  In network analysis, developing a unified theoretical framework that can
compare methods under different models is an interesting problem. This paper
proposes a partial solution to this problem. We summarize the idea of using
separation condition for a standard network and sharp threshold of
Erd\"os-R\'enyi random graph to study consistent estimation, compare
theoretical error rates and requirements on network sparsity of spectral
methods under models that can degenerate to stochastic block model as a
four-step criterion SCSTC. Using SCSTC, we find some inconsistent phenomena on
separation condition and sharp threshold in community detection. Especially, we
find original theoretical results of the SPACL algorithm introduced to estimate
network memberships under the mixed membership stochastic blockmodel were
sub-optimal. To find the formation mechanism of inconsistencies, we
re-establish theoretical convergence rates of this algorithm by applying recent
techniques on row-wise eigenvector deviation. The results are further extended
to the degree corrected mixed membership model. By comparison, our results
enjoy smaller error rates, lesser dependence on the number of communities,
weaker requirements on network sparsity, and so forth. Furthermore, separation
condition and sharp threshold obtained from our theoretical results match
classical results, which shows the usefulness of this criterion on studying
consistent estimation.
