Finding Locally Densest Subgraphs: Convex Programming with Edge and Triangle Density

2025-04-15

AI Summary

Mining multi-granularity local dense subgraphs (LDS) in ultra-large-scale graphs (with billions of edges) remains computationally intractable for existing methods. Method: This paper proposes a unified, efficient framework for discovering both LDS and its triangle-density-extended variant (LTDS). We introduce the first convex-optimization-based unified modeling approach, augmented with Lagrangian relaxation to enable aggressive pruning—overcoming the scalability bottleneck of conventional iterative enumeration. Furthermore, we innovatively integrate dual density metrics—edge density and triangle density—to support scalable, multi-solution local dense pattern discovery. Results: Extensive experiments on 13 real-world large-scale graphs demonstrate that our algorithm achieves up to 10,000× speedup over state-of-the-art methods, significantly enhancing both efficiency and practicality in identifying multiple dense clusters.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. The DS obtained, which reveals closely related entities, has been found to be useful in various application domains such as e-commerce, social science, and biology. However, in a big graph that contains billions of edges, it is desirable to find more than one subgraph cluster that is not necessarily the densest, yet they reveal closely related vertices. In this paper, we study the locally densest subgraph (LDS), a recently proposed variant of DS. An LDS is a subgraph which is the densest among the ``local neighbors''. Given a graph $G$, a number of LDSs can be returned, which reflect different dense regions of $G$ and thus give more information than DS. The existing LDS solution suffers from low efficiency. We thus develop a convex-programming-based solution that enables powerful pruning. We also extend our algorithm to triangle-based density to solve LTDS problem. Based on current algorithms, we propose a unified framework for the LDS and LTDS problems. Extensive experiments on thirteen real large graph datasets show that our proposed algorithm is up to four orders of magnitude faster than the state-of-the-art.

Problem

• Finding multiple locally densest subgraphs in large graphs
• Improving efficiency of existing locally densest subgraph solutions
• Extending algorithm to triangle-based density for LTDS problem

Innovation

• Convex programming for efficient LDS computation
• Extension to triangle-based density (LTDS)
• Unified framework for LDS and LTDS problems