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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.21669 |
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| _version_ | 1866918170073235456 |
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| author | Dreveton, Maximilien Liu, Elaine Siyu Grossglauser, Matthias Thiran, Patrick |
| author_facet | Dreveton, Maximilien Liu, Elaine Siyu Grossglauser, Matthias Thiran, Patrick |
| contents | This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Graph Clustering without Edge Density Signals Dreveton, Maximilien Liu, Elaine Siyu Grossglauser, Matthias Thiran, Patrick Machine Learning This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches. |
| title | Optimal Graph Clustering without Edge Density Signals |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.21669 |