Rapidly Varying Completely Random Measures for Modeling Extremely Sparse Networks

TL;DR

This paper introduces a new class of completely random measures tailored for modeling extremely sparse networks, providing analytical tractability, interpretability, and practical algorithms for large-scale network analysis.

Contribution

The paper develops a novel CRM class with index of variation α, suitable for sparse networks, including tractable properties and algorithms, extending existing models with better scalability.

Findings

Models produce near-linear edge growth in networks

Algorithms enable efficient simulation and inference

Empirical results validate model applicability to real data

Abstract

Completely random measures (CRMs) are fundamental to Bayesian nonparametric models, with applications in clustering, feature allocation, and network analysis. A key quantity of interest is the Laplace exponent, whose asymptotic behavior determines how the random structures scale. When the Laplace exponent grows nearly linearly - known as rapid variation - the induced models exhibit approximately linear growth in the number of clusters, features, or edges with sample size or network nodes. This regime is especially relevant for modeling sparse networks, yet existing CRM constructions lack tractability under rapid variation. We address this by introducing a new class of CRMs with index of variation $\alpha\in(0,1]$, defined as mixtures of stable or generalized gamma processes. These models offer interpretable parameters, include well-known CRMs as limiting cases, and retain analytical tractability through a tractable Laplace exponent and simple size-biased representation. We analyze the asymptotic properties of this CRM class and apply it to the Caron-Fox framework for sparse graphs. The resulting models produce networks with near-linear edge growth, aligning with empirical evidence from large-scale networks. Additionally, we present efficient algorithms for simulation and posterior inference, demonstrating practical advantages through experiments on real-world sparse network datasets.