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In this survey paper it is illustrated how spectral clustering methods forunweighted graphs are adapted to the dense and sparse regimes. WhereasLaplacian and modularity based spectral clustering is apt to dense graphs,recent results show that for sparse ones, the non-backtracking spectrum is thebest candidate to find assortative clusters of nodes. Here belief propagationin the sparse stochastic block model is derived with arbitrarily given modelparameters that results in a non-linear system of equations; with linearapproximation, the spectrum of the non-backtracking matrix is able to specifythe number $k$ of clusters. Then the model parameters themselves can beestimated by the EM algorithm.  Bond percolation in the assortative model is considered in the following twosenses: the within- and between-cluster edge probabilities decrease with thenumber of nodes and edges coming into existence in this way are retained withprobability $\beta$. As a consequence, the optimal $k$ is the number of thestructural real eigenvalues (greater than $\sqrt{c}$, where $c$ is the averagedegree) of the non-backtracking matrix of the graph. Assuming, theseeigenvalues $\mu_1 >\dots > \mu_k$ are distinct, the multiple phase transitionsobtained 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 nodesthemselves, which can be the base of the sparse spectral clustering. Simulationresults, as well as real life examples are presented.

Structure and Noise in Dense and Sparse Random Graphs: Percolated  Stochastic Block Model via the EM Algorithm and Belief Propagation with  Non-Backtracking Spectra