Graphon Theory

Generative graph models are central to understanding and simulating complex networks. However, existing approaches have complementary strengths and limitations. Mechanistic models offer interpretability but rely on instance-specific estimation methods. Deep generative models, on the other hand, offer amortized inference at the cost of interpretability and are largely limited to graph sizes seen during training. Scientific applications motivate a framework that retains the strengths of both paradigms. We bridge them by formulating both the generative model and parameter recovery in function space. A multifractal step graphon extends standard step graphons with a recursive construction that compactly parameterizes complex networks. This formulation admits a neural inverse operator to recover its parameters, enabling inference on unseen graph sizes. We evaluate our model, trained only on synthetic multifractal step graphon realizations, against both paradigms. Against a graph foundation model pretrained on empirical networks, our method achieves the best average performance on three of four metrics in a zero-shot graph-generation benchmark, indicating that the model transfers to real-world graphs. We also apply our method to single-observation networks, a regime largely inaccessible to deep models that require training corpora, where it performs comparably to an instance-specific method that optimizes on each graph. In a multi-subject EEG case study, the inferred parameters track a reversible change in brain state more sensitively than traditional network statistics. Together, these results indicate that mechanistic interpretability and amortized inference can be effectively unified in a generative graph model to enhance our understanding of complex networks.

Graph Transformers produce, for each attention head, a dense $n\times n$ matrix of learned pairwise interactions. We ask a fundamental question: do these attention-induced graphs converge to a stable limit object as $n$ grows, or does the learned interaction pattern remain unstructured and size-dependent? We answer this using dense graph limit theory, treating each attention matrix as a finite sample from an underlying kernel---an attention graphon---and studying concentration around this limit under the cut-distance. We derive a worst-case variance bound requiring no assumptions on the graphon, and a sharper regularity-aware bound based on nonparametric estimation theory. To operationalize the theory, we propose a canonicalize-then-block-average pipeline for estimating dataset-level attention graphons, and a variance-based diagnostic for testing whether attention admits a stable continuum description. Experiments across multiple graph benchmarks show that learned attention stabilizes to dataset-specific graphon structure on several datasets; that empirical cut-distance and cut-norm variance decreases with $n$ consistent with our bounds; and that attention graphons transfer to larger graph sizes with error decreasing in $n$.

Diffusion models represent a leading paradigm for graph generation, with notable impact in domains such as molecular design. Yet, scaling these models to large graphs remains an open problem. We approach this question in the dense-graph setting through the lens of graphons, the size-agnostic limit objects of dense graph sequences, to study how structural graph statistics behave across node-size scales. This perspective leads to DiPhon, a diffusion framework for size-scalable graph generation. Specifically, we formulate a continuous diffusion process on the graphon space via a Jacobi stochastic differential equation (SDE), and propose DiPhon, a discretized graph-level process that mimics these dynamics on finite graphs. We further derive the corresponding reverse-time process, which requires access to the marginal score. For the Jacobi process, this score interestingly admits a tractable form, which we estimate from data via graph denoising and plug into the reverse process to generate graph samples. We prove that DiPhon matches exactly the first moment of the marginal distributions induced by the continuous graphon process, and approximates the second moment up to a closed-form discrepancy. Thus, DiPhon inherits key size-agnostic statistical properties of the graphon dynamics, providing a principled route toward scalable graph generation. Empirically, we demonstrate this scalability by training on small graphs and generating progressively larger graphs at inference time, without retraining, while preserving their core topological properties.

Graph Neural Differential Equations (GNDEs) model continuous-time graph dynamics by parameterizing Neural ODE velocity fields with Graph Neural Networks. Their local, size-independent filters suggest a zero-shot size-transfer principle: train on a small graph and deploy on larger, similar graphs without retraining. We develop a quantitative theory for this principle on sparse random graphs sampled from graphons. We consider Graphon Neural Differential Equations (Graphon-NDEs) and adjoint Graphon-NDEs as the infinite-node limits of the forward and adjoint GNDE systems, and establish well-posedness. For an $n$-node random graph with sparsity parameter $α_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((α_n n)^{-1/2})$, up to logarithmic factors, with high probability. We also establish uniform-in-time convergence bounds for adjoint systems governing hidden-state and parameter gradients. We further study discretize-then-optimize (DTO) and optimize-then-discretize (OTD) training. Under explicit Euler discretization with $M$ steps, we show that DTO and OTD are asymptotically consistent, with hidden-state and local parameter-gradient discrepancies of orders $O(1/M)$ and $O(1/M^2)$, respectively, up to sparsity and logarithmic factors. Experiments on HSBM and tent graphons support the theoretical rates, while zero-shot transfer experiments across four graphon classes demonstrate accurate deployment of learned GNDEs on larger independently sampled graphs.

