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arxiv: 2607.00897 · v1 · pith:RAVIHTFHnew · submitted 2026-07-01 · 💻 cs.IT · math.IT· math.PR· math.ST· stat.TH

Recovery of Planted Subgraphs

Wasim Huleihel This is my paper

Pith reviewed 2026-07-02 05:47 UTC · model grok-4.3

classification 💻 cs.IT math.ITmath.PRmath.STstat.TH
keywords planted subgraph recoveryexact recoveryErdős–Rényi graphsminimal maximum subgraph densitystatistical thresholdscomputational lower boundsspectral algorithms
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The pith

Exact recovery of any planted subgraph in a random graph is possible precisely above a threshold set by its minimal maximum subgraph density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes sharp conditions under which an arbitrary planted subgraph Γ can be exactly recovered from a dense Erdős–Rényi graph G(n, q_n) with higher edge probability p_n inside Γ. Recovery succeeds with high probability if and only if the edge probability gap exceeds a value determined by the minimal maximum subgraph density of Γ, a quantity defined as the maximum subgraph density within the smallest induced balanced subgraph of Γ. Matching lower bounds show these conditions are also necessary. A spectral algorithm based on the adjacency matrix achieves efficient recovery in some regimes, while low-degree polynomial methods reveal computational-statistical gaps in others.

Core claim

The central claim is that the statistical threshold for exact recovery of an arbitrary planted subgraph Γ embedded in G(n, q_n) with edges inside Γ present at probability p_n > q_n is characterized by the minimal maximum subgraph density of Γ, defined as the maximum subgraph density of the smallest induced balanced subgraph of Γ, with matching upper and lower bounds establishing that this quantity fully determines when exact recovery is possible with high probability.

What carries the argument

minimal maximum subgraph density of Γ, which is the maximum subgraph density of the smallest induced balanced subgraph of the planted subgraph

If this is right

  • Exact recovery of Γ is possible with high probability precisely when the probability gap exceeds the minimal maximum subgraph density threshold.
  • Exact recovery is information-theoretically impossible below this threshold, as shown by the matching lower bounds.
  • A polynomial-time algorithm based on spectral properties of the adjacency matrix recovers the subgraph in regimes where the threshold permits statistical recovery.
  • There are parameter regimes where statistical recovery is possible but computationally hard, as established by low-degree polynomial lower bounds.
  • The threshold characterization and algorithmic results extend to semi-random models and to weaker notions of recovery.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same density-based threshold idea could be tested in other random graph models such as stochastic block models with multiple communities.
  • The quantity might provide a way to compare the recoverability of different subgraphs without simulating the full recovery process.
  • Real-world networks suspected to contain planted structures could be analyzed by computing this minimal maximum subgraph density to predict recoverability.
  • The computational hardness results suggest that approximation algorithms or heuristics may be needed for large instances even when information-theoretic recovery is feasible.

Load-bearing premise

The exact recovery threshold is fully determined by the minimal maximum subgraph density of the fixed planted subgraph Γ.

What would settle it

Finding a specific subgraph Γ where exact recovery succeeds with high probability below the predicted threshold or fails above it would falsify the claimed necessity and sufficiency of the minimal maximum subgraph density.

Figures

Figures reproduced from arXiv: 2607.00897 by Wasim Huleihel.

Figure 1
Figure 1. Figure 1: The observed graph G is the union of an Erd˝os–R´enyi graph and a planted subgraph Γ ⋆ n . In (a) Γ⋆ n is a bipartite subgraph, and in (b) Γ⋆ n is a path. We define the number of edges of the graph-cut as |H| ≜ |E|, and the number of non-selected vertices as |v(H)| ≜ |V \ S|. We note that throughout the paper, we use H to denote either a graph (identified with its edge set) or a graph-cut, depending on the… view at source ↗
read the original abstract

