图的动态结构聚类
Dynamic Structural Clustering on Graphs
摘要：
结构聚类（$DynClu$）是最流行的图聚类范例之一。在本文中，我们认为$StrClu$下两个常见的适应相似性，即Jaccard相似性和余弦相似性的动态图，$G = \langle V, E\rangle$，受到边缘插入和删除（更新）。目标是维护某些更新信息，以便在收到请求时，可以在$O(|V| + |E|)$时间内检索到~ $G$上的$StrClu$群集结果。国家的最先进的最坏情况下的成本是$O(|V|)$每次更新，我们提高了这个更新的时间约束显着的$ρ$近似的概念。具体来说，对于指定的故障概率$δ^*$和每个$M$更新序列（不需要事先知道$M$的值），我们的算法$DynELM$在线性空间中的任何时候都可以实现每次更新的$O(\log^2 |V| + \log |V| \cdot \log \frac{M}{δ^*})$摊销成本。此外，$DynELM$在\emph{each update}之后的任何时候都以至少$1 - δ^*$的概率提供对聚类质量的可证明的“三明治”保证。我们进一步发展$DynELM$到我们的最终算法，$DynStrClu$，它也支持集群组查询。给定$Q\subseteq V$，这将$Q$和每个$StrClu$集群的非空交集放入不同的组中。$DynStrClu$不仅实现了$DynELM$的所有保证，而且在$O(|Q|\cdot \log |V|)$时间内运行cluster-group-by查询。我们证明了我们的算法的性能，通过广泛的实验，在15个真实的数据集。实验结果证实，我们的算法比最先进的竞争对手的效率高出三个数量级，并且仍然提供高质量的结构聚类结果。此外，我们还研究了这两个相似点之间的差异。近似聚类结果的质量。
英文摘要：
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 $ρ$-approximate notion. Specifically, for a specified failure probability, $δ^*$, and every sequence of $M$ updates (no need to know $M$'s value in advance), our algorithm, $DynELM$, achieves $O(\log^2 |V| + \log |V| \cdot \log \frac{M}{δ^*})$ 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 $1 - δ^*$. 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.
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