Finding Locally Densest Subgraphs: Convex Programming with Edge and Triangle Density
Yi Yang, Chenhao Ma, Reynold Cheng, Laks V. S. Lakshmanan, Xiaolin Han
TL;DR
This work addresses the problem of finding top-$k$ locally densest subgraphs (LDS) and their triangle-based extension LTDS in large graphs. It introduces a convex-programming based framework that uses a new compact-number concept $\phi(u)$ to bridge LDS with a densest-subgraph CP formulation, enabling effective pruning via stable groups and reducing verification to smaller flow computations. The method, LDScvx, is extended to LTDS (LTDScvx), and the authors present a unified four-stage framework (Initial Reduction, Vertex Weight Updating, Graph Reduction and Division, Candidate Subgraph Extraction and Verification) to compare CP-based and max-flow-based approaches. Extensive experiments on 13 real-world datasets show LDScvx and LTDScvx outperform state-of-the-art by up to four orders of magnitude in speed, demonstrating scalable discovery of multiple dense regions with practical impact across domains such as social networks, biology, and e-commerce.
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.