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Selecting representative community partitions under modularity degeneracy: the STAR method

Francesca Grassetti, Rossana Mastrandrea

TL;DR

The proposed approach is model-agnostic and can be applied a posteriori to the output of any modularity-based community detection algorithm, and aims to identify a solution that best represents the structural features shared across degenerate partitions.

Abstract

Community detection based on modularity maximization is one of the most widely used approaches for uncovering mesoscale structures in complex networks. However, it is well known that the modularity function exhibits a highly degenerate optimization landscape: a large number of structurally distinct partitions attain close modularity values. This degeneracy raises issues of instability, reproducibility, and interpretability of the detected communities. We propose a simple and user-friendly post-processing method to address this problem by selecting a representative partition among the set of high-modularity solutions. The proposed approach is model-agnostic and can be applied a posteriori to the output of any modularity-based community detection algorithm. Rather than seeking the optimal partition in terms of modularity, our method aims to identify a solution that best represents the structural features shared across degenerate partitions. We compare our approach with consensus clustering methods, which pursue a similar objective, and show that the resulting partitions are highly consistent, while being obtained through a substantially simpler procedure that does not require additional optimization steps or external software packages. Moreover, unlike standard consensus clustering techniques, the proposed method can be applied to networks with both positive and negative edge weights, making it suitable for a wide range of applications involving signed networks and correlation-based systems, such as social, financial, and neuroscience networks. Overall, the method provides a practical and robust tool for handling degeneracy in modularity-based community detection, combining simplicity with broad applicability across different types of networks and real-world problems.

Selecting representative community partitions under modularity degeneracy: the STAR method

TL;DR

The proposed approach is model-agnostic and can be applied a posteriori to the output of any modularity-based community detection algorithm, and aims to identify a solution that best represents the structural features shared across degenerate partitions.

Abstract

Community detection based on modularity maximization is one of the most widely used approaches for uncovering mesoscale structures in complex networks. However, it is well known that the modularity function exhibits a highly degenerate optimization landscape: a large number of structurally distinct partitions attain close modularity values. This degeneracy raises issues of instability, reproducibility, and interpretability of the detected communities. We propose a simple and user-friendly post-processing method to address this problem by selecting a representative partition among the set of high-modularity solutions. The proposed approach is model-agnostic and can be applied a posteriori to the output of any modularity-based community detection algorithm. Rather than seeking the optimal partition in terms of modularity, our method aims to identify a solution that best represents the structural features shared across degenerate partitions. We compare our approach with consensus clustering methods, which pursue a similar objective, and show that the resulting partitions are highly consistent, while being obtained through a substantially simpler procedure that does not require additional optimization steps or external software packages. Moreover, unlike standard consensus clustering techniques, the proposed method can be applied to networks with both positive and negative edge weights, making it suitable for a wide range of applications involving signed networks and correlation-based systems, such as social, financial, and neuroscience networks. Overall, the method provides a practical and robust tool for handling degeneracy in modularity-based community detection, combining simplicity with broad applicability across different types of networks and real-world problems.
Paper Structure (15 sections, 11 equations, 3 figures, 1 algorithm)

This paper contains 15 sections, 11 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method
  3. Network properties
  4. Modularity maximization
  5. Similarity-based Top ARI Representative method
  6. Data
  7. Benchmarks with ground-truth
  8. Real Data
  9. Results
  10. Benchmark
  11. Real Data
  12. The World Trade Web
  13. FTSE100 stock market
  14. Discussion
  15. Conclusion

Figures (3)

  • Figure 1: The parameters of the LFR benchmark graphs are: average degree = 20, maximum degree = 50, minimum community size = 10, maximum community size = 50, degree exponent = 2, and community size exponent = 3.
  • Figure 2: World Trade Web 2015. Countries are colored according to the communities identified by (a) the STAR/consensus method and (b) the maximum modularity criterion.
  • Figure 3: Communities of FTSE100 market. Partitions are generated using the Louvain algorithm on the filtered correlation matrix (REF) for 150 runs. (a) Histogram of the modularity values associated to the 150 partitions. Community organization in sectors of (b) the representative partition obtained with the STAR method and (c) the partition associated to the maximum modularity value.