In this thesis we model the diffusion of information on social networks. A game played on a specific type of graph generator, the iterated local transitivity model, is examined. We study how the dynamics of the game change as the graph grows, and the relationship between properties of the game on a graph initially and properties of the game later in the graph’s development. We show that, given certain conditions, for the iterated local transitivity model it is possible to predict the existence of a Nash equilibrium at any point in the graph’s growth. We give sufficient conditions for the existence of Nash Equilibria on star graphs, cliques and trees. We give some results on potential games on the iterated local transitivity model. Chapter 2 provides an introduction to graph properties, and describes various early graph models. Chapter 3 describes some models for online social networks, and introduces the iterated local transitivity model which we use later in the thesis. In Chapter 4 various models for games played on networks are examined. We study a model for competitive information diffusion on star graphs, cliques and trees, and we provide conditions for the existence of Nash Equilibria on these. This model for competitive information diffusion is studied in detail for the iterated local transitivity model in Chapter 5. We discuss potential games in Chapter 6 and their existence on the iterated local transitivity model. We conclude with some suggestions on how to extend and develop upon the work done in this thesis.