  Spectral clustering is one of the most popular algorithms for community
detection in network analysis. Based on this rationale, in this paper we give
the convergence rate of eigenvectors for the adjacency matrix in the $l_\infty$
norm, under the stochastic block model (BM) and degree corrected stochastic
block model (DCBM), adding some mild and rational conditions. We also extend
this result to a more general model, presented based on the DCBM such that the
value of random variables in the adjacency matrix is not 0 or 1, but an
arbitrary real number. During the process of proving the above conclusion, we
obtain the relationship of the eigenvalues in the adjacency matrix and the
corresponding `population' matrix, which vary in dimension from the
community-wise edge probability matrix. Using that result, we can give an
estimate of the number of the communities in a known set of network data.
Meanwhile we proved the consistency of the estimator. Furthermore, according to
the derivation of proof for the convergence of eigenvectors, we propose a new
approach to community detection -- Spectral Clustering based on Difference of
Ratios of Eigenvectors (SCDRE). Our simulation experiments demonstrate the
superiority of our method in community detection.
