Stochastic Networks and Processes
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Empirical observations show that many real-world networks share similar features. They are small worlds, in the sense that typical distances between two parts of a network are much smaller than the size of the entire network. Moreover, many real-world networks are scale-free, in the sense that there is a high variability in the number of connections of the elements of the networks. Examples include social networks, populations and computer networks.
Various stochastic models have been proposed for such networks and have been shown to model them well. In this course, we discuss empirical findings of real-world networks, and describe some of the random graph models proposed for them.
We can also consider the functionality of many real-world processes of great practical importance and interest as stochastic processes on random graphs. Examples of such stochastic processes include the spread of an epidemic through population, and information diffusion on social networks. In this course we will study limit results and approximations for such doubly random processes. We will in particular study such diffusions on branching processes and provide a rigorous justification for the fact that their speed is exponential.
The course will begin with a review of some probabilistic methods required for the analysis of stochastic networks and stochastic processes evolving on them. We then define and analyse discrete- and continuous-time branching processes. In the second half of the course, we move to the models of random graphs. We will study Erdos-Renyi graphs, generalised random graphs, configuration models and preferential-attachment models.