LEARNING LATENT COMMUNITY STRUCTURES IN NETWORK-BASED DATA

In this thesis we study two models that incorporate latent group structure related to networks. In particular, for the first part we introduce a new multitype recursive tree model called Community Modulated Recursive Tree (CMRT) that assigns group labels to vertices in a way similar to the popular stochastic block model for random graphs. Then we introduce a closely related population dependent branching process, and proceed to derive some of CMRT's asymptotic properties based on that, including limiting degree distribution, a tightness result for maximal degree and almost sure convergence of height. For the second part, we study a collection of random processes driven by certain latent community structure in a network and show that global optimum of K-means criterion can recover the groups exactly with high probability given enough observations across time. We shall also discuss other algorithms, and their performance is assessed in a simulation study. For the third part, we focus on a vector autoregressive model driven by stochastic block model, as a special case under the framework considered in the second part, but with change points. We show that this model can be studied under the structural break framework, given that the community structure is fixed and known (or can be recovered from algorithms). We also propose an algorithm for the general case where both communities and edge probabilities change across time, and its performance is compared with other methods in numerical experiments.Doctor of Philosoph