Subtractive random forests

TL;DR

This work introduces the subtractive random forest (SuRF), a random-attachment model on the integers where the positive indices attach to ancestors at distance $Z_n$, producing a forest of rooted trees. The authors prove a sharp survival-extinction dichotomy governed by the tail of $Z$: if $\mathbb{E}[Z] < \infty$ there is a unique infinite tree and the total size of finite trees is finite (finite in expectation when $\mathbb{E}[Z^2] < \infty$); if $\mathbb{E}[Z] = \infty$ all trees are almost surely finite. They provide detailed analyses of expected tree sizes, the number of trees, leaves, degrees, and the height, including subadditivity properties, CLTs for depth-1 counts, tightness/maximum degree bounds, and height bounds that reveal sublinear growth in the finite-mean regime. The results connect to online recommendation dynamics by showing how long-run behavior collapses to either a single dominant tree or complete extinction depending on the attachment distribution. Overall, the paper delivers a comprehensive probabilistic characterization of SuRF’s global structure and growth dynamics.

Abstract

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$ are attached sequentially such that the parent of vertex $n$ is $n-Z_n$, where the $Z_n$ are i.i.d.\ random variables taking values in $\mathbb N$. We study several characteristics of the resulting random forest. In particular, we establish bounds for the expected tree sizes, the number of trees in the forest, the number of leaves, the maximum degree, and the height of the forest. We show that for all distributions of the $Z_n$, the forest contains at most one infinite tree, almost surely. If ${\mathbb E} Z_n < \infty$, then there is a unique infinite tree and the total size of the remaining trees is finite, with finite expected value if ${\mathbb E}Z_n^2 < \infty$. If ${\mathbb E} Z_n = \infty$ then almost surely all trees are finite.

Figures (1)

• Figure 1: An example of the subtractive attachment process (up to time $5$) and the resulting forest. Here $Z_1=3$, $Z_2=2$, $Z_3=1$, $Z_4=6$, and $Z_5=3$.

Theorems & Definitions (29)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Proposition 1
• proof
• Remark 1
• Remark 2