Efficient Identity and Position Graph Embedding via Spectral-Based Random Feature Aggregation

Irwin King; Jiahong Liu; Meng Qin

REVIEW 3 major objections 6 minor 66 references

Efficient Identity and Position Graph Embedding via Spectral-Based Random Feature Aggregation

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Random noise, propagated through a parameter-free spectral graph filter in a single forward pass, yields embeddings that match or beat trained GNNs and dedicated baselines on both identity and position tasks—provided the filter sign…

desk verdict Useful empirical paper with a real but overstated quality-efficiency claim; test-set hyperparameter tuning and missing error bars are the main issues. read the letter →

arxiv 2505.20992 v1 pith:AK3TJZEU submitted 2025-05-27 cs.LG cs.SI

Meng Qin , Jiahong Liu , Irwin King This is my paper

classification cs.LGcs.SI

keywords EfficientgraphembeddingNodeidentitypositionSpectralneuralnetworksRandomfeatureaggregationsignalprocessingDegreecorrectionUnsupervised

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that identity and position graph embeddings—the two standard kinds of topology-only node embeddings—can be produced without any training, feature extraction, or eigen-decomposition. The method, random feature aggregation (RFA), feeds Gaussian random noise into a parameter-free spectral graph filter and reads off the output after a single forward propagation through $K$ layers. The paper argues that the filter's sign is a task selector: a low-pass filter amplifies community structure and yields position embeddings, while a high-pass filter amplifies degree variation and yields identity embeddings. On ten datasets, the two RFA variants stay within the top three on F1 quality and achieve the best or near-best quality-versus-time trade-off, while several trained GNN baselines time out or run out of memory on the largest graphs. If this holds, topology-only graph embedding becomes a near-instant, parameter-free inference routine.

What carries the argument

The load-bearing mechanism is random feature aggregation (RFA): a parameter-free spectral GNN backbone. Each layer applies the graph convolution $g(\tilde{\Lambda}) = (\delta+\alpha)I_N - \alpha\tilde{\Lambda}$ to the current node-feature matrix, where $\tilde{\Lambda}$ is the eigenvalue matrix of the degree-corrected normalized Laplacian $L_\tau = I_N - D_\tau^{-1/2}AD_\tau^{-1/2}$ with $D_\tau = D + \tau I_N$; the sign of $\alpha$ chooses low-pass (positions) or high-pass (identities), and the degree-correction term $\tau$ squeezes the spectrum toward 1, an effect the Gershgorin circle theorem explains. Stacking $K$ layers with a nonlinear activation and row or column normalization turns the initial noise into informative embeddings while keeping the whole procedure one forward pass. The argument works because low-frequency eigenvectors are the relaxed solution of normalized cut, while high-frequency eigenvectors concentrate degree variation so that nodes with identical ego-nets receive identical coordinates.

What would settle it

On a fresh graph with known structural roles and known communities, freeze a single RFA(H) and RFA(L) configuration chosen only on validation data, then run both variants; if the high-pass variant cannot separate nodes with identical ego-nets, or the low-pass variant cannot separate communities at above-chance F1, the paper's spectral identity/position claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that node identities and node positions are carried by opposite ends of the graph spectrum. Low-frequency eigenvectors of the normalized Laplacian, which solve the relaxed normalized-cut objective, encode community structure and therefore positions; high-frequency eigenvectors encode local degree variation that distinguishes rooted subgraphs and therefore identities. RFA operationalizes this by replacing learned GNN weights with the fixed spectral kernel $g(\tilde{\Lambda}) = (\delta+\alpha)I_N - \alpha\tilde{\Lambda}$, feeding Gaussian noise $\Theta \sim \mathcal{N}(0,1/d)$ through $K$ such layers with nonlinear activation and normalization, and using the output as the embedding. The RFA(H) variant takes $\alpha<0$ (high-pass) and produces identity embeddings; RFA(L) takes $\alpha>0$ (low-pass) and produces position embeddings. The paper's experiments report top-three F1 quality on all ten datasets and the best or near-best normalized trade-off score between quality and inference time, with RFA completing in seconds to minutes where several trained GNN baselines time out or run out of memory.

Load-bearing premise

The load-bearing premise is that the per-dataset choices of dimension, degree-correction strength, layer count, activation, and normalization were not effectively tuned to the test sets; if those choices were made after seeing test performance, the reported quality advantage may not transfer to new graphs.

