Transport on randomly evolving trees

TL;DR

This paper models the dynamic process of transport on evolving random trees, analyzing node states, correlations, and survival probabilities over time using age-dependent branching processes.

Contribution

It introduces a new model of random tree evolution with living and dead nodes, deriving joint distributions and correlations, and analyzing end-node properties and survival probabilities.

Findings

Correlation between living and dead nodes changes abruptly at early stages

Derived explicit expressions for survival probability of trees

Analyzed stochastic properties of end-nodes over time

Abstract

The time process of transport on randomly evolving trees is investigated. By introducing the notions of living and dead nodes a model of random tree evolution is constructed which describes the spreading in time of objects corresponding to nodes. By using the method of the age-dependent branching processes we derive the joint distribution function of the number of living and dead nodes, and determine the correlation between these node numbers as a function of time. Also analyzed are the stochastic properties of the end-nodes; and the correlation between the numbers of living and dead end-nodes is shown to change its character suddenly at the very beginning of the evolution process. The survival probability of random trees is investigated and expressions are derived for this probability.