We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to their own cluster and more reliably classified than the rest. This can be regarded as a discrete version of the Hot Spots conjecture and should be useful in applications. We give a rigorous proof for the stochastic block model and several examples.

中文翻译：

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重新审视谱聚类：隐藏在 Fiedler 向量中的信息

我们对图上的聚类问题感兴趣：已知如果有两个底层簇，那么邻接矩阵的第二大特征值对应的特征向量的符号可以可靠地重建这两个簇。我们认为，特征向量分别具有最大和最小条目的顶点异常强烈地连接到它们自己的集群，并且比其他顶点更可靠地分类。这可以看作是热点猜想的离散版本，在应用中应该很有用。我们给出了随机块模型的严格证明和几个例子。