Editorial extensions

If this is right

Identity and position embeddings become a simple inference routine: draw Gaussian noise, apply the chosen filter for $K$ layers, and return the output—no optimizer, no eigen-decomposition, no feature extraction.
The filter sign is a task selector: choose $\alpha>0$ when the downstream task rewards community membership, and $\alpha<0$ when it rewards structural role; this gives a principled way to decide which topology property a spectral method will capture.
Inference cost is roughly linear in the number of edges and nodes, allowing graphs with more than a million nodes and a hundred million edges to be embedded in seconds on a GPU and in minutes on a CPU.
A small degree-correction parameter $\tau$ improves quality for both filter types, while a $\tau$ that is too large degrades it, giving a cheap tuning knob with a clear spectral interpretation.
For topology-only inputs, trained GNNs fed one-hot degree features need not be the default choice: the paper's results indicate that an untrained spectral backbone can match or exceed their embedding quality at a fraction of the cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: Because the filter sign separates the two embedding tasks, the same backbone could be reused across different graphs without retraining; a testable prediction is that a rule based only on the downstream task (role classification versus community detection) will select the correct variant.
Inference: The degree-correction term acts as spectrum shaping—it pulls the eigenvalues of the Laplacian together—so it could be adopted by any spectral GNN as a cheap regularizer, not just by this parameter-free backbone.
Inference: Because RFA needs no labels, it can serve as a feature generator for semi-supervised or few-shot settings: run the random propagation once, then train only the downstream classifier on a small labeled set.
Inference: Since RFA's input is isotropic Gaussian noise, its output is a deterministic function of the random seed; ensembling or concatenating a few seeds is a natural, architecture-free way to reduce variance and possibly raise quality further.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the empirical core is real and the paper is worth a serious referee, but the central quality-efficiency claim is softer than the abstract suggests because the per-dataset hyperparameters in Tables 8/9 look like they were chosen after seeing test labels, and there are no error bars.

What's new: RFA combines random noise inputs, a parameter-free spectral filter, and a degree-corrected convolution to produce identity and position embeddings in one feed-forward pass. The observation that high-pass filters give identity-like embeddings and low-pass filters give position-like embeddings is a genuinely fresh result, and the experiments are broad: ten datasets, eighteen baselines, plus ablation and scalability studies. The method is also very fast in practice, and the code is public.

The soft spots. First, the test-set tuning issue is real and load-bearing. The paper describes splitting labeled nodes for the downstream classifier, but never describes a validation split for choosing d, tau, K, activation, and normalization. Tables 8 and 9 show irregular settings across datasets (e.g., RFA(H) uses exp with z-norm on Europe but exp without normalization on USA, with K from 2 to 12), which is exactly the pattern you get when you pick settings per dataset to maximize reported F1. This doesn't undermine the claim that RFA is cheap or that the filter-frequency correlation exists, but it does undermine "a better trade-off between quality and efficiency," because a fair comparison requires the same model-selection protocol for RFA and the baselines. The stress-test note is on target.

Second, the paper reports means over 10 runs but no standard deviations, so we can't tell whether the quality gaps are meaningful.

Third, the theoretical support for the high-frequency/identity link is a heuristic argument (Remark 3.5), not a proof, and the paper itself admits that rigorous guarantees are future work. That's fine for a KDD paper, but the abstract's "we find" should be read as "we observe empirically."

None of these are fatal. The method is simple, reproducible, and a useful baseline for anyone doing unsupervised identity/position embedding. For a graph-ML reading group, it's a good paper to discuss. I'd send it to peer review, but I'd ask for a validation-based hyperparameter selection procedure, error bars, and a softened abstract. The paper deserves serious referee time.

Referee Report

3 major / 6 minor

Summary. This paper proposes random feature aggregation (RFA), an unsupervised graph embedding method that uses a parameter-free spectral GNN backbone with random Gaussian input, a degree-corrected graph convolution, nonlinear activation, and normalization. Two variants—RFA(H) with a high-pass filter (δ=0.1, α=−1) and RFA(L) with a low-pass filter (δ=0.1, α=1)—are designed to produce identity and position embeddings, respectively, through a single K-layer forward propagation with no training. The paper motivates the design with a graph-signal-processing observation on small examples that high- and low-frequency eigenvectors encode node identities and positions, and reports experiments on 10 datasets against 18 baselines, claiming better or competitive quality with orders of magnitude lower inference time, summarized by a normalized trade-off score (NToS). Theoretical remarks connect low-frequency eigenvectors to relaxed normalized cut and high-frequency eigenvectors to degree variation and the Weisfeiler-Lehman test, but these are heuristic. The code is publicly available.

