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- Wednesday, 13.10.2021, 11:30 (Online Event)
- Seminar Interacting Random Systems
Tejas Iyer, WIAS Berlin:
Degrees of fixed vertices and power law degree distributions in preferential attachment trees with neighbourhood influence (online talk)
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Motivated by the structure of complex networks such as the internet, we consider a growing model of preferential attachment trees with neighbourhood influence, where vertices arrive one at a time, are equipped with independent weights, and connect to existing vertices with probability proportional to their fitness function: a function of their own weight and the weights of their neighbours. In this model we prove almost sure limiting statements for the proportion of vertices with a given degree having weight belonging to a given measurable set, and the proportion of edges in the tree with endpoint belonging to a measurable set. We show that under certain conditions, the latter quantity demonstrates a condensation phenomenon, in which a positive proportion of edges in the network accumulate among those of weight that confers maximal reinforcement of fitness. Finally, we derive almost sure limiting statements for the log of the degree of a fixed vertex in this model, and prove that in this model the degree distribution behaves like a power law - a ubiquitous feature of many real-world complex networks.
Seminar Interacting Random Systems (Online Event)