Network sampling based inference for subgraph counts and clustering coefficient in a Stochastic Block Model framework with some extensions to a sparse case

Sampling is frequently used to collect data from large networks. In this article we provide valid asymptotic prediction intervals for subgraph counts and clustering coefficient of a population network when a network sampling scheme is used to observe the population. The theory is developed under a model based framework, where it is assumed that the population network is generated by a Stochastic Block Model (SBM). We study the effects of induced and ego-centric network formation, following the initial selection of nodes by Bernoulli sampling, and establish asymptotic normality of sample based subgraph count and clustering coefficient statistic under both network formation methods. The asymptotic results are developed under a joint design and model based approach, where the effect of sampling design is not ignored. In case of the sample based clustering coefficient statistic, we find that a bias correction is required in the ego-centric case, but there is no such bias in the induced case. We also extend the asymptotic normality results for estimated subgraph counts to a mildly sparse SBM framework, where edge probabilities decay to zero at a slow rate. In this sparse setting we find that the scaling and the maximum allowable decay rate for edge probabilities depend on the choice of the target subgraph. We obtain an expression for this maximum allowable decay rate and our results suggest that the rate becomes slower if the target subgraph has more edges in a certain sense. The simulation results suggest that the proposed prediction intervals have excellent coverage, even when the node selection probability is small and unknown SBM parameters are replaced by their estimates. Finally, the proposed methodology is applied to a real data set.