Unified Dense Subgraph Detection: Fast Spectral Theory Based Algorithms

Wenjie Feng, Shenghua Liu, Danai Koutra, Xueqi Cheng

Abstract: How can we effectively detect fake reviews or fraudulent links on a website? How can we spot communities that suddenly appear based on users’ interactions? And how can we efficiently find the minimum cut in a large graph? All of these are related to the finding of dense subgraphs, a significant primitive problem in graph analysis with extensive applications across various domains. In this paper, we focus on formulating the problem of the densest subgraph detection and theoretically compare and contrast several correlated problems. Moreover, we propose a unified framework, GenDS , for the densest subgraph detection, provide some theoretical analysis based on the network flow and spectral graph theory, and devise simple and computationally efficient algorithms, SpecGDS and GepGDS , to solve it by leveraging the spectral properties and greedy search. We conduct thorough experiments on 40 real-world networks with up to 1.47 billion edges from various domains. We demonstrate that our SpecGDS yields up to $58.6 \ \times$ speedup and achieves better or approximately equal-quality solutions for the densest subgraph detection compared to the baselines. GepGDS also reveals some properties of generalized eigenvalue problems for the GenDS . Also, our methods scale linearly with the graph size and are proven effective in applications such as finding collaborations that appear suddenly in an extensive, time-evolving co-authorship network.