Understanding the fundamental limits of recovering planted subgraphs in random graphs is a central challenge in high-dimensional statistics and theoretical computer science. While existing work has largely focused on special subgraph families such as cliques, bicliques, or dense blocks, the exact recovery of a general planted subgraph in Erd\H{o}s--R\'enyi random graphs remains poorly understood. In this paper, we study the exact recovery of an arbitrary planted subgraph $\Gamma = \Gamma_n$ embedded in a dense Erd\H{o}s--R\'enyi random graph $\mathcal{G}(n,q_n)$, where edges within $\Gamma$ are present independently with probability $p_n > q_n$. Our main results identify sharp conditions under which exact recovery is possible with high probability, and we establish matching lower bounds showing the necessity of these conditions. The resulting statistical threshold is characterized by a new graph-theoretic quantity, which we term the \emph{minimal maximum subgraph density}. This quantity is defined as the maximum subgraph density of the smallest induced balanced subgraph of $\Gamma$. We then turn to the problem of recovery under polynomial-time constraints. We propose a computationally efficient recovery algorithm that applies to arbitrary planted subgraphs and analyze its performance in terms of certain spectral properties of the adjacency matrix. In addition, we derive computational lower bounds for recovery using the low-degree polynomial framework, establishing regimes where recovery is statistically possible but computationally hard. Finally, we consider several extensions of our setting, including recovery in semi-random models and weaker notions of recovery.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies exact recovery of an arbitrary planted subgraph Γ embedded in a dense Erdős–Rényi graph G(n, q_n) with intra-subgraph edge probability p_n > q_n. It claims that the exact-recovery threshold is sharply characterized by a new graph-theoretic quantity termed the minimal maximum subgraph density (defined as the maximum subgraph density of the smallest induced balanced subgraph of Γ), with matching upper and lower bounds. It also proposes a spectral algorithm for polynomial-time recovery, derives computational lower bounds via the low-degree polynomial method, and considers extensions including semi-random models and weaker recovery notions.

Significance. If the central claims hold, the results would substantially generalize prior work on special cases (cliques, bicliques) to arbitrary planted subgraphs, with the new density quantity providing a clean, graph-theoretic characterization of the statistical threshold. The combination of matching bounds, an efficient spectral algorithm, and low-degree computational hardness results would represent a notable advance in random-graph recovery problems. The parameter-free nature of the threshold (no free parameters listed in the model) and the explicit definition of the characterizing quantity are strengths.

minor comments (3)
  1. [Abstract / §2] The abstract defines the minimal maximum subgraph density but does not specify the precise notion of 'balanced subgraph' or 'induced'; this definition should be stated explicitly in §2 or §3 with a formal equation to avoid ambiguity for general Γ.
  2. [Algorithm section] The spectral algorithm is described in terms of 'certain spectral properties of the adjacency matrix' without an explicit statement of the eigenvector or eigenvalue threshold used; adding the precise condition (e.g., Eq. (X) in the algorithm section) would improve clarity.
  3. [Model section] Notation for the sequences p_n and q_n is introduced but the regime (dense vs. sparse) and any assumptions on their growth rates relative to n are not summarized in one place; a short table or remark collecting these would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted, and we appreciate the recognition of the generalization to arbitrary planted subgraphs and the new density quantity. No specific major comments were provided in the report, so we have no points requiring direct rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the minimal maximum subgraph density directly as a graph-theoretic function of the planted subgraph Γ (maximum subgraph density of its smallest induced balanced subgraph) and states that this quantity characterizes the exact-recovery threshold, with matching upper and lower bounds established. This is a standard definitional characterization of a threshold rather than a self-referential loop or fitted parameter renamed as a prediction. No equations, self-citations, or ansatzes in the abstract or described claims reduce the central result to its own inputs by construction. The derivation is self-contained against the external graph structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on standard random-graph model assumptions and the newly introduced density quantity; no explicit free parameters or invented physical entities are stated.

axioms (1)
  • domain assumption Edges in the host graph and inside the planted subgraph are present independently.
    Standard modeling choice stated in the abstract for the Erdős–Rényi setting.
invented entities (1)
  • minimal maximum subgraph density no independent evidence
    purpose: Characterizes the exact-recovery statistical threshold
    Newly defined graph-theoretic quantity introduced in the paper; no independent evidence supplied in abstract.

pith-pipeline@v0.9.1-grok · 5803 in / 1304 out tokens · 27563 ms · 2026-07-02T05:47:53.977812+00:00 · methodology

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Reference graph

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