A Cluster Model for Growth of Random Trees

Abstract

We first consider the growth of trees by probabilistic attachment of new vertices to leaves. This leads to a growth model based on vertex clusters and probabilities assigned to clusters. This model turns out to be readily applicable to attachment at any depth of the tree, hence the paper evolves to a general study of tree growth by cluster-based attachment. Drawing inspiration from the concept of intrinsic vertex fitness due to Bianconi and Barab\'asi, we introduce vertex mass as an additive intrinsic vertex attribute. Unlike Bianconi and Barab\'asi who used fitness as a vertex degree multiplier in the context of growth by preferential attachment, we treat vertex mass as a fundamental probabilistic construct whose additivity plays a primary role. Notably, independent mass distributions induce a distribution on the sum of such masses through Laplace convolution. In this way, clusters of vertices inherit their mass distributions from vertices within the cluster. Our main contribution is a novel theorem for the joint distribution of cluster masses, conditioned on their respective distributions. As described by Ferguson and Kingman in the context of distributions on general measures, the choice of gamma conditioning distributions leads to the Dirichlet distribution. Beyond gamma conditioning distributions, our theorem allows other choices, such as the fat-tailed stable distributions with infinite mean. We discuss L\'evy conditioning distributions as a gamma alternative, the L\'evy distribution being a notable instance of the stable family. We conclude with a theorem giving the analytic marginals of the normalised distribution conditioned on the L\'evy distribution.