Optimal Graph Clustering without Edge Density Signals

TL;DR

This work defines the Popularity-Adjusted Block Model (PABM), which generalizes SBM and DCBM by introducing distinct intra- and inter-cluster popularity parameters. It provides a sharp information-theoretic characterization of the optimal clustering error under PABM via Chernoff-type divergences, showing that cluster recovery can succeed even when edge-density signals vanish, provided intra- and inter-cluster popularities differ. The authors prove lower and upper bounds that match up to second-order terms and reveal novel phenomena such as non-monotone error rates with inter-cluster edges and the necessity of higher-rank spectral embeddings, with the rank of the expected adjacency matrix ranging from $k$ to $k^2$. Numerical experiments on synthetic and real data confirm that spectral methods utilizing up to $k^2$ eigenvectors outperform traditional $k$-eigenvector approaches, validating the practical value of the theoretical insights for complex network clustering. The results highlight how local degree-pattern differences can enhance separability beyond global edge densities, informing future algorithm design and model selection in network analysis.

Abstract

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.

Figures (7)

• Figure 1: Performance of graph clustering on homogeneous PABM, where the matrix $P$ is given in Equation \ref{['eq:experiments_homogeneous_PABM']}. We sampled graphs with $n=900$ vertices in $k=3$ clusters of same size, average edge density $\rho = 0.05$. In both figures, the $\lambda^{\rm in}_i$ are iid sampled from $\operatorname{Uni}(1-c,1+c)$ and $\lambda^{\rm out}_i = 1$ for all $i$. Accuracy is averaged over 15 realizations, and error bars show the standard errors.
• Figure 2: Effect of the embedding dimension on the performance of graph clustering on homogeneous PABM, where the matrix $P$ is given in Equation \ref{['eq:experiments_homogeneous_PABM']}. We sampled graphs with $n=900$ vertices in $k=3$ clusters of same size, average edge density $\rho = 0.05$. In both figures, the $\lambda^{\rm in}_i$ are iid sampled from $\operatorname{Uni}(0,2)$. Accuracy is averaged over 15 realizations, and error bars show the standard errors.
• Figure 3: Optimal error rate on PABM with homogeneous interactions. The matrix $P$ is given in Equation \ref{['eq:experiments_homogeneous_PABM']}, and we let $n=900$ vertices, $k=3$ clusters of same size, average edge density $\rho = 0.05$, and interaction probabilities $p = \rho$ and $q = \xi p$. In both figures, the quantities $\lambda^{\rm in}_i$ are iid sampled from $\mathcal{D}_{\rm in} = \operatorname{Uni}(1-c,1+c)$ and the $\lambda^{\rm out}_i$ are all equal to one. In Figure \ref{['fig:optimal_error_rate_uniform_ones_varying_c']}, we let $\xi = 1$ an vary $c$, while in Figure \ref{['fig:optimal_error_rate_uniform_ones_varying_xi']} we let $c = 0.8$ and we vary $\xi$. The optimal error rates are computed using the formula obtained in Proposition \ref{['prop:recovery_rate_pabm']}.
• Figure 4: Numerical values obtained for the optimal error rate $\frac{1}{n} \sum_i \exp(- \mathrm{Chernoff} (i,z^*))$ on PABM with homogeneous interactions. The matrix $P$ is given in Equation \ref{['eq:experiments_homogeneous_PABM']}, and we let $n=900$ vertices, $k=3$ clusters of same size, average edge density $\rho = 0.05$, and interaction probabilities $p = \rho$ and $q = \xi p$. In both figures, the quantities $\lambda^{\rm in}_i$ and $\lambda^{\rm out}$ are iid sampled from a distribution $\mathcal{D}$. Figure \ref{['fig:optimal_error_rate_other_distributions_exponential']}: $\mathcal{D}$ is the exponential distribution with mean 1. Figure \ref{['fig:optimal_error_rate_other_distributions_logNormal']}: $\mathcal{D}$ is the log-normal distribution with parameters $(\mu, \sigma) = (-1/2,1)$ (chosen so that the mean of the distribution is $1$).
• Figure 5: Performance of graph clustering on homogeneous PABM, where the matrix $P$ is given in Equation \ref{['eq:experiments_homogeneous_PABM']}. We sampled graphs with $n=900$ vertices in $k=3$ clusters of same size, average edge density $\rho = 0.05$. In both figures, the $\lambda^{\rm in}_i$ are iid sampled from $\operatorname{Uni}(1-c,1+c)$ and $\lambda^{\rm out}_i = 1$ for all $i$. Accuracy is averaged over 15 realizations, and error bars show the standard errors.

Theorems & Definitions (13)

• Theorem 1: Lower-bound on the clustering error
• Theorem 2: Achievability
• Proposition 3
• Example 1
• Example 2
• proof : Proof of Theorem \ref{['thm:lower_bound']}
• Lemma 4: Lemma C.5 in avrachenkov2020community
• Lemma 5
• proof
• Lemma 6