[Paper Review] Planting trees in graphs, and finding them back

Laurent Massoulié, Ludovic Stephan+1 more|arXiv (Cornell University)|Nov 5, 2018

Forensic Fingerprint Detection Methods4 citations

TL;DR

This paper studies detection and reconstruction of planted tree structures—specifically line graphs and D-ary trees—within sparse Erdős-Rényi random graphs. It establishes phase transitions showing that detection becomes feasible when tree size exceeds a threshold, while reconstruction remains impossible for sublinear tree sizes, even when detection is easy, revealing a key discrepancy absent in low-rank planted structures like dense subgraphs.

ABSTRACT

In this paper we study detection and reconstruction of planted structures in Erdős-Rényi random graphs. Motivated by a problem of communication security, we focus on planted structures that consist in a tree graph. For planted line graphs, we establish the following phase diagram. In a low density region where the average degree $λ$ of the initial graph is below some critical value $λ_c=1$, detection and reconstruction go from impossible to easy as the line length $K$ crosses some critical value $f(λ)\ln(n)$, where $n$ is the number of nodes in the graph. In the high density region $λ>λ_c$, detection goes from impossible to easy as $K$ goes from $o(\sqrt{n})$ to $ω(\sqrt{n})$, and reconstruction remains impossible so long as $K=o(n)$. For $D$-ary trees of varying depth $h$ and $2\le D\le O(1)$, we identify a low-density region $λλ_D$, but confirm only the following part of this picture: Detection is easy for $D$-ary trees of size $ω(\sqrt{n})$, while at best only partial reconstruction is feasible for $D$-ary trees of any size $o(n)$. These results are in contrast with the corresponding picture for detection and reconstruction of {\em low rank} planted structures, such as dense subgraphs and block communities: We observe a discrepancy between detection and reconstruction, the latter being impossible for a wide range of parameters where detection is easy. This property does not hold for previously studied low rank planted structures.

Motivation & Objective

• To understand the phase transitions in detection and reconstruction of planted tree structures in sparse Erdős-Rényi graphs.
• To compare the feasibility of detection and reconstruction for tree-based planted structures against previously studied low-rank structures like dense subgraphs.
• To identify conditions under which detection becomes feasible while reconstruction remains impossible, particularly for sublinear tree sizes.
• To establish a complete phase diagram for both detection and reconstruction in the case of planted line graphs and D-ary trees.
• To investigate whether hard phases exist in the detection and reconstruction of tree-structured signals, contrasting with known results for low-rank models.

Proposed method

• Analyzes detection and reconstruction in the sparse Erdős-Rényi random graph model G(n, λ/n) with fixed average degree λ.
• Uses probabilistic analysis and concentration inequalities to derive thresholds for detection and reconstruction feasibility.
• Applies hypergeometric distribution modeling to estimate expected overlap between reconstructed and planted tree nodes, conditional on partial tree structure.
• Employs a recursive tree structure analysis to bound the expected number of correctly recovered nodes in the final layer of D-ary trees.
• Derives phase transitions based on asymptotic behavior in n, particularly focusing on thresholds involving ln(n), ln(ln(n)), and √n scaling.
• Uses a conditional probability framework under P₁ to evaluate reconstruction performance given partial knowledge of the tree's upper layers.

Experimental results

Research questions

• RQ1 At what threshold size does detection of a planted line graph become feasible as a function of the graph's average degree λ?
• RQ2 Why is reconstruction impossible for planted line graphs of size o(n), even when detection is easy?
• RQ3 What is the critical height h* for D-ary trees such that detection transitions from impossible to feasible?
• RQ4 Can full reconstruction of a D-ary tree be achieved when the tree height h exceeds h*, or is partial reconstruction the best possible outcome?
• RQ5 How do the detection and reconstruction thresholds for tree-structured signals compare to those for low-rank planted structures like dense subgraphs or communities?

Key findings

• For planted line graphs in low-density regions (λ < 1), detection and reconstruction become feasible when the line length K exceeds K* = ln(n)/ln(1/λ).
• In high-density regions (λ > 1), detection becomes feasible when K transitions from o(√n) to ω(√n), but reconstruction remains impossible for K = o(n).
• For D-ary trees, detection becomes feasible when tree height h exceeds h* = g(D)ln(ln(n)), with g(D) depending on D.
• Even when detection is feasible for D-ary trees, only partial reconstruction is possible: at least a constant fraction of nodes cannot be correctly identified.
• In high-density regions (λ > λ_D), detection is easy for D-ary trees of size ω(√n), but reconstruction remains impossible for any size o(n).
• The paper confirms that for D-ary trees, reconstruction must fail on a non-vanishing fraction of nodes, even when detection is feasible, highlighting a fundamental asymmetry between detection and reconstruction.