QUANTIFYING UNCERTAINTY IN RANDOM ALGEBRAIC OBJECTS USING DISCRETE METHODS
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This thesis consists of two parts. Part 1 is concerned with the study of random algebraic objects and Part 2 deals with statistical modeling for networks. Part 1 begins with the study of random monomial ideals. We define several models for generating random monomial ideals, illustrate their connection with models of random simplicial complexes, and study the behavior of various algebraic invariants of interest (e.g., Krull dimension and first Betti numbers) in the ER-type model. Next we consider a model for random numerical semigroups. In order to understand their properties, we introduce a family of simplicial complexes whose algebraic and combinatorial properties encode probabilistic information about random semigroups from the model. In Part 2, we introduce two exponential random graph models. The first is the shell distribution model. The sufficient statistics of this model are related to the k-cores of a network, which is a graph-theoretic concept designed to capture connectivity information in a more refined way than node degrees. We study the theoretical properties of the shell distribution model, develop an MCMC algorithm for sampling from the model, give an algorithm for sampling from the space of graphs with a fixed shell distribution, and present several simulation studies. The second model is the edge-degeneracy model, whose sufficient statistics are related to the density of edges in the graph. For this model, we prove several theoretical results concerning the model polytope and how it governs the asymptotic behavior of the model as the parameters diverge along infinite rays.