Randomly evolving trees II

TL;DR

This paper derives generating functions and analyzes stochastic properties of randomly evolving trees, revealing how end-node variance and survival probability behave differently across subcritical, critical, and supercritical regimes.

Contribution

It introduces new analytical expressions for the probability distribution and survival probability of evolving trees, enhancing understanding of their dynamic behavior.

Findings

Relative variance of end-nodes peaks at critical points.

Survival probability decreases as 1/x in critical evolution.

Fluctuations in tree lifetime become large near criticality.

Abstract

Generating function equation has been derived for the probability distribution of the number of nodes with $k \ge 0$ outgoing lines in randomly evolving special trees. The stochastic properties of end-nodes (k=0) have been analyzed, and it was shown that the relative variance of the number of end-nodes vs. time has a maximum when the evolution is either subcritical or supercritical. On the contrary, the time dependence of the relative dispersion of the number of dead end-nodes shows a minimum at the beginning of the evolution independently of its type. For the sake of better understanding of the evolution dynamics the survival probability of random trees has been investigated, and asymptotic expressions have been derived for this probability in the cases of subcritical, critical and supercritical evolutions. In critical evolution it was shown that the probability to find the tree lifetime larger than x, is decreasing to zero as 1/x, if x tends to infinity. Approaching the critical state it has been found the fluctuations of the tree lifetime to become extremely large, and so near the critical state the average lifetime could be hardly used for the characterization of the process.