[Paper Review] A Local Algorithm for Finding Well-Connected Clusters

Zeyuan Allen Zhu, Silvio Lattanzi+1 more|arXiv (Cornell University)|Apr 30, 2013

Complex Network Analysis Techniques36 references28 citations

TL;DR

This paper proposes a novel random-walk-based local algorithm for identifying well-connected clusters in large graphs, with runtime dependent only on the output cluster size. It achieves superior theoretical guarantees in clustering accuracy and conductance—especially for well-connected clusters—demonstrating tighter analysis and empirical validation on synthetic and real-world data.

ABSTRACT

Motivated by applications of large-scale graph clustering, we study random-walkbased local algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, both in terms of the clustering accuracy and the conductance of the output set. We also prove that our analysis is tight, and perform empirical evaluation to support our theory on both synthetic and real data. More specifically, our method outperforms prior work when the cluster is well-connected. In fact, the better it is well-connected inside, the more significant improvement we can obtain. Our results shed light on why in practice some random-walk-based algorithms perform better than its previous theory, and help guide future research about local clustering. 1.

Motivation & Objective

• Address the gap in theoretical guarantees for local graph clustering algorithms, particularly in handling well-connected clusters.
• Develop a method that achieves better clustering accuracy and conductance than prior work, especially when clusters are highly internally connected.
• Provide a tight theoretical analysis that explains why some random-walk-based algorithms outperform their theoretical bounds in practice.
• Guide future research in local clustering by establishing stronger theoretical foundations and empirical validation.

Proposed method

• Leverages a random-walk-based approach to explore local neighborhoods, focusing only on the cluster size rather than the entire graph.
• Introduces a modified walk process that enhances the likelihood of identifying well-connected clusters with high conductance.
• Employs a theoretical analysis framework that bounds the conductance and accuracy of the output cluster using walk properties.
• Uses a stopping criterion based on local walk statistics to ensure the output set is both well-connected and of high quality.
• Applies a tight analysis technique that proves the theoretical bounds are optimal, demonstrating the method's superiority.
• Validates the method empirically on both synthetic graphs and real-world datasets to support theoretical claims.

Experimental results

Research questions

• RQ1 How can local clustering algorithms achieve better theoretical guarantees in conductance and accuracy for well-connected clusters?
• RQ2 Why do some random-walk-based algorithms perform better in practice than their theoretical analysis suggests?
• RQ3 Can a tighter theoretical analysis be developed to explain and improve the performance of local clustering methods?
• RQ4 To what extent does the internal connectivity of a cluster affect the performance of local clustering algorithms?
• RQ5 How does the proposed method compare empirically to prior work on real and synthetic graph data?

Key findings

• The proposed algorithm achieves significantly better conductance and clustering accuracy than previous methods, particularly for well-connected clusters.
• The theoretical analysis is proven to be tight, confirming that the bounds cannot be improved further under the same assumptions.
• Empirical results on synthetic and real-world graphs confirm that the method outperforms prior work, with the improvement increasing as cluster connectivity increases.
• The method explains the empirical success of existing random-walk-based algorithms, which often surpass theoretical predictions in practice.
• The results suggest that future local clustering algorithms should prioritize enhancing internal cluster connectivity detection for improved performance.
• The algorithm’s runtime scales only with the output cluster size, making it efficient for large-scale graphs.