Mass distribution exponents for growing trees

TL;DR

This paper analyzes the asymptotic mass distribution in growing trees with attachment probabilities based on vertex valence, deriving explicit scaling exponents and identifying different regimes, supported by exact solutions and simulations.

Contribution

It introduces explicit expressions for mass distribution exponents in growing trees with valence-dependent attachment, revealing multiple regimes and broad distributions.

Findings

Mass distribution is broad with multiple regimes.

Explicit scaling exponents are derived for different regimes.

Results are supported by exact solutions and numerical simulations.

Abstract

We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that measures the repartition of the mass of large trees between their different subtrees. This distribution is shown to be a broad distribution and we derive explicit expressions for scaling exponents that characterize its behavior when one subtree is much smaller than the others. We show in particular the existence of various regimes with different values of these mass distribution exponents. Our results are corroborated by a number of exact solutions for particular solvable cases, as well as by numerical simulations.