Rapidly Varying Completely Random Measures for Modeling Extremely Sparse Networks

2025-05-19

AI Summary

Existing completely random measures (CRMs) struggle to model extremely sparse networks where the number of edges grows nearly linearly, particularly when their Laplace exponents exhibit approximate linearity—leading to analytical intractability. Method: We propose a novel class of CRMs with fast-varying Lévy intensities (α ∈ (0,1]), constructed via a mixture of stable and generalized gamma processes. We develop a posterior inference algorithm based on size-biased representations and Laplace exponent analysis within an MCMC framework. Contribution/Results: Our CRM is the first to simultaneously achieve analytical tractability, interpretable parametrization, and strict containment of classical CRM limiting cases. Theoretically, we establish precise asymptotic characterizations and rigorously embed our model into the Caron–Fox sparse graph framework. Empirically, it significantly outperforms conventional CRMs on real-world sparse networks, delivering marked improvements in goodness-of-fit, predictive accuracy, and computational stability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• 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
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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 $alphain(0,1]$, 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.

Problem

• Modeling extremely sparse networks with rapidly varying CRMs
• Lack of tractable CRM constructions under rapid variation
• Developing interpretable, tractable CRMs for near-linear edge growth

Innovation

• New class of CRMs with index of variation
• Mixtures of stable or gamma processes
• Efficient algorithms for simulation and inference