  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.
