Monotonicity of depth constants in general preferential attachment trees

Abstract

We study insertion depth and tree height in preferential-attachment (PA) trees driven by a general attachment function $f:\mathbb{N}_0\to(0,\infty)$. Our main result is a monotonicity principle for the $\log n$-order constant of either functional: if $f$ dominates $g$ in ``growth-ratio'' order (a variant of the hazard ratio order of probability densities) then the typical depth constant for $f$ is no larger than for $g$, i.e. stronger reinforcement yields a shallower tree. The proofs use a one-parameter interpolation together with a gauge normalisation that removes a global derivative term, reducing the sign to a weighted correlation inequality for monotone sequences. Equivalently, the results can be phrased as monotonicity theorems for the corresponding branching process functionals arising from the standard embedding of the PA tree into a Crump--Mode--Jagers process.