We propose a topological framework for comparing trained Graph Neural Networks (GNNs) by mapping the Stochastic Block Models (SBMs) induced on the graphon-signal space of a Message Passing Neural Network (MPNN) onto the unit $n$-sphere $\sphere^{n-1}\subset\R^n$. The construction rests on three classical pillars: the compactness of the cut-distance graphon space $(\Wo,\cutdist)$ \citep{lovasz2006limits,lovasz2012large}, the Frieze--Kannan weak regularity lemma together with its graphon-signal extension due to \citet{levie2023graphon}, and the Lipschitz continuity of MPNNs with respect to the cut-distance. We show that, for any prescribed tolerance $\varepsilon>0$, a trained MPNN $Φ$ acting on a sufficiently large graph factors (up to $\varepsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map $Ψ_n\colon[0,1]\to\sphere^{n-1}$ that places the SBM regions on disjoint spherical caps. This produces a problem-agnostic, low-dimensional ``fingerprint'' of a trained GNN that is amenable to visual inspection and to nearest-neighbour search across model zoos, enabling transfer-learning candidate retrieval without retraining. We discuss the obstruction posed by concentration of measure in high dimension -- a phenomenon directly relevant to LLM-scale embeddings. We close with five concrete future research directions: hyperbolic and Grassmannian alternatives to the spherical model, Gromov--Wasserstein distances on graphon-signals as an isometry-free alternative to the $n$-sphere map, an information-geometric (Fisher) reformulation of the SBM manifold, persistent-homology fingerprints of layer-wise embedding clouds, and a spectral-distance baseline derived from the graphon eigendecomposition.

Graph foundation models (GFMs) aim to reuse a single backbone across diverse graph domains, yet their transfer is often uneven and can exhibit negative transfer. While most prior work improves transfer through architectural or adaptation choices, we ask a data-centric question: which properties of two graph domains determine how much a fixed representation model changes its outputs? Using a graphon-based continuous limit for dense graphs, we show that for both set-based and message-passing tokenizations, any Lipschitz backbone admits an explicit decomposition of cross-domain output shift into (i) graph-specific finite-sample approximation terms and (ii) an intrinsic, relabeling-invariant domain discrepancy capturing structural mismatch. A key ingredient is positional-encoding (PE) stability: we establish stability guarantees for spectral PEs and highlight contrasting behaviors of eigenvector- versus subspace-based PEs. Experiments on synthetic and real graphs validate the theory and translate the decomposition into guidance for data curation in GFM transfer.

Latent-position random graph models usually treat the node set as fixed once the sample size is chosen, while graphon-based and random-measure constructions allow more randomness at the cost of weaker geometric interpretability. We introduce Intensity Dot Product Graphs (IDPGs), which extend Random Dot Product Graphs by replacing a fixed collection of latent positions with a Poisson point process on a Euclidean latent space. This yields a model with random node populations, RDPG-style dot-product affinities, and a population-level intensity that links continuous latent structure to finite observed graphs. We define the heat map and the desire operator as continuous analogues of the probability matrix, prove a spectral consistency result connecting adjacency singular values to the operator spectrum, compare the construction with graphon and digraphon representations, and show how classical RDPGs arise in a concentrated limit. Because the model is parameterized by an evolving intensity, temporal extensions through partial differential equations arise naturally.

A recent paper, ``A Graphon-Signal Analysis of Graph Neural Networks'', by Levie, analyzed message passing graph neural networks (MPNNs) by embedding the input space of MPNNs, i.e., attributed graphs (graph-signals), into a space of attributed graphons (graphon-signals). Based on extensions of standard results in graphon analysis to graphon-signals, the paper proved a generalization bound and a sampling lemma for MPNNs. However, there are some missing ingredients in that paper, limiting its applicability in practical settings of graph machine learning. In the current paper, we introduce several refinements and extensions to existing results that address these shortcomings. In detail, 1) we extend the main results in the paper to graphon-signals with multidimensional signals (rather than 1D signals), 2) we extend the Lipschitz continuity to MPNNs with readout with respect to cut distance (rather than MPNNs without readout with respect to cut metric), 3) we improve the generalization bound by utilizing robustness-type generalization bounds, and 4) we extend the analysis to non-symmetric graphons and kernels.

We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves mapping the original graphs to their line graphs. We show that graphs satisfying a particular property, which we call the square-degree property are sparse, but give rise to dense line graphs. This enables the use of results on graph limits of dense graphs to derive convergence. In particular, star graphs satisfy the square-degree property resulting in dense line graphs and non-zero graphons of line graphs. We demonstrate empirically that we can distinguish different numbers of stars (which are sparse) by the graphons of their corresponding line graphs. Whereas in the original graphs, the different number of stars all converge to the zero graphon due to sparsity. Similarly, superlinear preferential attachment graphs give rise to dense line graphs almost surely. In contrast, dense graphs, including Erdos-Renyi graphs make the line graphs sparse, resulting in the zero graphon.