Network Cross-Validation for Nested Models by Edge-Sampling

In the network literature, a wide range of statistical models has been proposed to exploit structural patterns in the data. Therefore, model selection between different models is a fundamental problem. However, there remains a lack of systematic theoretical understanding for this problem when comparing across different model classes. In this paper, to address this challenging problem, we propose a penalized edge-sampling cross-validation framework for nested network model selection. By incorporating a model complexity penalty into the evaluation process, our method effectively mitigates the overfitting tendency of cross-validation and adapts to varying model structures. This framework supports comparisons among widely used models, including stochastic block models (SBMs), degree-corrected SBMs (DCBMs), and graphon models, providing the first consistency guarantees for model selection across these settings to our knowledge. Empirical evaluations, including both simulated data and the ``Political Books'' network, demonstrate that our method yields stable and accurate performance across various scenarios.