In this survey paper it is illustrated how spectral clustering methods for unweighted graphs are adapted to the dense and sparse regimes. 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 arbitrarily 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. Simulation results, as well as real life examples are presented.

翻译：本综述论文阐述了如何将无权图的谱聚类方法适配于稠密与稀疏两种机制。基于拉普拉斯矩阵和模块度的谱聚类方法适用于稠密图，而近期研究结果表明，对于稀疏图，非回溯谱是发现节点同配性聚类的最佳候选方案。本文推导了稀疏随机块模型中具有任意给定模型参数的置信传播方法，得到一个非线性方程组；通过线性近似，非回溯矩阵的谱能够确定聚类数量$k$。随后，模型参数本身可通过EM算法进行估计。在同配性模型中，我们通过以下两种意义考虑边渗流现象：簇内与簇间连边概率随节点数量递减，且以此方式产生的连边以概率$\beta$被保留。因此，最优$k$值即为图的非回溯矩阵中结构性实特征值（大于$\sqrt{c}$，其中$c$为平均度）的数量。假设这些特征值$\mu_1 >\dots > \mu_k$互异，则得到关于$\beta$的多个相变点$\beta_i =\frac{c}{\mu_i^2}$；进一步地，在$\beta_i$处可检测的聚类数量为$i$（$i=1,\dots ,k$）。本文还讨论了用于节点分类的膨胀-收缩技术，该技术可作为稀疏谱聚类的基础。文中同时展示了仿真结果与真实案例。