Graphons of Line Graphs

A large graph is often available only in part: a crawl stopped by its budget, a panel, a partial dump. When the sampled fraction $s$ is known by design the total edge count follows from $\hat e=e_s/s^2$ and no model is needed. We treat the case where $s$ is unknown and the population size is known. Our main result is a reduction: under a sparse exchangeable (graphex) model the expected non-isolated fraction obeys $n_s/n_1\to s^{1+σ}$, so the sampling rate becomes estimable once the tail index $σ$ is, and substituting it back gives $e_s(n_1/n_s)^{2/(1+σ)}$ -- the same estimator, with the design quantity inferred. Estimating global edge cardinality in a sparse graph is therefore, in expectation, tail-index estimation, and the quadratic graphon estimator is the case $σ=0$: it fails by an identity rather than by a fit ($260\%$ median error against $27\%$). We bound the finite-size error of the substitution and show the reduction is modular in the tail-index estimator --- filled with a published closed-form one it reaches $21.7\%$ over $13$ networks and $39$ sampling budgets with no fitting at all. Fitting a full graphex additionally returns the degree distribution at any size and a generative object, in a representation where sparsity is a coordinate and the interpolation path is dictated rather than chosen. Two limits are exact: rank-one graphexes have transitivity fixed by the degree profile, so high-clustering graphs lie outside the class; and under snowball or random-walk crawls every method here fails, the design-based oracle worst of all ($7.8\%$ to $588\%$).

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 $n$.

Recovering structural information from noisy high-dimensional data is a fundamental task in statistical inference. We investigate the recovery thresholds for a graph hidden in a randomly weighted complete graph. Specifically, an unknown graph $H^* \in H_n$ is chosen uniformly at random, and hidden in a complete graph of $n$ vertices as follows: the weight of an edge $e \in H$ is distributed independently according to $P_n$; otherwise the weight is distributed independently according to $Q_n$. The goal is to recover almost all of $H$ from these edge weights. Assuming a local Lipschitzness of the Rényi divergence between distributions $P_n$ and $Q_n$, and a mild density condition for the graphs $H_n$, we give a unified characterization of the information-theoretic limit for recovering almost all of $H$ (also known as almost exact recovery). Our characterization connects the KL divergence between $P_n$ and $Q_n$ to the logarithm of the first moment threshold of $H$ in the Erdős-Rényi random graph model $G(n,p)$. Our lower bound also extends to the task of partial recovery, in which only a constant $λ$-fraction of $H$ needs to be recovered. Last but not least, for certain Bernoulli and Exponential regimes, and for Gaussian distributions, we are able to show an All-or-Nothing (AoN) threshold phenomenon at the exponential scale.