Bipartite and directed random graphs with arbitrary degree distributions

This work aims to study some mathematical properties of random graphs with arbitrary vertex degree distributions through the theory of generating functions. The investigated results include average component sizes, average numbers of vertices a certain distance away from a randomly chosen one and average distances between vertices, together with the analysis of the point at which giant components appear. After having provided examples with different vertex probability distributions, we give mathematical proof of some results for unipartite undirected graphs. Subsequently, we move to analyze bipartite graphs and introduce the network of directors of companies belonging to the Fortune 1000 ranking. Eventually, directed graphs are considered, and the representative example of the World Wide Web is given. In both these two cases of networks we present some numerical experiments to compare the theoretical results to the ones coming from known databases.