TCSAlgBENCH / diffusion-network-alignment-an-efficient-algorithm-and-explicit-probability-bounds /theorem-tree-test.md
| # 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](https://arxiv.org/abs/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} | |