Random (di)(multi)graphs and networks have been studied by mathematicians at least since the famous work of Erdős & Rényi of 1959. Their model has been studied from two main angles. Firstly, people have analysed the resulting finite structures, looking at their degree distributions and other properties. Secondly, a remarkable theorem establishes that there is a unique infinite structure which any Erdős-Rényi process approaches with probability one. Model theorists know this object simply as "the random graph".
Recently, network scientists looking to model real-world structures such as the world wide web have turned away from Erdős-Rényi processes and looked to other models. In particular, the "preferential attachment" paradigm of Barabási and Albert in 1999 has been intensively studied in recent years, and is a good fit for many real-world networks. However, little work has been done on the infinite limits of such processes. I will report on some recent progress on this question.