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Advances in Urn Models and Applications to Self-similar Bipolar Networks Open Access

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We study a class of tenable Polya urn schemes and processes. We concentrate on balanced urn models, and characterize their properties. As an application, we employ these findings in the study random self-similar bipolar networks.Chapter 1 introduces the history of urns models and graph theory. To conform with the aim and scope of the dissertation, we focus on the evolution of Polya urn models and random networks. Chapter 2 gives some key techniques and theories that we frequently use in this study, such as partial differential equations (for urns embedded in real time) and analytic methods. Chapter 3 considers balanced triangular urn schemes with both deterministic and random flavor. We characterize the exact moments of balanced triangular urn schemes via an elementary approach, and obtain the distribution by analytic method. Chapter 4 is about Poissonized triangular (reducible) urns on two colors. We analyze the number of balls of different colors in the urn after a certain period of time has elapsed. We show that for balanced processes in this class, a different scaling is needed for each color to produce a nontrivial limit, contrary to the distributions in the usual irreducible urns which only require the same scaling for both colors. For the dominant color, we get exact moments, while relaxing the balance condition. We also consider the balanced triangular processes with random replacement matrices. Chapter 5 gives a novel approach to studying Poissonized tenable and balanced urns on two colors. We demonstrate our method via a process obtained by embedding a generalized Polya-Eggenberger urn (the Bagchi-Pal urn) into continuous time. We analyze the number of balls of different colors in the urn after a certain period of time. The asymptotic mixed moments of the process are computed. We show that the limiting distributions of the scaled variables underlying the Bagchi-Pal urn are gamma. We analyze the process obtained by embedding randomized Play-the-Winner scheme. Chapter 6 investigate several aspects of a self-similar evolutionary process that builds a random bipolar network from building blocks that are themselves small bipolar networks. We characterize admissible outdegrees in the history of the evolution. We obtain the limit distribution of the polar degrees (when suitably scaled) characterized by its sequence of moments. We also obtain the asymptotic joint multivariate normal distribution of the number of nodes of small admissible outdegrees. Five possible substructures arise, and each has its own parameters (mean vector and covariance matrix) in the multivariate distribution. Chapter 7 is about proposed future work on Polya schemes and processes.

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