Seeded Graph Matching Overview

• Seeded graph matching is a method to reconstruct hidden bijections by using pre-matched seed vertices as anchors between two correlated graphs.
• It employs large neighborhood statistics and ℓ-hop witness strategies to amplify signal-to-noise ratios, ensuring precise matching in sparse and noisy settings.
• Recent approaches integrate deep learning and covariate-assisted models to enhance scalability and accuracy in heterogeneous and large-scale networks.

Seeded graph matching refers to the class of algorithms and theoretical frameworks for reconstructing a hidden correspondence (bijection) between two graphs, given noisy structural information and partial side-information in the form of a set of pre-matched vertex pairs (seeds). Seeds serve as anchors that can dramatically reduce the combinatorial complexity and algorithmic hardness of the general graph matching problem, enabling efficient and highly accurate recovery of the global alignment in both random-graph models and numerous practical applications.

1. Formal Model and Problem Definition

In the seeded graph matching problem, two graphs G₁=(V₁,E₁) and G₂=(V₂,E₂), of (typically) equal size |V₁|=|V₂|=n, are observed. There exists an unknown ground-truth bijection π:V₁→V₂ encoding how the vertices of G₁ correspond to those of G₂. The canonical objective is to recover π or an approximation thereof, typically by minimizing the number of adjacency disagreements:

$$\min_{\pi : \text{consistent with seeds}} \sum_{u,v \in V_1} |A_{uv} - B_{\pi(u)\pi(v)}|,$$

where A and B are the adjacency matrices of G₁ and G₂, and π is required to agree with the set of seed pairs S = {(u,v)}. Formally, for the seeded Quadratic Assignment Problem (QAP) (Fishkind et al., 2012):

$$Q^* = \arg\min_{Q \in \Pi_n : Q_{S,S} = I_s} \|A - Q^T B Q\|_F^2$$

In the correlated Erdős–Rényi (ER) model (Mossel et al., 2018), pairs of graphs (G₁,G₂) are generated from a shared “parent” G₀~G(n,p) via independent edge-subsampling at rate s, and G₂ is relabeled by an unknown π; the seeds give π(i) for a subset of indices i∈I₀.

2. Information-Theoretic Thresholds for Seeded Matching

In the correlated ER setting, strong impossibility and achievability results establish concrete seed-sparsity thresholds for exact recovery. A classical counting argument shows that perfect alignment—seeded or not—is impossible unless the intersection graph has no isolated vertices, i.e., the edge density satisfies

$n p s^2 \geq \log n + \Omega(1).$

Moreover, even the maximum-likelihood estimator fails unless the fraction of seeded nodes α satisfies α≈1 when n p s^2−\log n=O(1). Therefore, the information-theoretic threshold for (seeded or seedless) exact recovery is

$n p s^2 \simeq \log n.$

In terms of seeds, (Mossel et al., 2018) demonstrates that polynomial-time seeded matching in the very sparse regime requires as few as k = n^{3\epsilon} seeds (for p \leq n^{-1+\epsilon}, ε<1/6), and only k = Ω(\log n) in the dense regime (p \gg \log n / n).

3. Algorithmic Methods: Large Neighborhood Statistics and Structural Expansions

The core algorithmic paradigm in information-theoretically optimal seeded matching is based on “large neighborhood statistics” (Mossel et al., 2018). For each candidate pair (u,v), one compares the sets of seeds in their respective ℓ-hop neighborhoods. The signal-to-noise amplification achieved through large neighborhoods enables discrimination between true and spurious pairs even at vanishingly small seed rates.

• Sparse Regime (Algorithm 1: ℓ-hop Witness):

For p\leq n^{-1+\epsilon}, select ℓ ≈ ((1/2−ε) log n)/log(n p s^2), so that the ℓ-neighborhood contains about n^{1/2−ε} vertices. Each candidate pair is accepted if there exist m vertex-disjoint paths of length ℓ from both u and v to m common seeds. Candidate checking reduces to a max-flow in an auxiliary graph of size O(n^{1/2−ε}).

• Dense Regime (Algorithm 2: (d−1)-hop Witness):

Set d = ⌊1/a⌋ + 1 for n p = b n^a, b=Θ(1), a∈(0,1]. For each unseeded vertex, compute the number of seeds in the (d−1)-hop neighborhoods overlapping under the candidate match; assign each vertex to the candidate maximizing this seed-overlap.

• Enhanced Algorithm for Ultra-Sparse:

When p is as small as polylog(n)/n, direct ℓ-hop witness counts are too noisy. One refines the comparison by minimaxing over small deletions to decorrelate neighborhoods; this allows maintaining the same seed size bounds at higher computational cost.

