Correlation detection in trees for planted graph alignment

TL;DR

This work proposes MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem, and proves MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible.

Abstract

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.

Figures (10)

• Figure 1: A sample from model ${\mathsf{G}}(n,q=\lambda/n,s)$ with $n=11$, $\lambda=1.9$, $s=0.7$.
• Figure 2: Diagram of the $(\lambda,s)$ regions where partial recovery is known to be impossible (ganassali2021impossibility), IT-feasible (Wu2021SettlingTS), or easy (Ganassali20a and this paper). In the orange region, though partial graph alignment is IT-feasible, one-sided detectability is impossible in the tree correlation detection problem, and partial graph alignment is conjectured to be hard (this paper).
• Figure 3: A rooted tree $t$ of depth $d=4$ (the root is highlighted in yellow).
• Figure 4: Samples from models $\mathbb{P}^{(\lambda)}_{d}$ and $\mathbb{P}^{(\lambda,s)}_{d}$, with $\lambda = 1.8$, $s=0.8$, and $n=5$. The root node is highlighted in yellow. Labels are not shown.
• Figure 5: Example of a transition described hereabove, with $\lambda = 1.85$, $s=0.85$, at depth $d=5$. The original tree $t$ is drawn on the left. On the right, $t'$ is obtained as follows: first extracting a $s-$subsampling $\tau$ of $t$ (dashed blue edges are deleted), and drawing a $(\lambda,s)-$augmentation of $\tau$ -- first attaching new children to all vertices of $\tau$ (dark red nodes with thick edges), and attaching new Galton-Watson trees to these new children (light red nodes with standard edges). Labels are not shown.

Theorems & Definitions (55)

• Remark 1: A colored view
• Remark 2
• Theorem 1: Correlation detection in trees
• Remark 3: On condition $(v)$
• Theorem 2: Consequences for one-sided partial graph alignment
• Conjecture 1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1: Recursive formula for $L_d$