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Complementary Strengths: Combining Geometric and Topological Approaches for Community Detection

Jelena Losic

TL;DR

This paper addresses the lack of a universal approach to community detection by proposing a hybrid framework that combines geometric projections (spectral or force-directed embeddings) with Topological Data Analysis via ToMATo. By embedding networks into low-dimensional spaces where density basins correspond to communities, and applying a persistence-based clustering, the method leverages both geometric and topological strengths. Results show the hybrid approach can match or exceed traditional modularity-based methods on modular networks and outperform them on certain real-world networks, depending on the embedding. The work advocates for adaptive, geometry-informed hybrid algorithms that select projection strategies based on network structure to improve robustness, interpretability, and scalability in community detection.

Abstract

The optimal strategy for community detection in complex networks is not universal, but depends critically on the network's underlying structural properties. Although popular graph-theoretic methods, such as Louvain, optimize for modularity, they can overlook nuanced, geometric community structures. Conversely, topological data analysis (TDA) methods such as ToMATo are powerful in identifying density-defined clusters in embedded data but can be sensitive to initial projection. We propose a unified framework that integrates both paradigms to take advantage of their complementary advantages. Our method uses spectral embedding to capture the network's geometric skeleton, creating a landscape where communities manifest as density basins. The ToMATo algorithm then provides a topologically-grounded and parameter-aware method to extract persistent clusters from this landscape. Our comprehensive analysis across synthetic benchmarks shows that this hybrid approach is highly robust: it performs on par with Louvain on modular networks. These results argue for a new class of hybrid algorithms that select strategies based on network geometry, moving beyond one-size-fits-all solutions.

Complementary Strengths: Combining Geometric and Topological Approaches for Community Detection

TL;DR

This paper addresses the lack of a universal approach to community detection by proposing a hybrid framework that combines geometric projections (spectral or force-directed embeddings) with Topological Data Analysis via ToMATo. By embedding networks into low-dimensional spaces where density basins correspond to communities, and applying a persistence-based clustering, the method leverages both geometric and topological strengths. Results show the hybrid approach can match or exceed traditional modularity-based methods on modular networks and outperform them on certain real-world networks, depending on the embedding. The work advocates for adaptive, geometry-informed hybrid algorithms that select projection strategies based on network structure to improve robustness, interpretability, and scalability in community detection.

Abstract

The optimal strategy for community detection in complex networks is not universal, but depends critically on the network's underlying structural properties. Although popular graph-theoretic methods, such as Louvain, optimize for modularity, they can overlook nuanced, geometric community structures. Conversely, topological data analysis (TDA) methods such as ToMATo are powerful in identifying density-defined clusters in embedded data but can be sensitive to initial projection. We propose a unified framework that integrates both paradigms to take advantage of their complementary advantages. Our method uses spectral embedding to capture the network's geometric skeleton, creating a landscape where communities manifest as density basins. The ToMATo algorithm then provides a topologically-grounded and parameter-aware method to extract persistent clusters from this landscape. Our comprehensive analysis across synthetic benchmarks shows that this hybrid approach is highly robust: it performs on par with Louvain on modular networks. These results argue for a new class of hybrid algorithms that select strategies based on network geometry, moving beyond one-size-fits-all solutions.

Paper Structure

This paper contains 22 sections, 4 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Topological Data Analysis for Clustering
  3. Multi-scale Feature Extraction
  4. Robustness to Noise and Outliers
  5. Sensitivity to Complex Cluster Shapes
  6. Complementarity with Conventional Methods
  7. Informative Data Representations
  8. Community detection in networks
  9. Problem definition
  10. Challenges in Community Detection and How TDA Addresses Them
  11. Methodology
  12. Projection via Force-Directed Layout and ToMATo
  13. Graph-Based Clustering using Spectral Projection and ToMATo
  14. Graph Representation and Spectral Embedding
  15. Density Estimation using Kernel Density Estimation
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Figure illustrates the clustering outcomes obtained on the LFR benchmark network using two different approaches, both visualized in the same spectral embedding space.
  • Figure 2: Figure illustrates peristence diagram and Persistence barcode of cora full dataset