Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).

Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k “stepwise” eigenvectors (i.e., vectors that have constant entries over the components corresponding to the same set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k-th and the ($k+1$)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability.

光谱技术已被证明是最有效的图聚类方法。但是，通常它们需要对合适矩阵（通常是图的拉普拉斯矩阵）的主要特征向量进行显式计算。

最近的工作（例如，Becchetti等人，SODA 2017）建议，至少在某些情况下，观察应用于初始随机向量的幂方法的时间演化可能会提供有关前两个特征向量跨越的空间的足够信息，以便无需显式特征向量计算即可恢复隐藏分区。而Becchetti等人的结果。适用于表现出很强规则性的完美平衡分区和/或图，我们将其方法扩展到包含隐藏k分区并以适度形式的体积规则性为特征的图。我们表明，类ķ -量经常性的图表是最大的一类无向图（可能加权）的图表，其转移矩阵承认ķ“逐步”特征向量（即，在与隐藏分区的相同集合相对应的组件上具有恒定条目的向量）。为获得此结果，我们强调了体积规则性与马尔可夫链的集总性之间的联系。此外，我们证明如果逐步特征向量是与图上随机游走的转换矩阵的前k个最大特征值以及第k个和（$ķ+\mathrm{1个}$）th特征值足够大，Becchetti等人的平均动力学。以对数时间恢复图的底层社区结构的可能性很高。