Random graphs with correlation structure
In this thesis we consider models of random graphs where, unlike in the classical models G (n, p) the probability of an edge arising can be correlated with that of other edges arising. Attention focuses on graphs whose vertices are each assigned a colour (type) at random and where edges between differently coloured vertices subsequently arise with different probabilities (so-called RRC graphs), especially the special case with two colours. Various properties of these graphs are considered, often by comparing and contrasting them with the classical model with the same probability of each particular edge existing. Topics examined include the probabilities of trees and cycles, how the joint probability of two subgraphs compares with the product of their probabilities, the number of edges in the graph (including large deviations results), connectedness, connectivity, the number and order of complete graphs and cliques, and tournaments with correlation structure.