Research Challenge: Tree correlation detection threshold via order statistic
Source paper: Diffusion-Network Alignment: An Efficient Algorithm and Explicit Probability Bounds
Reference: arXiv:2606.12879 — Theorem tree.test
Task: Prove the theorem stated below. This document is self-contained: all definitions and notation needed to understand the statement are included. The proof is deliberately omitted.
Overview
For correlated Galton-Watson-type trees, once the parameters exceed Otter's-constant thresholds, a likelihood-based test statistic grows super-exponentially with depth with probability approaching the tree's survival probability, enabling detection of the correlation.
Definitions and setup
The following definitions and notation (quoted from the source paper) are referenced by the theorem.
correlated tree pair distribution (— Section 3 (Preliminary), 'Independent and correlated Galton–Watson tree distributions')
Correlated tree pair distribution $P^{(\lambda,s,\td{s})}d$: Under this distribution, two correlated trees $\ct$ and $\td{\ct}$ with marginal distributions $\gw{d,\lambda s}$ and $\gw_{d,\lambda\td{s}}$ are generated. To generate the two correlated trees, an intersection tree $\ct^*$ is first sampled from distribution $\gw_{d,\lambda s\td{s}}$. Then for each $k\in{0,\ldots,d-1}$, we independently sample a random variable $X^+v\sim\poi(\lambda s(1-\td{s}))$ for each vertex $v$ at depth $k$ in $\ct^*$, and add $X_v$ new children to $v$. For each newly added vertex $u$ at depth $k$, a rooted tree $t_u$ is independently sampled from the distribution $\gw{d-k,\lambda s}$ and attached to $u$ as a subtree. After attaching $t_u$ for each new vertex $u$, the resulting rooted tree is $\ct$. To obtain tree $\td{\ct}$, we again start with the intersection tree $\ct^*$, and independently repeat the process of adding new children and attaching new subtrees, but instead with $X_v^+\sim\poi(\lambda \td s(1-s))$ and $t_u\sim\gw_{d-k,\lambda \td{s}}$.
parameter choices l and \dmax (— Section 4.1 (Proposed algorithm))
In the analysis of the proposed algorithm, we choose parameters $l=\lfloor\sqrt{\log n}\rfloor$ and $\dmax=\left\lfloor\frac{1-\epsilon}{\log(\lambda q)}\log n\right\rfloor$, where $\epsilon>0$ is an arbitrarily small constant.
sequence (p_d)_{d\ge 0} (— Section 4.2 (Main results), eqs. (2)–(3))
We recursively define the sequence $(p_d){d\ge 0}$ as follows: First define $p_0=\P(X\ge 3)$, where $X\sim\poi(\lambda sq(1-p^ext{\lambda sq}-\eps'))$. Then for each $d\ge 1$, define $p_{d}=\P(2Y+Z\ge 3)$, where $Y\sim\poi(\lambda sq p_{d-1})$, $Z\sim\poi(\lambda sq (1-p^ext_{\lambda sq}-\eps'-p_{d-1}))$, and $Y$ and $Z$ are independent.
Otter's tree-counting constant (— Theorem (thm:guarantee), Section 4.2)
Let $\alpha\approx0.338$ denote Otter's tree-counting constant.
likelihood ratio for tree correlation tests (— Section 3 (Preliminary), 'Likelihood ratio test', eq. (1))
In the tree correlation test problem, we are given two rooted trees $\ct$ and $\tdct$, and the goal is to distinguish the two hypotheses $H_0: (\ct,\tdct)\sim Q^{(\lambda,s,\td{s})}_d$ and $H_1: (\ct,\tdct)\sim P^{(\lambda,s,\td{s})}d$. ... For the two trees $\ct$ and $\tdct$, their likelihood ratio is given by $L{d}^{(\lambda,s,\td{s})}(\ct,\tdct):=\frac{P^{(\lambda,s,\td{s})}_d(\ct,\tdct)}{Q^{(\lambda,s,\td{s})}_d(\ct,\tdct)}.$
extinction probability (— Section 3 (Preliminary), 'Poisson Galton–Watson trees')
We define the extinction probability $p^{ext}{d,\lambda}$ as the probability that there exist no vertices at distance $d$ from the root under the law $\gw_\lambda$, and define $p^{ext}\lambda:=\lim{d\rightarrow\infty}p^{ext}{d,\lambda}$.
