A Generative Model for Extremely Sparse Edge-Exchangeable Networks

Abstract: We propose a graph generative model for sequences of extremely sparse, edge-exchangeable networks. Models for sparse graphs often face a trade-off between desirable properties like exchangeability and the ability to capture the sparsity observed in real-world networks. While models based on vertex or edge exchangeability have successfully generated sparse graphs, achieving the "extremely sparse" regime, where the number of edges scales near-linearly with the number of nodes, has remained a challenge. Recently, a novel Completely Random Measure (CRM) was introduced, demonstrating that this rate could be achieved within the vertex-exchangeable framework of Caron and Fox. This paper extends this work by demonstrating that this new CRM can be integrated into the alternative edge-exchangeable framework to achieve extreme sparsity.

• The paper introduces a RapidBeta process that generalizes edge-exchangeable models to achieve near-linear edge growth in extremely sparse networks.
• It employs a Bayesian nonparametric framework with rapidly varying CRMs and a partitioned thinning scheme to control network sparsity and enable exact sampling.
• Simulations demonstrate that the model generates richer internal structures and power-law degree distributions while preserving permutation invariance.

Generative Modeling of Extremely Sparse Edge-Exchangeable Networks

Background and Motivation

Sparsity is a fundamental empirical property observed across a wide spectrum of real-world networks, where the number of edges $E$ scales far below the quadratic maximum $\mathcal{O}(N^2)$ possible for $N$ nodes. Exchangeability, a desirable modeling symmetry, is often incompatible with sparsity: the Aldous-Hoover theorem dictates that any node-exchangeable graph model must be either dense or degenerate. State-of-the-art generative models for sparse graphs—such as the Kallenberg exchangeable Caron-Fox framework [Caron & Fox, 2017], and preferential attachment mechanisms—have only partially addressed the "extremely sparse" scenario, where the edge count grows near-linearly with node count. Recent advances utilizing rapidly varying CRMs have enabled the Caron-Fox framework to reach this regime [Kilian et al., 2025]. This paper (2606.22105) generalizes the methodology to edge-exchangeable models, presenting a rigorous construction for generating extremely sparse networks while preserving permutation invariance to edge arrival order.

Edge-Exchangeable Models and Bayesian Nonparametrics

Edge-exchangeable modeling treats the sequence of observed edges as invariant to their arrival permutations, placing it within the broader spectrum of statistical symmetries introduced by Crane and Dempsey, and later formalized in [Cai et al., 2016]. The generative mechanism comprises latent vertices $\mathbb{N}$, with associated edge probabilities $\{w_{i,j}\}$ and random labeling. The edge formation process is independent, following an explicit Bernoulli or binomial construction. Central to the Bayesian nonparametric approach is the adoption of CRMs, with the Poisson point process realization controlled by the rate measure $v$; its choice directly governs the asymptotic scaling law of the generated graph sequence. The model can be adapted—via marginalization and thresholding—to produce simple graphs while retaining key structural properties.

Achieving Extreme Sparsity Through Rapid Variation

The principal theoretical result is the generalization of Cai et al.'s sparsity theorem to encompass rapidly varying tail measures of the CRM. When the rate measure $v(w) \sim cw^{-2}l(w^{-1})$ as $w\to 0$ (with $l$ slowly varying and integrability conditions satisfied), the number of edges scales as $E = \mathcal{O}(N l_1(N))$ almost surely, where $\mathcal{O}(N^2)$0 also denotes a slowly varying function. This includes logarithmic families and other iterated variations, affording modeling flexibility. The construction thereby achieves the "extremely sparse" regime: edge counts exhibit near-linear scaling with node count up to slowly varying corrections, surpassing the sparse scaling rates of earlier edge-exchangeable models.

The RapidBeta Process: Explicit Construction and Sampling Strategies

The paper introduces the RapidBeta process—a CRM with rate measure $\mathcal{O}(N^2)$1 parametrized to engineer rapid tail decay. Parametric control via shape and truncation parameters allows fine-tuning of the asymptotic and finite-sample behavior, affecting the emergence and prominence of extreme sparsity. Simulation of such processes is rendered nontrivial by the infinite nature of the underlying Poisson point realization. The authors provide a partitioned thinning scheme for exact sampling from a truncated RapidBeta process. The correctness proof leverages Kingman's thinning theorem; truncation-induced error vanishes logarithmically as the threshold decreases. Diagnostics validate both marginal tail and global distributional behavior, demonstrating alignment between empirical and theoretical statistics.

Comparative Simulations and Emergent Structural Properties

Simulated graph sequences from the RapidBeta edge-exchangeable model display a stronger sparsity than those produced by the three-parameter Beta process [Cai et al., 2016]. The Barabási-Albert model achieves even sparser scaling but forfeits exchangeability. Notably, the RapidBeta construction leads to richer internal structure, including the observed formation of a giant (yet not forever connected) component, in contrast to pathological connectivity outcomes in strict linear-scaling models [Janson, 2018]. Power-law degree distribution and small-world properties are expected but remain to be formally characterized. The model enables a broad class of sparse graph structures, facilitating statistical inference under exchangeability and promoting realistic modeling of large-scale networks.

Implications, Limitations, and Future Directions

The theoretical advances reconcile extreme sparsity with edge-exchangeability, expanding the repertoire of network models available for statistical analysis of massive datasets. Practically, the RapidBeta process opens avenues for flexible generative modeling and scalable simulation algorithms, subject to ongoing refinement of truncation error bounds and inference methodology. The model's capacity to represent realistic network heterogeneity—while preserving permutation invariance—has implications for both statistical network theory and applications across domains. Future work will entail formal analysis of degree distributions, clustering coefficients, and explicit inference procedures, as well as the exploration of structural phase transitions in the giant component. Extending theoretical guarantees to E-truncation thresholds, as suggested by Li & Campbell [2021], remains an open challenge.

Conclusion

This paper establishes the first practical edge-exchangeable generative model capable of achieving extreme sparsity, building on rapidly varying CRMs and the explicit RapidBeta process construction. The theoretical results and simulation methodology provided not only expand the modeling landscape but also yield practical algorithms for sampling and network analysis in the extremely sparse regime (2606.22105). The approach balances statistical symmetry with realistic network structure, and paves the way for further research in sparse network inference and generative modeling.