These paradigms are extended to non-homogeneous and attributed models by leveraging analogous “witness” statistics over relevant neighborhoods, with similar percolation and cascading mechanisms seen in scale-free and power-law graphs (Yu et al., 2021).

4. Theoretical Guarantees and Proof Techniques

The fundamental achievability proofs (Mossel et al., 2018) rest on:

• Neighborhood Expansion and Concentration: In G₁*∧G₂~G(n,p s^2) with n p s^2≫log n, the ℓ-neighborhood grows like a Galton–Watson tree, ensuring separate (non-overlapping) neighborhoods and distinct seed signatures with high probability.
• Seed-Amplification Via Large Neighborhoods: The expected number of seeds in the neighborhood of a true pair (u,u) is ≍(n p s^2)^ℓ, versus much smaller O((n p s)^{2ℓ}/n) for a false pair (u,v), enabling thresholding.
• Union Bounds and Concentration: Chernoff, Hoeffding, and multivariate-polynomial concentration control error probabilities over O(n^2) candidate pairs.

These methods extend directly into information-theoretic mutual information thresholds when signatures or structural “fingerprints” are used for matching (Shirani et al., 2017, Shirani et al., 2020). In the communication-information-theoretic approach, a seed’s adjacency pattern with respect to candidates encodes I(X;X') nats per seed; thus, providing Λ_n ≥ (2 log n)/I(X;X') seeds suffices for vanishing-error recovery in polynomial time (Shirani et al., 2017, Shirani et al., 2020).

5. Extensions: Partial Seeding, Side Information, and Power-Law Networks

Recent work addresses scenarios with only partially-correct seeds (Yu et al., 2020), ambiguous seed information (Shariatnasab et al., 2021), and seconds where only a shortlist of possible matches (ambiguity sets) is available for each vertex. These frameworks generalize the standard seeded problem to mixed or “soft” seed settings and reveal phase transitions in permitted error rates and required seed density for exact recovery.

For networks with power-law degree distribution, D-hop witnesses and degree-stratified matching strategies achieve seed requirement reductions from n^{1/2+\epsilon} to polylogarithmic levels; for β ∈ (2,3), D > (4−β)/(3−β) and m = Ω((\log n)^{4−β}) suffice for perfect matching of a constant fraction of vertices (Yu et al., 2021).

6. Empirical Performance and Scalability

Evaluations on both synthetic and large real-world networks confirm that the aforementioned seeded algorithms not only achieve optimal or near-optimal seed efficiencies but are also computationally scalable:

• The ℓ-hop witness algorithm (Mossel et al., 2018) and PPR-based heavy hitter methods (Zhang et al., 2018) operate in polynomial time (e.g., O(n^3) or better with parallelization), with empirical F1-scores and match accuracy dramatically superior to prior percolation or greedy baselines.
• Parallel iterative-repair matchers such as IRMA (Babayov et al., 2022) deliver major gains in precision, recall, and wall-clock time in scale-free or low-overlap regimes, with empirical F1 gains of 20–40 points over one-pass algorithms.

7. Extensions: Deep Learning and Covariate-Assisted Methods

State-of-the-art neural approaches use seed-aware message passing, multi-hop matching features, and seed propagation for supervised and semi-supervised seeded graph matching (Yu et al., 2022, Chen et al., 2021). These designs encode combinatorial witness statistics directly in neural architectures, enabling transferability and strong empirical accuracy with few seeds.

Moreover, the incorporation of node and edge covariates through generalized linear models, as in covariate-assisted seeded matching (Dawn et al., 12 Dec 2025), has further improved alignment accuracy and sample complexity in heterogeneous networks. By learning the structural–covariate regression on seed pairs and solving a modified QAP, these approaches robustly outperform structure-only baselines.

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References:

• "Seeded Graph Matching via Large Neighborhood Statistics" (Mossel et al., 2018)
• "Efficient and High-Quality Seeded Graph Matching: Employing High Order Structural Information" (Zhang et al., 2018)
• "The Power of D-hops in Matching Power-Law Graphs" (Yu et al., 2021)
• "IRMA: Iterative Repair for graph MAtching" (Babayov et al., 2022)
• "Seeded Graph Matching" (Fishkind et al., 2012)
• "A Concentration of Measure Approach to Correlated Graph Matching" (Shirani et al., 2020)
• "Seeded Graph Matching: Efficient Algorithms and Theoretical Guarantees" (Shirani et al., 2017)
• "Covariate-assisted graph matching" (Dawn et al., 12 Dec 2025)
• "SeedGNN: Graph Neural Networks for Supervised Seeded Graph Matching" (Yu et al., 2022)

These works together define and advance the state of seeded graph matching for both theoretical and applied large-graph inference.