness result assures that any sequence of such generalized graph limits has a convergent subsequence, a foundational property that underpins further analytic and combinatorial investigations.