This paper studies clustering algorithms for dynamically evolving graphs
$\{G_t\}$, in which new edges (and potential new vertices) are added into a
graph, and the underlying cluster structure of the graph can gradually change.
The paper proves that, under some mild condition on the cluster-structure, the
clusters of the final graph $G_T$ of $n_T$ vertices at time $T$ can be well
approximated by a dynamic variant of the spectral clustering algorithm. The
algorithm runs in amortised update time $O(1)$ and query time $o(n_T)$.
Experimental studies on both synthetic and real-world datasets further confirm
the practicality of our designed algorithm.

该研究论文探讨了用于动态演变图的聚类算法，证明了在某些簇结构的条件下，最终图的簇可以被谱聚类算法的动态变体很好地近似，并运行时间复杂度为 O (1) 和查询时间复杂度为 o (n_T)，实验证明了该算法的实用性。

带有可证明近似保证的动态谱聚类

Dynamic Spectral Clustering with Provable Approximation Guarantee

We consider the problem of recovering hidden communities in the Labeled
Stochastic Block Model (LSBM) with a finite number of clusters, where cluster
sizes grow linearly with the total number $n$ of items. In the LSBM, a label is
(independently) observed for each pair of items. Our objective is to devise an
efficient algorithm that recovers clusters using the observed labels. To this
end, we revisit instance-specific lower bounds on the expected number of
misclassified items satisfied by any clustering algorithm. We present
Instance-Adaptive Clustering (IAC), the first algorithm whose performance
matches these lower bounds both in expectation and with high probability. IAC
consists of a one-time spectral clustering algorithm followed by an iterative
likelihood-based cluster assignment improvement. This approach is based on the
instance-specific lower bound and does not require any model parameters,
including the number of clusters. By performing the spectral clustering only
once, IAC maintains an overall computational complexity of $\mathcal{O}(n
\text{polylog}(n))$. We illustrate the effectiveness of our approach through
numerical experiments.

本文介绍了一种针对 Labeled Stochastic Block Model 中包含多个聚类的问题的 Instance-Adaptive Clustering (IAC) 算法，通过一次谱聚类算法和迭代的基于 likelihood 的聚类算法提高聚类的效率。