  A fundamental problem in network data analysis is to test Erd\"{o}s-R\'{e}nyi
model $\mathcal{G}\left(n,\frac{a+b}{2n}\right)$ versus a bisection stochastic
block model $\mathcal{G}\left(n,\frac{a}{n},\frac{b}{n}\right)$, where $a,b>0$
are constants that represent the expected degrees of the graphs and $n$ denotes
the number of nodes. This problem serves as the foundation of many other
problems such as testing-based methods for determining the number of
communities (\cite{BS16,L16}) and community detection (\cite{MS16}). Existing
work has been focusing on growing-degree regime $a,b\to\infty$
(\cite{BS16,L16,MS16,BM17,B18,GL17a,GL17b}) while leaving the bounded-degree
regime untreated. In this paper, we propose a likelihood-ratio (LR) type
procedure based on regularization to test stochastic block models with bounded
degrees. We derive the limit distributions as power Poisson laws under both
null and alternative hypotheses, based on which the limit power of the test is
carefully analyzed. We also examine a Monte-Carlo method that partly resolves
the computational cost issue. The proposed procedures are examined by both
simulated and real-world data. The proof depends on a contiguity theory
developed by Janson \cite{J95}.
