Empirical Bayes estimation for random dot product graph representation of the stochastic blockmodel

Network models are increasingly used to model datasets that involve interacting units, particularly random graph models where the vertices represent individual entities and the edges represent the presence or absence of a specified interaction between these entities. Finding inherent communities in networks (i.e. partitioning vertices with a more similar interaction pattern into groups) is considered to be a fundamental task in network analysis, which aids in understanding the structural properties of real-world networks. Despite a large amount of research on this task since the emergence of graphical representation of relational data, this still remains a challenge. In particular, within the statistical community, the use of the stochastic blockmodel for this task is currently of immense interest. Recent theoretical developments have shown that adjacency spectral embedding of graphs yields tractable distributional results. Specifically, a random dot product graph formulation of the stochastic blockmodel provides a mixture of multivariate Gaussians for the asymptotic distribution of the latent positions estimated by adjacency spectral embedding. The first part of this thesis seeks to employ this new theory to provide an empirical Bayes model for estimating block memberships of vertices in a stochastic blockmodel graph. Posterior inference is conducted using a Metropolis-within-Gibbs algorithm. Performance of the model is illustrated through Monte Carlo simulation studies and experimental results on a Wikipedia dataset. Results show performance gains over other alternative models that are considered. Instead of a complete classification of vertices via community detection, one may wish to discover whether vertices possess an attribute of interest. Given that this attribute is observed for a few vertices, the goal is to find other vertices that possess that same attribute. As an example, if a few employees in a company are known to have committed fraud, how can we identify others who may be complicit? This is a special case of community detection, known as vertex nomination, which has recently grown rapidly as a research topic. The second part of this thesis extends the empirical Bayes model for vertex nomination based on information contained in the graph structure. This yields promising simulation results as well as real-data results from an Enron email dataset. Recent studies have shown that information pertinent to vertex nomination exists not only in the graph structure but also in the edge attributes (Coppersmith and Priebe, 2012; Suwan et al., 2015). This motivates the third part of this thesis by further extending the model to exploit both graph structure and edge attributes for vertex nomination. Simulation studies confirm the benefit of doing so. However, the same benefit is not observed when the model is applied to the Enron email dataset; further investigations suggest that this is due to the data violating one of the model assumptions. [This page intentionally left blank] ACKNOWLEDGMENTS First and foremost, I would like to thank God Almighty for giving me the patience and strength to complete this thesis. I would then like to express my deepest gratitude to all my supervisors. This thesis would not have been achievable without the encouragement, patience and guidance I received from them, their efforts throughout this entire course has taught me a new set of skills which I never imagined I could possess. My biggest thanks go to Dr Dominic Lee for being the best supervisor, teacher and mentor I could ever ask for. Even after moving on from the world of academia, he still arranged regular meetings after his work hours to continue his support for me. I am very grateful for his precious time, effort, patience, support, and most importantly for believing in me. Without him I would not have courage and strength to acquire this highest level of qualification. A huge thank you to Professor Carey Priebe for his expertise and guidance throughout this process, in fact the ideas in this thesis mainly stem from the work of Professor Carey Priebe and his associates. I would like to thank his many associates who commented, shared their expertise, and contributed to making this thesis a worthwhile endeavor. I would also like to thank Dr Carl Scarrott for agreeing to be my supervisor, supporting, giving good advice from as early on as my undergraduate years, and encouraging me to pursue a PhD study when I doubted myself. Many thanks to all the postgraduate colleagues and staff of the School of Mathematics and Statistics at the University of Canterbury for making my time as a PhD student very enjoyable. I am especially grateful to have had Rachel Binny as my office-mate. I can not imagine having a better colleague to work next to, thank you for being a good friend and supporting me through the many ups and downs in the last three years. And of course I cannot forget Paul Brouwers and Steve Gourdie, who were very friendly and approachable whenever I needed their assistance with my programming and computing troubles. I need to also thank my mother, husband, children, in-laws, for their unconditional love, understanding, practical and emotional support, and for their confidence in me. For putting up with me when I was stressed out, and for being patient with me when I could not always give them the time they deserved. Lastly, I am grateful to the University of Canterbury for supporting me financially through a scholarship grant to complete this thesis, without which I would never have been able to take this huge leap forward in my career. [This page intentionally left blank]