Random Walk Models, Preferential Attachment, and Sequential Monte Carlo Methods for Analysis of Network Data

Networks arise in nearly every branch of science, from biology and physics to sociology and economics. A signature of many network datasets is strong local dependence, which gives rise to phenomena such as sparsity, power law degree distributions, clustering, and structural heterogeneity. Statistical models of networks require a careful balance of flexibility to faithfully capture that dependence, and simplicity, to make analysis and inference tractable. In this dissertation, we introduce a class of models that insert one network edge at a time via a random walk, permitting the location of new edges to depend explicitly on the structure of the existing network, while remaining probabilistically and computationally tractable. Connections to graph kernels are made through the probability generating function of the random walk length distribution. The limiting degree distribution is shown to exhibit power law behavior, and the properties of the limiting degree sequence are studied analytically with martingale methods. In the second part of the dissertation, we develop a class of particle Markov chain Monte Carlo algorithms to perform inference for a large class of sequential random graph models, even when the observation consists only of a single graph. Using these methods, we derive a particle Gibbs sampler for random walk models. Fit to synthetic data, the sampler accurately recovers the model parameters; fit to real data, the model offers insight into the typical length scale of dependence in the network, and provides a new measure of vertex centrality. The arrival times of new vertices are the key to obtaining results for both theory and inference. In the third part, we undertake a careful study of the relationship between the arrival times, sparsity, and heavy tailed degree distributions in preferential attachment-type models of partitions and graphs. A number of constructive representations of the limiting degrees are obtained, and connections are made to exchangeable Gibbs partitions as well as to recent results on the limiting degrees of preferential attachment graphs.