We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a (sub-)linear function of the indegree of the older vertex at that time. We provide a limit theorem with rates of convergence for the distribution of a vertex, chosen uniformly at random, as the number of vertices tends to infinity. To do so, we develop Stein's method for a new class of limting distributions including power-laws. Similar, but slightly weaker results are shown to be deducible using coupling techniques. Concentrating on a specific preferential attachment model we also show that the outdegree distribution asymptotically follows a Poisson law. In addition, we deduce a central limit theorem for the number of isolated vertices. We thereto construct a size-bias coupling which in combination with Stein’s method also yields bounds on the distributional distance.
| Identifer | oai:union.ndltd.org:uni-osnabrueck.de/oai:repositorium.ub.uni-osnabrueck.de:urn:nbn:de:gbv:700-201905171547 |
| Date | 17 May 2019 |
| Creators | Betken, Carina |
| Contributors | Prof. Dr. Hanna Döring, Prof. Dr. Adrian Röllin |
| Source Sets | Universität Osnabrück |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf, application/zip |
| Rights | Attribution 3.0 Germany, http://creativecommons.org/licenses/by/3.0/de/ |
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