Erdős–Rényi Subgraph Pair Model

• The Erdős–Rényi Subgraph Pair Model is a framework for generating correlated random graphs via independent edge subsampling from a base graph, enabling rigorous analysis of detection and recovery.
• It characterizes phase transitions and detection thresholds using densest subgraph statistics, balanced-load functions, and local weak convergence in sparse graph regimes.
• The model informs optimal strategies for partial vertex recovery and graph matching, revealing a gap between information-theoretic limits and current polynomial-time algorithms.

The Erdős–Rényi subgraph pair model is a foundational framework for studying correlated random graphs, with applications ranging from detection and recovery problems to the analysis of subgraph statistics in complex networks. This model and its variants underpin several sharp information-theoretic thresholds and limit theorems, with particular emphasis on sparsity, dense substructure detection, and fundamental limits for unlabeled graph inference.

1. Formal Definition of the Erdős–Rényi Subgraph Pair Model

The canonical Erdős–Rényi subgraph pair model proceeds as follows. Begin with a base graph $G_0 \sim \mathcal{G}(n,p)$, where each undirected edge $\{i,j\}\subset [n]$ is present independently with probability $p$. Two observed graphs $G_1$ and $G_2$ are generated by independent edge-subsampling: each edge in $G_0$ is retained in $G_1$ with probability $s$, and (independently) retained in $G_2$ also with probability $s$.

Formally, for each $\{i,j\}\subset [n]$0, introduce independent indicator variables $\{i,j\}\subset [n]$1, $\{i,j\}\subset [n]$2, and set

$\{i,j\}\subset [n]$3

The model naturally generalizes to cases where $\{i,j\}\subset [n]$4 is observed after a hidden vertex-permutation, enabling study of graph matching and detection under anonymization (Du, 17 Feb 2025).

This construction yields correlated Erdős–Rényi graphs with marginal edge probability $\{i,j\}\subset [n]$5 and Pearson correlation $\{i,j\}\subset [n]$6 between corresponding edges.

2. Probabilistic Structure and Key Quantities

For a fixed edge $\{i,j\}\subset [n]$7, denote the presence indicators as $\{i,j\}\subset [n]$8:

• Marginals: $\{i,j\}\subset [n]$9.
• Joint: $p$0.
• Covariance: $p$1.
• Variance: $p$2.
• Correlation: $p$3.

The intersection graph $p$4 has edge inclusion probability $p$5 and, in the sparse regime ($p$6), approximates $p$7.

A central combinatorial statistic is the balanced-load function $p$8 on $p$9, defined as the unique vertex-valued assignment satisfying a local equilibrium: for all $G_1$0, the set $G_1$1 maximizes $G_1$2 (Du, 17 Feb 2025). Empirically, $G_1$3 converges in distribution (under local weak convergence) to a deterministic law $G_1$4, whose right endpoint $G_1$5 gives the densest-subgraph density in $G_1$6.

3. Detection and Information-Theoretic Phase Transitions

The model supports rigorous analysis of detection thresholds between the correlated-subsampling law and two independent Erdős–Rényi graphs with matching marginals (Ding et al., 2022, Feng, 15 Jun 2025). The sharp detection threshold is controlled by the intersection graph's densest subgraph:

• Under the null hypothesis $G_1$7: $G_1$8 are independent samples from $G_1$9.
• Under the alternative $G_2$0: $G_2$1 are derived via independent $G_2$2-edge-subsampling from a common $G_2$3.

The main transition is in $G_2$4: for $G_2$5, there exists a unique $G_2$6 such that $G_2$7.

• If $G_2$8, then $G_2$9 (strong detection possible).
• If $G_0$0, then $G_0$1 (impossible).

The threshold is formulated as $G_0$2 (Ding et al., 2022). In the constant-average-degree regime, a further refinement yields $G_0$3, where $G_0$4 is Otter’s constant, resolving the last constant gap in previous work (Feng, 15 Jun 2025).

There is no known polynomial-time test matching this information-theoretic threshold in full, highlighting a computation–information disparity.

4. Partial Recovery and Asymptotic Overlap

The partial recovery problem concerns matching as many true vertex pairs as possible under a latent ground-truth permutation $G_0$5. For $G_0$6 with $G_0$7, let $G_0$8 denote the upper-tail of the balanced-load law.

The central theorem asserts that for any $G_0$9, $G_1$0, and all $G_1$1:

• There exists an estimator $G_1$2 achieving overlap at least $G_1$3 with high probability.
• No estimator achieves overlap exceeding $G_1$4 with nonvanishing probability.

Thus, the asymptotically optimal recoverable fraction of vertices is $G_1$5, except at atoms of $G_1$6 (Du, 17 Feb 2025). The upper and lower bounds coincide except on a countable exceptional set.

This result ties recoverability explicitly to the tail of the balanced-load distribution in $G_1$7, revealing connections between matching, subgraph densities, and local equilibrium in sparse random graphs.

5. Analytical Tools and Subgraph Statistics

The analysis leverages several advanced probabilistic and combinatorial techniques:

• Densest-subgraph statistics: The densest-subgraph density $G_1$8 controls detection thresholds (Ding et al., 2022).
• Second-moment and likelihood ratio methods: Sharpen impossibility and achieve phase transitions in detection problems.
• Tree and forest enumeration: Explicitly enters in the proof of tight thresholds, governed by Otter’s enumeration and the associated constant $G_1$9 (Feng, 15 Jun 2025).
• Local weak convergence and empirical distributions: Under sparse regimes, quantities like balanced load $s$0 have limiting distributions capturing the relevant asymptotic behavior (Du, 17 Feb 2025).
• Limit theorems for joint subgraph counts: Functional and local limit theorems for subgraph counts and their covariances characterize fluctuations at the moderate and dense regimes (Sah et al., 2024, Hazra et al., 3 Feb 2025).

6. Broader Implications and Related Results

The Erdős–Rényi subgraph pair model mediates a unifying role between detection, partial recovery, hypothesis testing, and subgraph enumeration:

• The optimality and tightness of balanced-load and densest-subgraph-based statistics extend to related tasks such as community detection and proportional graph existence.
• The connection to local equilibrium conditions and the role of Poisson-Galton–Watson approximations provide bridges to other domains in random graph theory.
• Technical results in this area have resolved longstanding open problems, such as those posed by Wu–Xu–Yu, by eliminating constant-factor gaps and achieving sharp thresholds (Ding et al., 2022, Feng, 15 Jun 2025, Du, 17 Feb 2025).

Ongoing work explores the computational barriers to practical algorithms matching the information-theoretic optimality, the precise behavior of the load distribution $s$1, and the universality of these thresholds in broader random graph settings.

7. Summary Table: Key Regimes and Results

Regime	Main Threshold/Quantity	Principle Result
$s$2	$s$3	Sharp detection/overlap threshold (Ding et al., 2022, Du, 17 Feb 2025)
$s$4 fixed	$s$5	Exact strong-detection boundary (Feng, 15 Jun 2025)
Partial recovery	$s$6	Optimal overlap fraction (Du, 17 Feb 2025)

The structure of the Erdős–Rényi subgraph pair model has proven to be both tractable and rich, supporting a wide array of theoretical developments with precise statistical and combinatorial characterizations.