This paper presents a detailed theory of random graphs with arbitrary degree distributions, including directed and bipartite graphs. It addresses the structure of social networks and the internet, where degree distributions differ significantly from the Poisson distribution typically studied in random graphs. The authors derive exact expressions for the phase transition at which a giant component first forms, the mean component size, the size of the giant component, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. They apply their theory to real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 500 company directors, showing that in some cases random graphs with appropriate degree distributions predict real-world behavior accurately, while in others there is a measurable discrepancy, suggesting the presence of additional social structure. The paper also extends the theory to directed and bipartite graphs, introducing generating functions to calculate statistical properties of these graph types. It discusses the asymptotic form of the cluster size distribution, the mean component size, the phase transition, and the giant component. The authors also calculate the average path length between vertices and show that it scales logarithmically with the size of the graph. The paper concludes with simulation results confirming the theoretical predictions for the size of the giant component in graphs with various degree distributions.This paper presents a detailed theory of random graphs with arbitrary degree distributions, including directed and bipartite graphs. It addresses the structure of social networks and the internet, where degree distributions differ significantly from the Poisson distribution typically studied in random graphs. The authors derive exact expressions for the phase transition at which a giant component first forms, the mean component size, the size of the giant component, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. They apply their theory to real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 500 company directors, showing that in some cases random graphs with appropriate degree distributions predict real-world behavior accurately, while in others there is a measurable discrepancy, suggesting the presence of additional social structure. The paper also extends the theory to directed and bipartite graphs, introducing generating functions to calculate statistical properties of these graph types. It discusses the asymptotic form of the cluster size distribution, the mean component size, the phase transition, and the giant component. The authors also calculate the average path length between vertices and show that it scales logarithmically with the size of the graph. The paper concludes with simulation results confirming the theoretical predictions for the size of the giant component in graphs with various degree distributions.