Anomalous scaling in an age-dependent branching model

TL;DR

This paper introduces a family of age-dependent branching models where the probability of branching decreases with age, revealing a transition in tree depth scaling that aligns with biological evolution patterns.

Contribution

It presents a novel one-parametric model showing a transition in tree depth scaling, linking age-dependent branching to critical phenomena in biological evolution.

Findings

Tree depth scales logarithmically or algebraically depending on the parameter.

At the critical point, tree depth scales as (log n)^2.

Model aligns with observed biological evolution trends.

Abstract

We introduce a one-parametric family of tree growth models, in which branching probabilities decrease with branch age $\tau$ as $\tau^{-\alpha}$. Depending on the exponent $\alpha$, the scaling of tree depth with tree size $n$ displays a transition between the logarithmic scaling of random trees and an algebraic growth. At the transition ($\alpha=1$) tree depth grows as $(\log n)^2$. This anomalous scaling is in good agreement with the trend observed in evolution of biological species, thus providing a theoretical support for age-dependent speciation and associating it to the occurrence of a critical point.