Scale-free random branching trees in supercritical phase

Abstract

We study the size and the lifetime distributions of scale-free random branching trees in which k branches are generated from a node at each time step with probability qk ∼ k−γ. In particular, we focus on finite-size trees in a supercritical phase, where the mean branching number C = ∑kkqk is larger than 1. The tree-size distribution p(s) exhibits a crossover behaviour when 2 < γ < 3. A characteristic tree size sc exists such that for s ≪ sc, p(s) ∼ s−γ/(γ−1) and for s ≫ sc, p(s) ∼ s−3/2exp(−s/sc), where sc scales as ∼(C − 1)−(γ−1)/(γ−2). For γ > 3, it follows the conventional mean-field solution, p(s) ∼ s−3/2exp(−s/sc) with sc ∼ (C − 1)−2. The lifetime distribution is also derived. It behaves as ℓ(t) ∼ t−(γ−1)/(γ−2) for 2 < γ < 3, and ∼t−2 for γ > 3 when branching step t ≪ tc ∼ (C − 1)−1, and ℓ(t) ∼ exp(−t/tc) for all γ > 2 when t ≫ tc. The analytic solutions are corroborated by numerical results.