2021/06/12 by Nomvelo Sibisi, Sibisi, Nomvelo
Agricultural and Biological Sciences · Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Plant and animal studies #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2106.06779
openalex publication_date 2021/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Barabási, we introduce vertex mass as an additive intrinsic vertex attribute. Unlike Bianconi and Barabási 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 Lévy conditioning distributions as a gamma alternative, the Lévy 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 Lévy distribution.