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.