Partial orders and monotonicity of logarithmic depth and height in preferential attachment trees

TL;DR

This paper investigates how the structure of preferential attachment trees changes with different attachment functions, focusing on the monotonicity of depth and height, and introduces conditions under which these properties hold.

Contribution

The paper identifies conditions under which the monotonicity of logarithmic depth and height in PA trees is guaranteed, extending understanding beyond growth-ratio dominance.

Findings

Counterexample showing growth-ratio dominance is insufficient for monotonicity

Profile-order assumptions ensure monotonicity of depth and height

Explicit conditions relate attachment functions to tree height and depth

Abstract

We study preferential attachment (PA) trees with general attachment functions. PA suggests an intuitive monotonicity: if high-degree vertices are rewarded more strongly, then the resulting tree should become shallower. We examine this principle through the constants governing two natural logarithmically growing observables, the insertion depth of the newest vertex and the height of the whole tree. Growth-ratio dominance (GRD) is the natural order on attachment functions, but we provide an explicit counterexample showing that GRD is not sufficient for either depth or height monotonicity at the level of logarithmic constants. The missing input is a dual tail-order condition on certain measures associated with the CMJ/BRW embedding of the PA tree. Under these profile-order assumptions we prove the expected monotonicity results.