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Random tree growth by vertex splitting

David, François and Dukes, Mark and Jónsson, Thordur and Stefánsson, Sigurdur Örn (2009) Random tree growth by vertex splitting. Journal of Statistical Mechanics: Theory and Experiment, 2009 (4). ISSN 1742-5468

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Abstract

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalizes the preferential attachment model and Ford's α-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from 1 to ∞, depending on the parameters of the model.