On the depth of depth-weighted trees

TL;DR

The paper analyzes the depth of depth-weighted trees where each new vertex attaches to a prior vertex with probability proportional to a weight function of that vertex's depth, covering convergent, periodic, slowly growing, exponential, and super-exponential regimes. It develops a unifying framework based on two continuous-time representations—a Crump–Mode–Jagers–type branching process and a Cox-process profile model—to translate the discrete growth into tractable continuous dynamics, allowing precise depth asymptotics through stopping-time analysis and concentration techniques. Key results include a.s. depth $d(T_n)$ scaling as $\Theta(\log n)$ for boundedly varying $f$, a phase transition to linear height for exponential growth with $c>1$, and a sharp, $I_n$-driven description in the super-exponential regime where all but a finite number of vertices lie on a single infinite path. These findings validate and strengthen prior conjectures, clarify phase transitions in depth behavior, and provide rigorous, versatile tools for analyzing hierarchical network growth models with depth-based attraction.

Abstract

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability proportional to $f$(distance between $u$ and $r$). This work is dedicated to a systematic analysis of the depth of DWT($f$). Namely, we provide precise analytic expressions of the typical depth of DWT($f$) for convergent, periodic, slowly growing, and (super-)exponentially growing weight functions. Furthermore, for bounded or exponentially growing $f$, we determine the typical depth up to a multiplicative constant, thus confirming and strengthening a conjecture of Leckey, Mitsche and Wormald.

Figures (9)

• Figure 1: Simulations of DWT($f$) for different weight functions $f$. The big red dot denotes the root and the blue paths realise the depth of the tree, that is, the length of the longest path from the root to a leaf.
• Figure 2: A numerical approximation of the constant $\beta$ predicted in Question \ref{['Q:subexp']} (in black) for the tree with weight function $f(k)={\mathrm e}^{ck/\log(k+2)}$ and $n=20.000$. Here, $c$ ranges between $0.1$ and $2.5$ with increments of $0.1$, where we take an average over $50$ samples. The red line serves as a comparison.
• Figure 3: A log-log plot of the growth of the depth of the tree with weight function $f(k)={\mathrm e}^{ck/\log(k+2)}$. We have taken an average over 50 samples for each value of $c$.
• Figure 4: A numerical approximation of the predicted constant $\nu$ in Conjecture \ref{['conj:exp']}. The figure shows the average value of $d(T_n)/n$ for $n=15.000$ and weight function $f(k)=c^k$. The plot is obtained by taking an average over $50$ independent samples for each $c$ between $1.1$ and $15$ with increments of $0.1$.
• Figure 5: The construction of the times $\beta_{m,k}$. In the first iteration, the process runs for $\beta_{1,k}$ units of time, at which point the tree (in red) reaches depth $k$. In the next iteration, we let the whole tree grow for $\beta_{2,k}$ units of time, at which point the subtree (in blue) reaches depth $k$. Notice that by time $\beta_{2,k}$ the whole tree may have already reached a greater depth.

Theorems & Definitions (64)

• Definition 1.1: Depth-weighted tree (DWT)
• Theorem 1.2: Theorem 2.3 in LeckMitWor20
• Proposition 1.3
• Theorem 1.4
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Remark 1.10
• Conjecture 1.12