Spectral Analysis of Randomly Generated Networks With Prescribed Degree Sequences

Network science attempts to capture real-world phenomenon through mathematical models. The underlying model of a network relies on a mathematical structure called a graph. Having seen its early beginnings in the 1950\u27s, the field has seen a surge of interest over the last two decades, attracting interest from a range of scientists including computer scientists, sociologists, biologists, physicists, and mathematicians. The field requires a delicate interplay between real-world modeling and theory, as it must develop accurate probabilistic models and then study these models from a mathematical perspective. In my thesis, we undertake a project involving computer programming in which we generate random network samples with fixed degree sequences and then record properties of these samples. We begin with a real-world network, from which we extract a sample of at least one-hundred vertices through the use of snowball sampling. We record the degree sequence, D, of this sample and then generate random models with this same degree sequence. To generate these models, we use a well-known graph algorithm, the Havel-Hakimi algorithm, to produce an initial non-random sample G_D. We then run a Monte Carlo Markov Chain (MCMC) on the sample space of graphs with this degree sequence beginning at G_D in order to produce a random graph in this space. Denote this random graph by H_D. Lastly, we compute the eigenvalues of the Laplacian matrix of H_D, as these eigenvalues are intricately connected with the structure of the graph. In doing this, we intend to capture local properties of the network captured by the degree sequence alone. The programming of this project is done in Python and Matlab