Effective Indexing for Dynamic Structural Graph Clustering

Download PDF

Abstract

Graph clustering is a fundamental data mining task that clusters vertices into different groups. The structural graph clustering algorithm ($SCAN$) is a widely used graph clustering algorithm that derives not only clustering results, but also special roles of vertices like hubs and outliers. In this paper, we consider structural graph clustering under Jaccard similarity on dynamic graphs. The state-of-the-art index-based solution focuses on static graphs and takes prohibitive update costs to maintain the index. Recently, an approximate dynamic structural graph clustering algorithm under Jaccard similarity is proposed, reducing the expected update cost to $O\left(\log^2{n}+ \log{n}\cdot \log{(M/p_f)}\right)$, which guarantees that the returned clustering result satisfies the approximation definition with probability $1-p_f$ after up to $M$ updates. However, their solution needs to fix the input parameters while the parameter settings of SCAN usually need to be fine-tuned to achieve good clustering results. Motivated by these limitations, we present a study on devising effective index structures for SCAN algorithm on dynamic graphs. Similar to the state-of-the-art dynamic scheme, our main idea to reduce the time complexity is still by bringing approximation to clustering results. However, our solution does not need to fix the input parameters. To achieve this, our solution includes two key components. The first is to maintain a bottom-$k$ sketch for each vertex so that the similarities of affected vertices can be easily updated. The second key is a bucketing strategy that allows us to update clustering results and roles of vertices efficiently. Our theoretical analysis shows that our proposed algorithm achieves $O(\log{n}\cdot \log{\frac{M+m}{p_f}})$ expected update cost and guarantees to return approximate clustering result with probability $1-p_f$ after up to $M$ updates. Extensive experiments show that our solution is up to two orders of magnitude faster than the state-of-the-art index-based solution while still achieving high-quality clustering results.