Random Dot Product Graphs: a Flexible Model for Complex Networks

For her patience, understanding, and love to Naomi. iii ACKNOWLEDGEMENTS I would first of all like to thank the School of Mathematics and the ACO program for providing me this opportunity. I would especially like to thank Prof. Luca Dieci for his support and understanding when things were not going quite as planned. Also, I would like to thank the staff of the School of Mathematics for their phenomenal support and their ability of making administrative headaches disappear. I would like to thank my committee for the time and effort and for all they have taught me directly and indirectly. I would especially like to thank Prof. Linyuan Lu for his thoughtful remarks regarding this thesis and for taking the time to serve as reader. I would like to thank all of my friends in the ACO program and the School of Mathematics , especially Mark, Mitch, Csaba, Dave, and Noah. Thank you guys for listening to me spout my ideas and helping refine them. I also thank my wife and family for trying to understand what I do and why I do it, and encouraging me to see it through to the end. I also thank the people too numerous to name whose constant belief in me has kept me going throughout this process. I especially want to thank Prof. Tom Trotter for accepting me as a " half "-student. I am deeply thankful for all you have taught me by word and example, both as a mathematician and as a person. Most of all I would like to thank my advisor, Milena Mihail, for accepting someone she barely knew with an out of the blue idea for a thesis topic, for having the patience and perserverence to see me through the process even when it seemed to be going nowhere, and for having the insight to know what I needed, even when I didn't know myself. Thank you. iv SUMMARY Over the last twenty years, as biological, technological, and social networks have risen in prominence and importance, the study of complex networks has attracted researchers from a wide range of fields. As a result, there is a large and diverse body of literature concerning the properties and development of models for complex networks. However, many of the models that have been previously developed, although quite successful at capturing many observed properties of complex networks, have failed to capture …