Abstract
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 a separation condition for a standard network and sharp threshold of the Erdös-Rényi random graph to study consistent estimation, and compare theoretical error rates and requirements on the network sparsity of spectral methods under models that can degenerate to a 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. In particular, we find that the original theoretical results of the SPACL algorithm introduced to estimate network memberships under the mixed membership stochastic blockmodel are sub-optimal. To find the formation mechanism of inconsistencies, we re-establish the theoretical convergence rate 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. The separation condition and sharp threshold obtained from our theoretical results match the classical results, so the usefulness of this criterion on studying consistent estimation is guaranteed. Numerical results for computer-generated networks support our finding that spectral methods considered in this paper achieve the threshold of separation condition.