Scale-free random branching tree in supercritical phase

Abstract

We study the size and the lifetime distributions of scale-free random branching tree in which $k$ branches are generated from a node at each time step with probability $q_k\sim k^{-\gamma }$. In particular, we focus on finite-size trees in a supercritical phase, where the mean branching number $C={\sum }_{k}k{q}_k$ is larger than 1. The tree-size distribution $p(s)$ exhibits a crossover behavior when $2<\gamma <3$; A characteristic tree size $s_c$ exists such that for $s\ll s_c$, $p(s)\sim s^{-\gamma /(\gamma -1)}$ and for $s\gg s_c$, $p(s)\sim s^{-3/2}\exp (-s/s_c)$, where $s_c$ scales as $\sim (C-1{)}^{-(\gamma -1)/(\gamma -2)}$. For $\gamma >3$, it follows the conventional mean-field solution, $p(s)\sim s^{-3/2}\exp (-s/s_c)$ with $s_c\sim (C-1{)}^{-2}$. The lifetime distribution is also derived. It behaves as $\ell (t)\sim t^{-(\gamma -1)/(\gamma -2)}$ for $2<\gamma <3$, and $\sim t^{-2}$ for $\gamma >3$ when branching step $t\ll t_c\sim (C-1{)}^{-1}$, and $\ell (t)\sim \exp (-t/t_c)$ for all $\gamma >2$ when $t\gg t_c$. The analytic solutions are corroborated by numerical results.

6 pages, 6 figures

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