We consider several families of random graphs that grow in time by the addition of vertices and edges in some 'uniform' manner These families are natural starting points for modelling real-world networks that grow in time. Recently, it has been shown (heuristically and rigorously) that such models undergo an 'infinite-order phase transition': as the density parameter increases above a certain critical value, a 'giant component' emerges, but the speed of this emergence is extremely slow. In this paper we shall present some of these results and investigate the connection between the existence of a giant component and the connectedness of the final infinite graph.
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