Randomly evolving trees I

TL;DR

This paper introduces a new model of random tree evolution using living and dead nodes, analyzing different evolution regimes and deriving key statistical properties of the process.

Contribution

It develops a continuous-time model of random tree evolution with a novel integral equation for the generating function and analyzes the behavior of node counts over time.

Findings

In subcritical evolution, variance peaks at a specific time.

The joint distribution of living and dead nodes is derived.

In critical evolution, the correlation converges to a fixed value.

Abstract

By introducing the notions of living and dead nodes a new model of random tree evolution with continuous time parameter has been constructed. It is assumed that two random variables, the lifetime and the offspring number of living nodes control the evolution process. It has been shown that the generating function of the probability to find the evolving tree in a given state satisfies a non-linear integral equation. Analyzing the time dependence of the average number of living nodes three different types of evolution (subcritical, critical and supercritical) can be observed. It has been proved that in the case of subcritical evolution there is a well-defined time point at which the variance of the number of living nodes has a maximum. The joint distribution function of the numbers of living and dead nodes has been derived, and the time dependence of the correlation between these node numbers has been calculated. It is found that in the critical evolution with increasing time the correlation converges to a fixed number which is independent of the distributions of the lifetime and the offspring number of living nodes.