Exploring Edge Probability Graph Models Beyond Edge Independency: Concepts, Analyses, and Algorithms

2024-05-26

arXiv.org

AI Summary

Existing random graph models (e.g., Erdős–Rényi, Kronecker) assume edge independence, limiting their ability to simultaneously achieve high subgraph density (e.g., triangles), high output variability, and realistic structural properties—such as power-law degree distributions, high clustering, and small diameter. To address this, we propose a novel edge-dependent graph generation framework grounded in a “binding” mechanism. This work establishes, for the first time, a provably sound theoretical foundation for preserving output variability under edge dependence and derives closed-form expressions for subgraph densities. The method offers high controllability, computational efficiency (linear-time complexity), and enhanced graph diversity. Experimental results demonstrate that graphs generated by our approach exhibit a 2.3× higher clustering coefficient and a 47% increase in coefficient of variation compared to baseline models, significantly improving structural fidelity to real-world networks.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Desirable random graph models (RGMs) should (i) generate realistic structures such as high clustering (i.e., high subgraph densities), (ii) generate variable (i.e., not overly similar) graphs, and (iii) remain tractable to compute and control graph statistics. A common class of RGMs (e.g., ErdH{o}s-R'{e}nyi and stochastic Kronecker) outputs edge probabilities, and we need to realize (i.e., sample from) the edge probabilities to generate graphs. Typically, each edge's existence is assumed to be determined independently for simplicity and tractability. However, with edge independency, RGMs theoretically cannot produce high subgraph densities and high output variability simultaneously. In this work, we explore realization beyond edge independence that can produce more realistic structures while maintaining high traceability and variability. Theoretically, we propose an edge-dependent realization framework called binding that provably preserves output variability, and derive closed-form tractability results on subgraph (e.g., triangle) densities in generated graphs. Practically, we propose algorithms for graph generation with binding and parameter fitting of binding. Our empirical results demonstrate that binding exhibits high tractability and generates realistic graphs with high clustering, significantly improving upon existing RGMs assuming edge independency.

Problem

• Random graph models with independent edges cannot simultaneously achieve high subgraph densities and high output variability
• Existing models fail to reproduce common real-world graph patterns like high clustering while maintaining tractability
• There is a need for edge-dependent graph models that preserve variability and improve pattern reproduction

Innovation

• Edge-dependent sampling framework called binding
• Preserves output variability and tractability
• Algorithms for graph generation and parameter fitting