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An Efficient Quadratic Penalty Method for a Class of Graph Clustering Problems

Wenshun Teng, Qingna Li

TL;DR

This work formulates a class of graph clustering problems as semi-assignment optimization tasks with a block-structured quadratic objective. By reformulating the problem into a sparse-constrained relaxation, the authors develop two penalty-based solvers: the quadratic penalty method (QP-GC) and the quadratic penalty regularized method (QPR-GC), solved respectively by an active-set projected Newton method and a spectral projected gradient approach. Through extensive experiments on synthetic graphs (PPM and SBM) and real-world networks (e.g., Karate Club, US Football), the methods demonstrate strong clustering quality and scalability, with QP-GC excelling on large-scale graphs and QPR-GC offering efficiency on smaller problems. The study also benchmarks against Gurobi and Boltzmann machines, showing competitive or superior performance, and provides practical guidance on metric choices and method selection for different graph sizes. Overall, the paper contributes a unified optimization framework for graph clustering with robust, scalable solvers and compelling empirical validation.

Abstract

Community-based graph clustering is one of the most popular topics in the analysis of complex social networks. This type of clustering involves grouping vertices that are considered to share more connections, whereas vertices in different groups share fewer connections. A successful clustering result forms densely connected induced subgraphs. This paper studies a specific form of graph clustering problems that can be formulated as semi-assignment problems, where the objective function exhibits block properties. We reformulate these problems as sparse-constrained optimization problems and relax them to continuous optimization models. We then apply the quadratic penalty method and the quadratic penalty regularized method to the relaxation problem, respectively. Extensive numerical experiments demonstrate that both methods effectively solve graph clustering tasks for both synthetic and real-world network datasets. For small-scale problems, the quadratic penalty regularized method demonstrates greater efficiency, whereas the quadratic penalty method proves more suitable for large-scale cases.

An Efficient Quadratic Penalty Method for a Class of Graph Clustering Problems

TL;DR

This work formulates a class of graph clustering problems as semi-assignment optimization tasks with a block-structured quadratic objective. By reformulating the problem into a sparse-constrained relaxation, the authors develop two penalty-based solvers: the quadratic penalty method (QP-GC) and the quadratic penalty regularized method (QPR-GC), solved respectively by an active-set projected Newton method and a spectral projected gradient approach. Through extensive experiments on synthetic graphs (PPM and SBM) and real-world networks (e.g., Karate Club, US Football), the methods demonstrate strong clustering quality and scalability, with QP-GC excelling on large-scale graphs and QPR-GC offering efficiency on smaller problems. The study also benchmarks against Gurobi and Boltzmann machines, showing competitive or superior performance, and provides practical guidance on metric choices and method selection for different graph sizes. Overall, the paper contributes a unified optimization framework for graph clustering with robust, scalable solvers and compelling empirical validation.

Abstract

Community-based graph clustering is one of the most popular topics in the analysis of complex social networks. This type of clustering involves grouping vertices that are considered to share more connections, whereas vertices in different groups share fewer connections. A successful clustering result forms densely connected induced subgraphs. This paper studies a specific form of graph clustering problems that can be formulated as semi-assignment problems, where the objective function exhibits block properties. We reformulate these problems as sparse-constrained optimization problems and relax them to continuous optimization models. We then apply the quadratic penalty method and the quadratic penalty regularized method to the relaxation problem, respectively. Extensive numerical experiments demonstrate that both methods effectively solve graph clustering tasks for both synthetic and real-world network datasets. For small-scale problems, the quadratic penalty regularized method demonstrates greater efficiency, whereas the quadratic penalty method proves more suitable for large-scale cases.
Paper Structure (29 sections, 7 theorems, 31 equations, 6 figures, 12 tables, 3 algorithms)

This paper contains 29 sections, 7 theorems, 31 equations, 6 figures, 12 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Semi-assignment optimization model for graph clustering
  3. Optimization model based on semi-assignment constraints
  4. Different distance metrics
  5. Burt's distance $D^{B}$
  6. Jaccard distance $D^{J}$
  7. Otsuka-Ochiai distance $D^{O}$
  8. Equivalent formulation of
  9. Continuous relaxation of and two methods
  10. Relaxation problem
  11. Quadratic penalty method
  12. Quadratic penalty regularized method
  13. Numerical Results
  14. Evaluation of clustering quality
  15. Synthetic graphs
  16. ...and 14 more sections

Key Result

proposition thmcounterproposition

For the matrix $D$ defined by $D^{B}$, $D^{J}$ or $D^{O}$ as above, it holds that $D_{ij}\geq 0$, $D_{ii}= 0$, $D_{ij}=D_{ji}$, $i=1,\cdots,n$, $j=1,\cdots,n$. In other words, $D$ is nonnegative, symmetric and has zero diagonal elements.

Figures (6)

  • Figure 1: A schematic representation of a network with cluster structure. In this network, there are three clusters of densely connected vertices (indicated by solid circles), which are represented by yellow, purple, and green dashed circles. The density of connections between these clusters is much lower
  • Figure 2: A visual representation of three PPM graph models after clustering with our method
  • Figure 3: Dendrogram visualization of intra cluster density distributions: ground truth ${\rm P_{intra}}$ versus $\bar{\kappa}_{intra}$ by QP-GC and QPR-GC for twelve SBM models
  • Figure 4: The radar plot representation of ${\rm P_{intra}}$ and $\bar{\kappa}_{intra}$ by QP-GC, QPR-GC, Gurobi and BM for twelve SBM models
  • Figure 5: The friendship network derived from Zachary's karate club, as described in this paper. Nodes of different colors represent membership in different factions. The red squares in the figure indicate the nodes of clustering error
  • ...and 1 more figures

Theorems & Definitions (8)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • remark thmcounterremark
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4