Markov branching in the vertex splitting model

作者：Stefansson Sigurdur Orn^*

来源：Journal of Statistical Mechanics: Theory and Experiment , 2012, P04018.

DOI：10.1088/1742-5468/2012/04/P04018

• rigorous results in statistical mechanics
• growth processes
• random graphs
• networks

We study a special case of the vertex splitting model which is a recent model of randomly growing trees. For any finite maximum vertex degree D, we find a one parameter model, with parameter alpha is an element of [0, 1] which has a so-called Markov branching property. When D = infinity we find a two parameter model with an additional parameter gamma is an element of [0, 1] which also has this feature. In the case D = 3, the model bears resemblance to Ford's alpha-model of phylogenetic trees and when D = infinity it is similar to its generalization, the alpha gamma-model. For alpha = 0, the model reduces to the well known model of preferential attachment.
In the case alpha > 0, we prove convergence of the finite volume probability measures, generated by the growth rules, to a measure on infinite trees which is concentrated on the set of trees with a single spine. We show that the annealed Hausdorff dimension with respect to the infinite volume measure is 1/alpha. When gamma = 0 the model reduces to a model of growing caterpillar graphs in which case we prove that the Hausdorff dimension is almost surely 1/alpha and that the spectral dimension is almost surely 2/(1 + alpha). We comment briefly on the distribution of vertex degrees and correlations between degrees of neighbouring vertices.

• 出版日期2012-4