Moments of general time dependent branching processes with applications

TL;DR

This paper establishes conditions for boundedness in certain moments of time-dependent branching processes and applies these results to analyze the growth of maximal degree in a related random graph model.

Contribution

It provides new sufficient conditions for $L_k$ boundedness in Crump-Mode-Jagers processes and applies these to understand degree growth in a time-dependent random graph.

Findings

Maximal degree grows at the same rate as the degree of a fixed vertex.

Established $L_k$ boundedness conditions for time-dependent branching processes.

Applied theoretical results to a graph process motivated by collaborations.

Abstract

In this paper, we give sufficient conditions for a Crump-Mode-Jagers process to be bounded in $L_k$ for a given $k>1$. This result is then applied to a recent random graph process motivated by pairwise collaborations and driven by time-dependent branching dynamics. We show that the maximal degree has the same rate of increase as the degree process of a fixed vertex.