CoLaS: Copula-Seeded Sparse Local Graphs with Tunable Assortativity, Persistent Clustering, and a Degree-Tail Dichotomy

TL;DR

The paper introduces CoLaS, a flexible graph model that separates node popularity and locality, enabling tunable degree mixing, persistent clustering, and a degree-tail dichotomy, with theoretical guarantees and a method for calibration.

Contribution

It presents CoLaS, a novel copula-based latent-variable model for sparse graphs with controllable degree mixing and clustering, and develops theory and methods for degree and tail behavior.

Findings

Degrees converge to mixed-Poisson limits.

Degree tails are light under fixed-range local kernels.

Proposed CoLaS-HT extends to power-law degrees while maintaining sparsity.

Abstract

Empirical networks are typically sparse yet display pronounced degree variation, persistent transitivity, and systematic degree mixing. Most sparse generators control at most two of these features, and assortativity is often achieved by degree-preserving rewiring, which obscures the mechanism-parameter link. We introduce CoLaS (copula-seeded local latent-space graphs), a modular latent-variable model that separates marginal specifications from dependence. Each node has a popularity variable governing degree heterogeneity and a latent geometric location governing locality. A low-dimensional copula couples popularity and location, providing an interpretable dependence parameter that tunes degree mixing while leaving the chosen marginals unchanged. Under shrinking-range locality, edges are conditionally independent, the graph remains sparse, and clustering does not vanish. We develop sparse-limit theory for degrees, transitivity, and assortativity. Degrees converge to mixed-Poisson limits and we establish a degree-tail dichotomy: with fixed-range local kernels, degree tails are necessarily light, even under heavy-ailed popularity. To recover power-law degrees without sacrificing sparsity or locality, we propose CoLaS-HT, a minimal tail-inheriting extension in which effective connection ranges grow with popularity. Finally, under an identifiability condition, we provide a consistent one-graph calibration method based on jointly matching transitivity and assortativity.