Dynamic Structural Clustering on Graphs

TL;DR

This paper introduces efficient algorithms for maintaining and querying dynamic graph clusterings based on structural similarity measures, significantly improving update times while ensuring clustering quality.

Contribution

The paper presents the first algorithms with sub-logarithmic amortized update time for dynamic structural clustering on graphs, with provable quality guarantees and practical efficiency.

Findings

Algorithms achieve up to 1000x faster updates than previous methods.

The methods maintain clustering quality with high probability after each update.

Experimental results confirm significant efficiency and quality improvements over existing approaches.

Abstract

Structural Clustering ($DynClu$) is one of the most popular graph clustering paradigms. In this paper, we consider $StrClu$ under two commonly adapted similarities, namely Jaccard similarity and cosine similarity on a dynamic graph, $G = \langle V, E\rangle$, subject to edge insertions and deletions (updates). The goal is to maintain certain information under updates, so that the $StrClu$ clustering result on~$G$ can be retrieved in $O(|V| + |E|)$ time, upon request. The state-of-the-art worst-case cost is $O(|V|)$ per update; we improve this update-time bound significantly with the $\rho$-approximate notion. Specifically, for a specified failure probability, $\delta^*$, and every sequence of $M$ updates (no need to know $M$'s value in advance), our algorithm, $DynELM$, achieves $O(\log^2 |V| + \log |V| \cdot \log \frac{M}{\delta^*})$ amortized cost for each update, at all times in linear space. Moreover, $DynELM$ provides a provable "sandwich" guarantee on the clustering quality at all times after \emph{each update} with probability at least $1 - \delta^*$. We further develop $DynELM$ into our ultimate algorithm, $DynStrClu$, which also supports cluster-group-by queries. Given $Q\subseteq V$, this puts the non-empty intersection of $Q$ and each $StrClu$ cluster into a distinct group. $DynStrClu$ not only achieves all the guarantees of $DynELM$, but also runs cluster-group-by queries in $O(|Q|\cdot \log |V|)$ time. We demonstrate the performance of our algorithms via extensive experiments, on 15 real datasets. Experimental results confirm that our algorithms are up to three orders of magnitude more efficient than state-of-the-art competitors, and still provide quality structural clustering results. Furthermore, we study the difference between the two similarities w.r.t. the quality of approximate clustering results.