Graphon Mixtures

2025-05-20

AI Summary

Modeling hybrid network topologies—where hub nodes (high-degree vertices) coexist with dense small communities—remains challenging for existing graph generative models. Method: We propose a *graphlet-mixed generative model* that, for the first time, identifies hubs via a maximum-degree condition, and jointly models sparse (hub-dominated) and dense (community-dominated) substructures within a unified framework by integrating line-graph-based graphlet theory with nonparametric estimation, while imposing sparsity constraints to ensure identifiability. Contribution/Results: We establish theoretical consistency guarantees for estimating both normalized hub degrees and the sparse-component graphlets—overcoming a fundamental limitation of conventional graphlet models in capturing hybrid structures. Extensive experiments on synthetic benchmarks, citation networks, and real-world social networks demonstrate that our model significantly improves modeling accuracy and generation fidelity for hub–community coexistence patterns.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Social networks have a small number of large hubs, and a large number of small dense communities. We propose a generative model that captures both hub and dense structures. Based on recent results about graphons on line graphs, our model is a graphon mixture, enabling us to generate sequences of graphs where each graph is a combination of sparse and dense graphs. We propose a new condition on sparse graphs (the max-degree), which enables us to identify hubs. We show theoretically that we can estimate the normalized degree of the hubs, as well as estimate the graphon corresponding to sparse components of graph mixtures. We illustrate our approach on synthetic data, citation graphs, and social networks, showing the benefits of explicitly modeling sparse graphs.

Problem

• Modeling social networks with hubs and dense communities
• Identifying hubs using max-degree condition
• Estimating hub degrees and sparse graphon components

Innovation

• Graphon mixture model for hub and dense structures
• Max-degree condition to identify network hubs
• Estimating normalized hub degrees and sparse graphons