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Preferential attachment networks were popularised by Barabási and Albert in the late 1990s as models for complex systems and real-world networks; examples suggested included social networks, the World Wide Web and biological networks. The key principle is that the probability that a vertex's probability of acquiring new connections increases with its existing connectivity, something which is a natural assumption when modelling many systems.
One of the first probabilistic results obtained on preferential attachment was that, as predicted by Barabási and Albert, the basic preferential attachment rule produces a distribution of vertex degrees which asymptotically follows a power law, something which has been claimed for many real-world systems. There has also been much interest in the structural properties of these graphs, for example their diameter and local structure.
Further developments have considered modifications of the basic model. Examples of such modifications include the introduction of vertex fitnesses, non-linear relationships between connectivity and ability to attract new connections, geometric embeddings of the model, and versions which allow for vertices to be removed from the network as well as added. These developments are of interest for better modelling of some of the real-world systems which motivated the original work, but they also produce much interesting mathematics.
Collection created by Dr Jonathan Jordan (University of Sheffield)