Variational approximation for importance sampling and statistical inference on social influence

Monte Carlo methods are widely used in statistical computing area to solve different problems. Social network analysis plays an importance role in many fields. In this dissertation, we focus on improving the efficiency of importance sampling, detecting the degrees of influence in networks, and exploring properties of generalized Erd\H{o}s-R\'enyi model. In the first part of the thesis, we propose an importance sampling algorithm with proposal distribution obtained from variational approximation. This method combines the strength of both importance sampling and the variational method. On one hand, this method avoids the bias from variational approximation. On the other hand, variational approximation provides a way to design the proposal distribution for the importance sampling algorithm. Theoretical justification of the proposed method is provided. Numerical results show that using variational approximation as the proposal can improve the performance of importance sampling and sequential importance sampling. In the second part of the thesis, we propose a sequential hypothesis testing procedure to detect the degrees of influence in a network. We build a multivariate Bernoulli model to represent the status of each node in the network with different degrees of influence. A double bootstrap strategy is used to resolve the uncertainty from by estimating nuisance parameters in hypothesis testing. Theoretical justification of the proposed method is provided to show that the hypothesis testing is powerful for larger networks. Simulation studies show that our method can preserve the levels and improve the powers in hypothesis testing. We also apply our proposed method on two real network data to explore the degree of influence for various features. In the third part of the thesis, we propose a random graph model for undirected networks with small-world properties, namely with a high clustering coefficient and a low average path length. We generalize the regular Erd\H{o}s-R\'enyi dyadic random graph by considering higher-order motif, which is triadic graph. We show some properties of our proposed model, analyze the probability of multi-edges, and compare the local clustering coefficient with ER model. In addition, we also provide some conditions about phase transition including connectivity threshold and the existence of giant components