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

2025-12-22

AI Summary

Existing sparse network generation models struggle to jointly control degree heterogeneity, persistent transitivity (clustering), and systemic assortativity—particularly because assortativity often relies on degree-preserving rewiring, yielding opaque mechanisms. This paper proposes a copula-driven sparse graph generation framework that, for the first time, decouples node popularity (governing degree distribution) from geometric positions (governing local clustering). It explicitly parameterizes degree mixing via low-dimensional copulas, enabling coordinated control of all three properties. Theoretically, we establish joint convergence of degree distribution, clustering, and assortativity in the sparse limit; uncover a degree-tail bimodality phenomenon; and extend CoLaS-HT to preserve heavy-tailed degrees under sparsity constraints. Algorithmically, we introduce a unified single-graph calibration method that matches both transitivity and assortativity using only one observed graph—while guaranteeing sparsity, non-vanishing clustering, and tunable assortativity.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Web Mining and Content Analysis: Web data generation and simulation

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.

Problem

• Model sparse networks with degree variation and clustering
• Tune assortativity without altering marginal degree distributions
• Achieve power-law degrees while preserving sparsity and locality

Innovation

• Copula-seeded modular latent-variable model
• Interpretable dependence parameter tunes degree mixing
• Tail-inheriting extension for power-law degrees