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In-depth Analysis of Densest Subgraph Discovery in a Unified Framework

Yingli Zhou, Qingshuo Guo, Yi Yang, Yixiang Fang, Chenhao Ma, Laks Lakshmanan

TL;DR

This work introduces a unified three-stage framework for Densest Subgraph Discovery (DSD) that encompasses existing exact and approximation algorithms via graph reduction, vertex weight updating, and candidate extraction/verification. It conducts a large-scale, cross-domain empirical study of 12 undirected and 7 directed DSD methods, demonstrating that combining established techniques yields faster variants without sacrificing accuracy (up to 10x improvements). The framework is extended to directed graphs, and the results highlight the practical impact of graph reduction, simultaneous weight updates, and careful control of the search space in directed settings. The study provides actionable guidance for practitioners and identifies opportunities in distributed computing, GPU acceleration, privacy-aware DSD, and heterogeneous graphs. The authors also release code to enable reproducibility and further exploration.

Abstract

As a fundamental topic in graph mining, Densest Subgraph Discovery (DSD) has found a wide spectrum of real applications. Several DSD algorithms, including exact and approximation algorithms, have been proposed in the literature. However, these algorithms have not been systematically and comprehensively compared under the same experimental settings. In this paper, we first propose a unified framework to incorporate all DSD algorithms from a high-level perspective. We then extensively compare representative DSD algorithms over a range of graphs -- from small to billion-scale -- and examine the effectiveness of all methods. Moreover, we suggest new variants of the DSD algorithms by combining the existing techniques, which are up to 10 X faster than the state-of-the-art algorithm with the same accuracy guarantee. Finally, based on the findings, we offer promising research opportunities. We believe that a deeper understanding of the behavior of existing algorithms can provide new valuable insights for future research. The codes are released at https://anonymous.4open.science/r/DensestSubgraph-245A

In-depth Analysis of Densest Subgraph Discovery in a Unified Framework

TL;DR

This work introduces a unified three-stage framework for Densest Subgraph Discovery (DSD) that encompasses existing exact and approximation algorithms via graph reduction, vertex weight updating, and candidate extraction/verification. It conducts a large-scale, cross-domain empirical study of 12 undirected and 7 directed DSD methods, demonstrating that combining established techniques yields faster variants without sacrificing accuracy (up to 10x improvements). The framework is extended to directed graphs, and the results highlight the practical impact of graph reduction, simultaneous weight updates, and careful control of the search space in directed settings. The study provides actionable guidance for practitioners and identifies opportunities in distributed computing, GPU acceleration, privacy-aware DSD, and heterogeneous graphs. The authors also release code to enable reproducibility and further exploration.

Abstract

As a fundamental topic in graph mining, Densest Subgraph Discovery (DSD) has found a wide spectrum of real applications. Several DSD algorithms, including exact and approximation algorithms, have been proposed in the literature. However, these algorithms have not been systematically and comprehensively compared under the same experimental settings. In this paper, we first propose a unified framework to incorporate all DSD algorithms from a high-level perspective. We then extensively compare representative DSD algorithms over a range of graphs -- from small to billion-scale -- and examine the effectiveness of all methods. Moreover, we suggest new variants of the DSD algorithms by combining the existing techniques, which are up to 10 X faster than the state-of-the-art algorithm with the same accuracy guarantee. Finally, based on the findings, we offer promising research opportunities. We believe that a deeper understanding of the behavior of existing algorithms can provide new valuable insights for future research. The codes are released at https://anonymous.4open.science/r/DensestSubgraph-245A
Paper Structure (30 sections, 5 theorems, 9 equations, 27 figures, 17 tables, 6 algorithms)

This paper contains 30 sections, 5 theorems, 9 equations, 27 figures, 17 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Problem definitions
  4. The $\mathsf{CP}$ formulations of UDS problems
  5. A Unified Framework
  6. Comparison and Analysis Of DSD Algorithms for Undirected Graphs
  7. Graph reduction
  8. Vertex weight updating
  9. Candidate subgraph extraction and verification
  10. Optimizations
  11. Comparison and Analysis of DSD Algorithms for Directed Graphs
  12. Experiments
  13. Setup
  14. Evaluation of UDS algorithms
  15. Evaluation of DDS algorithms
  16. ...and 15 more sections

Key Result

lemma 1

Given an undirected graph $G$ let its UDS $\mathcal{D}(G)$ have density $\rho:=\rho^{*}_G$. Then $\mathcal{D}(G)$ is contained in the $\lceil \rho \rceil$-core.

Figures (27)

  • Figure 1: Examples of undirected and directed graphs.
  • Figure 2: An example of the core-based graph reduction.
  • Figure 3: An undirected Graph $G$ and its flow network $\mathcal{F}$.
  • Figure 4: Efficiency results of ($1+\epsilon$)-approximation algorithms on undirected graph.
  • Figure 5: Efficiency results of exact algorithms for UDS.
  • ...and 22 more figures

Theorems & Definitions (13)

  • definition 1: Density of undirected graph goldberg1984finding
  • definition 2: Directed graph densitykannan1999analyzingkhuller2009findingma2020efficientma2021directed
  • definition 3: $k$-core goldberg1984findingtsourakakis2015kfang2019efficient
  • lemma 1: fang2019efficient
  • Example 1
  • lemma 2: fang2019efficient
  • definition 4: $[ x, y ]$-core ma2020efficient
  • theorem 1: ma2020efficient
  • definition 5: $c$-biased densityma2022convex
  • definition 6: $c$-biased directed densest subgraph (DDS) ma2022convex
  • ...and 3 more