Structure-Aware Spectral Sparsification via Uniform Edge Sampling

TL;DR

This work tackles the scalability challenge of spectral clustering by showing that, on graphs with strong, well-defined cluster structure, uniform edge sampling can serve as an effective spectral sparsifier without any resistance-based preprocessing. The authors introduce the structure ratio $Υ(k)=\lambda_{k+1}/\rho_G(k)$ and prove that, when this ratio is large, uniformly sampling $m=O\left(\frac{\kappa^2}{(1-k/Υ(k))^2(1-\rho_G(k))^2}\,n\log n/\varepsilon^2\right)$ edges preserves the top $(n-k)$ eigenspace relevant for clustering, up to a small error. Central to the analysis are new rank-$n-k$ effective-resistance bounds for intra-cluster edges and a matrix Chernoff bound focused on the dominant eigenspace, which together bypass the need for leverage-score computations. Empirical results on hierarchical SBM and LFR graphs corroborate the theory, showing that uniform sampling can match or exceed resistance-based methods in well-clustered regimes, enabling scalable, structure-preserving spectral clustering. These findings offer a practical, provable pathway to scalable spectral methods in large graphs and connect coreset-like ideas to spectral sparsification under favorable clusterability conditions.

Abstract

Spectral clustering is a fundamental method for graph partitioning, but its reliance on eigenvector computation limits scalability to massive graphs. Classical sparsification methods preserve spectral properties by sampling edges proportionally to their effective resistances, but require expensive preprocessing to estimate these resistances. We study whether uniform edge sampling-a simple, structure-agnostic strategy-can suffice for spectral clustering. Our main result shows that for graphs admitting a well-separated $k$-clustering, characterized by a large structure ratio $Υ(k) = λ_{k+1} / ρ_G(k)$, uniform sampling preserves the spectral subspace used for clustering. Specifically, we prove that uniformly sampling $O(γ^2 n \log n / ε^2)$ edges, where $γ$ is the Laplacian condition number, yields a sparsifier whose top $(n-k)$-dimensional eigenspace is approximately orthogonal to the cluster indicators. This ensures that the spectral embedding remains faithful, and clustering quality is preserved. Our analysis introduces new resistance bounds for intra-cluster edges, a rank-$(n-k)$ effective resistance formulation, and a matrix Chernoff bound adapted to the dominant eigenspace. These tools allow us to bypass importance sampling entirely. Conceptually, our result connects recent coreset-based clustering theory to spectral sparsification, showing that under strong clusterability, even uniform sampling is structure-aware. This provides the first provable guarantee that uniform edge sampling suffices for structure-preserving spectral clustering.

Figures (4)

• Figure 1: Good Clusters: Error plots comparison between Uniform Sampling and Effective Resistance Sampling of strong clusters with varying values of $\gamma$. Shaded region denotes 1 sd over 20 runs.
• Figure 2: Poor Clusters: Error plots comparison between Uniform Sampling and Effective Resistance Sampling of bad clusters with varying values of $\gamma$. Shaded region denotes 1 sd over 20 runs.
• Figure 3: Hierarchical Clusters: Error plots comparison between Uniform Sampling and Effective Resistance Sampling of strong clusters with varying values of $\gamma$. Shaded region denotes 1 sd over 20 runs.
• Figure 4: LFR Network Clusters: Error plots comparison between Uniform Sampling and Effective Resistance Sampling of strong clusters with varying values of $\mu$. Shaded region denotes 1 sd over 20 runs.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 4.1: Structure Theorem macgregor_tighter_2022peng_partitioning_2015
• Theorem 4.2: Sparsification with Structure Preservation
• Theorem 4.3
• Definition 4.4: Rank $n-k$ Effective Resistance
• Lemma 4.5
• Lemma 4.6: Intercluster Edges
• Lemma 4.7
• Theorem 4.8: Chernoff via Uniform Sampling
• proof : Proof of Part 1