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Analyzing Modularity Maximization in Approximation, Heuristic, and Graph Neural Network Algorithms for Community Detection

Samin Aref, Mahdi Mostajabdaveh

TL;DR

This study critically evaluates modularity-based community detection by comparing ten inexact algorithms (including heuristics, GNN variants, and Bayan approximations) against an exact IP baseline across 104 networks. It reveals that average heuristics rarely reach optimal modularity (43.9%), while approximate Bayan and GNN methods perform better (82.3% and 68.7%, respectively), with Bayans approaching optimality more when using tighter tolerances. Importantly, near-optimal partitions are often substantially dissimilar to any optimal partition, underscoring a fundamental mismatch between high modularity and true community structure. The work argues for using approximation or exact optimization when feasible and encourages development of algorithms that balance accuracy with scalability for practical modularity-based community detection.

Abstract

Community detection, which involves partitioning nodes within a network, has widespread applications across computational sciences. Modularity-based algorithms identify communities by attempting to maximize the modularity function across network node partitions. Our study assesses the performance of various modularity-based algorithms in obtaining optimal partitions. Our analysis utilizes 104 networks, including both real-world instances from diverse contexts and modular graphs from two families of synthetic benchmarks. We analyze ten inexact modularity-based algorithms against the exact integer programming baseline that globally optimizes modularity. Our comparative analysis includes eight heuristics, two variants of a graph neural network algorithm, and nine variations of the Bayan approximation algorithm. Our findings reveal that the average modularity-based heuristic yields optimal partitions in only 43.9% of the 104 networks analyzed. Graph neural networks and approximate Bayan, on average, achieve optimality on 68.7% and 82.3% of the networks respectively. Additionally, our analysis of three partition similarity metrics exposes substantial dissimilarities between high-modularity sub-optimal partitions and any optimal partition of the networks. We observe that near-optimal partitions are often disproportionately dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of the commonly used modularity-based methods: they rarely produce an optimal partition or a partition resembling an optimal partition even on networks with modular structures. If modularity is to be used for detecting communities, we recommend approximate optimization algorithms for a more methodologically sound usage of modularity within its applicability limits.

Analyzing Modularity Maximization in Approximation, Heuristic, and Graph Neural Network Algorithms for Community Detection

TL;DR

This study critically evaluates modularity-based community detection by comparing ten inexact algorithms (including heuristics, GNN variants, and Bayan approximations) against an exact IP baseline across 104 networks. It reveals that average heuristics rarely reach optimal modularity (43.9%), while approximate Bayan and GNN methods perform better (82.3% and 68.7%, respectively), with Bayans approaching optimality more when using tighter tolerances. Importantly, near-optimal partitions are often substantially dissimilar to any optimal partition, underscoring a fundamental mismatch between high modularity and true community structure. The work argues for using approximation or exact optimization when feasible and encourages development of algorithms that balance accuracy with scalability for practical modularity-based community detection.

Abstract

Community detection, which involves partitioning nodes within a network, has widespread applications across computational sciences. Modularity-based algorithms identify communities by attempting to maximize the modularity function across network node partitions. Our study assesses the performance of various modularity-based algorithms in obtaining optimal partitions. Our analysis utilizes 104 networks, including both real-world instances from diverse contexts and modular graphs from two families of synthetic benchmarks. We analyze ten inexact modularity-based algorithms against the exact integer programming baseline that globally optimizes modularity. Our comparative analysis includes eight heuristics, two variants of a graph neural network algorithm, and nine variations of the Bayan approximation algorithm. Our findings reveal that the average modularity-based heuristic yields optimal partitions in only 43.9% of the 104 networks analyzed. Graph neural networks and approximate Bayan, on average, achieve optimality on 68.7% and 82.3% of the networks respectively. Additionally, our analysis of three partition similarity metrics exposes substantial dissimilarities between high-modularity sub-optimal partitions and any optimal partition of the networks. We observe that near-optimal partitions are often disproportionately dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of the commonly used modularity-based methods: they rarely produce an optimal partition or a partition resembling an optimal partition even on networks with modular structures. If modularity is to be used for detecting communities, we recommend approximate optimization algorithms for a more methodologically sound usage of modularity within its applicability limits.
Paper Structure (20 sections, 2 equations, 10 figures)

This paper contains 20 sections, 2 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Reviewing ten modularity-based algorithms
  3. Methods and Materials
  4. Modularity matrix of a graph
  5. Modularity of a partition
  6. Modularity maximization
  7. Optimal and sub-optimal partitions
  8. Sparse IP formulation of modularity maximization
  9. Measures for evaluating the algorithms
  10. Illustrative example
  11. Specifications of network data and computing resources
  12. Results
  13. Comparing partitions from different algorithms on one network
  14. The disproportionate cost of sub-optimality
  15. Distribution of partition similarity measures for each algorithm
  16. ...and 5 more sections

Figures (10)

  • Figure 1: A toy example demonstrating several sub-optimal and one optimal partition for a graph and the corresponding modularity, GOP, AMI, RMI, and ECS values. All information and values are related to the graph shown on top.
  • Figure 2: Modularity maximization for one network using six methods leading to six sub-optimal partitions (panels a-f) with increasing $Q$, different $k$, and different AMI values. Only the giant component is shown. (Magnify the high-resolution color figure on screen for more details.)
  • Figure 3: Modularity maximization for one network using six methods leading to one optimal partition (panel f) and five sub-optimal partitions (panels a-e) with increasing $Q$, different $k$, and different AMI values. Only the giant component is shown. (Magnify the high-resolution color figure on screen for more details.)
  • Figure 4: Global optimality percentage (GOP) and normalized adjusted mutual information (AMI) measured for each algorithm by comparing its results with (all) globally optimal partitions. (Magnify the high-resolution figure on screen for more details.)
  • Figure 5: Global optimality percentage (GOP) and normalized reduced mutual information (RMI) measured for each algorithm by comparing its results with (all) globally optimal partitions. (Magnify the high-resolution figure on screen for more details.)
  • ...and 5 more figures