  While there have been numerous sequential algorithms developed to estimate
community structure in networks, there is little available guidance and study
of what significance level or stopping parameter to use in these sequential
testing procedures. Most algorithms rely on prespecifiying the number of
communities or use an arbitrary stopping rule. We provide a principled approach
to selecting a nominal significance level for sequential community detection
procedures by controlling the tolerance ratio, defined as the ratio of
underfitting and overfitting probability of estimating the number of clusters
in fitting a network. We introduce an algorithm for specifying this
significance level from a user-specified tolerance ratio, and demonstrate its
utility with a sequential modularity maximization approach in a stochastic
block model framework. We evaluate the performance of the proposed algorithm
through extensive simulations and demonstrate its utility in controlling the
tolerance ratio in single-cell RNA sequencing clustering by cell type and by
clustering a congressional voting network.
