  The network data has attracted considerable attention in modern statistics.
In research on complex network data, one key issue is finding its underlying
connection structure given a network sample. The methods that have been
proposed in literature usually assume that the underlying structure is a known
model. In practice, however, the true model is usually unknown, and network
learning procedures based on these methods may suffer from model
misspecification. To handle this issue, based on the random matrix theory, we
first give a spectral property of the normalized adjacency matrix under a mild
condition. Further, we establish a general goodness-of-fit test procedure for
the unweight and undirected network. We prove that the null distribution of the
proposed statistic converges in distribution to the standard normal
distribution. Theoretically, this testing procedure is suitable for nearly all
popular network models, such as stochastic block models, and latent space
models. Further, we apply the proposed method to the degree-corrected mixed
membership model and give a sequential estimator of the number of communities.
Both simulation studies and real-world data examples indicate that the proposed
method works well.
