Discussion of ‘Network cross-validation by edge sampling’

Abstract

1. Introduction We congratulate the authors for a nice contribution to the literature on model selection and assessment for complex network data. The article of Li et al. (2020) enriches the collection of tools available for network data analysis, bringing together different fields such as low-rank matrix estimation, cross-validation and network modelling. This discussion will focus on the V -fold variant of the edge sampling cross-validation method and its combination with other recently developed cross-validation methods and network models. 2. V -fold implementation and comparison with block sampling V -fold cross-validation is perhaps the most popular variant of cross-validation. It partitions the training set into V disjoint folds and performs sample splitting validation by leaving out each fold as the validation set and using the remaining V − 1 folds for estimation. The final validated risk is obtained by aggregating the sample splitting validation risks over the folds. This idea can be extended straightforwardly to the edge sampling cross-validation framework, as mentioned in § 2.2 of Li et al. (2020). A related cross-validation method for network and other low-rank matrix estimation problems is the block sampling method developed by Chen & Lei (2018), where the sample splitting is realized by holding out a diagonal sub-block of the adjacency matrix. The simulation study in Li et al. (2020) includes some empirical comparisons between edge sampling and block sampling under various settings, and suggests that edge sampling may be preferable in more challenging settings. Given the nature of these algorithms, it is hard to imagine a difference in rates of convergence of the estimation error, and it seems more likely that the estimation errors would differ at a constant factor level. A possible explanation is the more efficient use of validation sample points in edge sampling. In V -fold edge sampling, each node pair appears exactly once as a validation sample point, so that the validation sample size is roughly n 2 / 2 ⁠ . In V -fold block sampling, on the other hand, only the node pairs in the diagonal sub-blocks are used for validation; hence the validation sample size for V -fold block sampling is approximately n 2 / ( 2 V ) ⁠ .