Whereas Laplacian and modularity based spectral clustering is apt to dense graphs, recent results show that for sparse ones, the non-backtracking spectrum is the best candidate to find assortative clusters of nodes. Here belief propagation in the sparse stochastic block model is derived with arbitrary given model parameters that results in a non-linear system of equations; with linear approximation, the spectrum of the non-backtracking matrix is able to specify the number $k$ of clusters. Then the model parameters themselves can be estimated by the EM algorithm. Bond percolation in the assortative model is considered in the following two senses: the within- and between-cluster edge probabilities decrease with the number of nodes and edges coming into existence in this way are retained with probability $\beta$. As a consequence, the optimal $k$ is the number of the structural real eigenvalues (greater than $\sqrt{c}$, where $c$ is the average degree) of the non-backtracking matrix of the graph. Assuming, these eigenvalues $\mu_1 >\dots > \mu_k$ are distinct, the multiple phase transitions obtained for $\beta$ are $\beta_i =\frac{c}{\mu_i^2}$; further, at $\beta_i$ the number of detectable clusters is $i$, for $i=1,\dots ,k$. Inflation-deflation techniques are also discussed to classify the nodes themselves, which can be the base of the sparse spectral clustering.

翻译：虽然基于拉普拉斯矩阵和模块度的谱聚类适用于稠密图，但最新研究表明，对于稀疏图而言，非回溯谱是检测节点同配聚类的最优选择。本文推导了具有任意给定模型参数的稀疏随机块模型中的置信传播算法，该算法形成一个非线性方程组；通过线性近似，非回溯矩阵的谱能够确定聚类数$k$。随后，模型参数本身可通过EM算法进行估计。我们从以下两个角度考虑同配模型中的键逾渗：簇内和簇间边概率随节点数增加而减小，且以这种方式产生的边以概率$\beta$保留。由此，最优$k$等于图中非回溯矩阵的结构性实特征值（大于$\sqrt{c}$，其中$c$为平均度数）的个数。假设这些特征值$\mu_1 >\dots > \mu_k$互异，则$\beta$对应的多个相变点为$\beta_i =\frac{c}{\mu_i^2}$；进一步，在$\beta_i$处可检测的聚类数为$i$（$i=1,\dots ,k$）。本文还讨论了用于节点分类的膨胀-收缩技术，这可为稀疏谱聚类奠定基础。