𝒢𝒲_{d,λ} (depth-d truncated Poisson Galton–Watson tree) (— Section 3 (Preliminary), 'Poisson Galton–Watson trees')
For each $d\ge 0$, we use $\gw_{d,\lambda}$ to denote the depth-$d$ truncation of $\gw_\lambda$, which is the distribution over the rooted trees obtained by generating a tree from $\gw_\lambda$ and remove all the vertices with distance greater than $d$ from the root.
𝒢𝒲_λ (Poisson Galton–Watson branching tree) (— Section 3 (Preliminary), 'Poisson Galton–Watson trees')
The Galton--Watson branching tree with offspring distribution $\poi(\lambda)$, denoted $\gw_{\lambda}$, is a distribution over unlabeled rooted trees. The tree has a distinguished vertex as its root. Then iteratively for each vertex $v$ in the tree, an independent random variable $X_v\sim\poi(\lambda)$ is sampled, representing the number of children attached to $v$.
Poi(μ) (Poisson distribution) (— Appendix, 'Notation Used in the Appendices')
We use $\poi(\mu)$ to denote the Poisson distribution with mean $\mu$
Q_d^{(λ,s,s̃)} (independent tree pair distribution) (— Section 3 (Preliminary), 'Independent and correlated Galton–Watson tree distributions')
Independent tree pair distribution $Q^{(\lambda,s,\td{s})}d$: Under this distribution, two unlabeled rooted trees $\ct$ and $\td{\ct}$ are sampled independently from distributions $\gw{d,\lambda s}$ and $\gw_{d,\lambda\td{s}}$ respectively.
The challenge
Hypotheses / assumptions:
- $s\td{s}>\alpha$
- $\lambda s\td{s}>1$
- $\lambda>\lambda'(s,\td{s})$
Statement to prove (Theorem tree.test):
Let $\alpha\approx0.3383$ denote Otter's tree counting constant. Suppose $s\td{s}>\alpha$, $\lambda s\td{s}>1$ and $\lambda>\lambda'(s,\td{s})$, where $\lambda'$ is a constant depending only on $s$ and $\td{s}$. Then the following inequality holds: $\liminf_{d\rightarrow \infty} P^{(\lambda,s,\td{s})}_d(L^{(\lambda,s,\td{s})}d(\ct,\td{\ct})>\exp(C(\lambda s \td{s})^d))\ge 1-p^{ext}{\lambda s\td{s}}$, where $C$ is a constant depending only on $\lambda,s$ and $\td{s}$.
Free variables in the statement: \lambda, s, \td{s}, d, C, \alpha
Your task: Prove the statement above, using only the definitions and assumptions provided. State any additional standard background results you invoke.
Referenced algorithms and equations
The following are quoted verbatim from the source paper for reference (algorithms, numbered equations, assumptions, and paper-specific parameters the theorem relies on).
$\lambda'(s,\td{s})$ (— Appendix (appd:LR_feasibility), Theorem (thm:tree_test))
Let $\alpha\approx0.3383$ denote Otter's tree counting constant. Suppose $s\td{s}>\alpha$, $\lambda s\td{s}>1$ and $\lambda>\lambda'(s,\td{s})$, where $\lambda'$ is a constant depending only on $s$ and $\td{s}$. Then the following inequality holds: \begin{equation} \liminf_{d\rightarrow \infty} P^{(\lambda,s,\td{s})}_d(L^{(\lambda,s,\td{s})}d(\ct,\td{\ct})>\exp(C(\lambda s \td{s})^d))\ge 1-p^{ext}{\lambda s\td{s}}, \end{equation}