Significance. If the reported results hold under a proper validation protocol, the paper makes a valuable empirical contribution: it shows that a training-free, random-input spectral aggregation can match or beat trained GNNs and dedicated structural embedding methods on unsupervised identity and position tasks, with a dramatic efficiency advantage. The connection between filter type and embedding property is interesting and potentially useful for practitioners. Strengths include the breadth of the evaluation (10 datasets, 18 baselines), the explicit efficiency and scalability analysis, and a public code release. The main caveat is that the quantitative claim currently rests on an evaluation protocol in which hyperparameters appear to be selected after observing test labels, and no uncertainty estimates are provided; until this is addressed, the significance of the empirical claim is not fully established.

major comments (3)

[§4.1, Tables 8-9] The hyperparameters of RFA (d, τ, K, activation, normalization) appear to be selected per dataset on the test portion, since no held-out validation split or selection procedure is described and the settings vary widely across datasets (e.g., K ranges from 2 to 14; exp with z-norm is used on Europe but exp without normalization on USA). Because all three quantitative claims—quality, efficiency trade-off, and NToS—are computed from these settings, the comparison is potentially inflated. Please report a validation-based selection protocol (e.g., choose hyperparameters on the 20% labeled training split or a separate validation split) and give the resulting test scores, or provide results for a fixed default hyperparameter setting.
[§4.1, Tables 3-4] No standard deviations or significance tests are reported for quality metrics, although differences between RFA and the best baseline are small on several datasets (e.g., Reality-Call micro-F1 85.88 vs. 85.72 for GMAE; Film micro-F1 60.03 vs. 59.51 for GMAE; PPI micro-F1 20.38 vs. 20.88 for SketchNE). Without repeated-measures variance or error bars, the claimed 'better quality' and the derived NToS improvements are not statistically substantiated. Please report means with standard deviations over the 10 random splits and, where relevant, paired significance tests.
[§3.4, Remark 3.5] The proof of Remark 3.5 does not establish the claim that high-frequency information captures node identities. Equation (8) only rewrites the eigenvector equation to express λ_r as a sum of degree-normalized neighbor terms; the subsequent step that high-frequency eigenvectors 'encode degree information' and that RFA 'may play a role similar to the WL test' relies on an unverified injectivity condition and the phrase 'if a high-frequency base is powerful enough.' Since the paper presents this as a theoretical interpretation and labels it a proof, please either develop the argument into a precise statement with explicit conditions or clearly label the remarks as intuition.

minor comments (6)

[§4.2] The phrase 'low- and high-pass information may respectively characterize node identities and positions' reverses the paper's own finding; it should read 'high- and low-pass information.'
[Figure 4] The legend entries in Figure 4 use 'LFA' instead of 'RFA' (e.g., 'LFA(L) w/ CPU') and should be corrected to 'RFA.'
[Appendix B] The USA dataset URL is identical to the Europe URL (europe-airports.edgelist); the USA link appears to be wrong and should be fixed.
[§2] There is a typo in the sentence about Figure 1: 'v1 ad v8' should be 'v1 and v8.'
[§4.3] The NToS computation should state explicitly how out-of-time and out-of-memory methods are treated in the min-max normalization, since their inclusion or exclusion affects the reported scores.
[§4.1] The statement that for baselines 'the best quality metrics reported' were used is too vague for reproducibility; please specify the searched parameter grids and normalization choices for each baseline.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: RFA is a parameter-free, label-free embedding procedure whose claims are empirical and self-contained.

full rationale

The core derivation of RFA is not circular: the embedding is computed by feeding random Gaussian noise through a fixed, learnable-parameter-free spectral GNN layer that is only a degree-corrected normalized adjacency propagation (Eqs. 1-4), with no use of labels or ground-truth identities/positions during embedding generation. The high/low-frequency investigation (Fig. 1 and Appendix A) is an empirical observation about eigenvectors of the graph Laplacian used to motivate the filter choice, not a quantity fitted from the downstream labels and then reported as a prediction. The paper explicitly frames theoretical guarantees as future work in Section 6 ('This study empirically validates... In our future research, we plan to explore rigorous guarantees...'), confirming that the central claim is presented as an experimental finding rather than a derivation from assumptions containing the conclusion. Self-citations (e.g., IRWE [31], RaftGP [13], and related community-detection papers) appear only as motivation or related work; none provides a load-bearing uniqueness theorem, ansatz, or fitted parameter. The per-dataset hyperparameters in Tables 8 and 9 (d, tau, K, activation, normalization) are a legitimate experimental-design concern, but they do not constitute circularity under the required standard: no equation reduces to its inputs, and no fitted parameter is renamed as a prediction. The empirical quality claims are therefore self-contained against the external benchmark datasets and baselines, and no circular step can be exhibited.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method itself is parameter-free at inference, but the reported results depend on per-dataset hyperparameters selected using test labels, which are effectively free parameters of the evaluation. No new physical or conceptual entities are introduced; the degree-corrected spectral convolution is a modification of an existing operation.

free parameters (5)

tau (degree correction term) = 0 to 20 across datasets (Tables 8-9)
Chosen per dataset based on downstream classification quality; Fig. 6 shows it significantly changes results.
K (number of layers) = 2 to 14 across datasets (Tables 8-9)
Tuned per dataset; quality improves with K but saturates, as shown in Fig. 5.
d (embedding dimension) = 64 to 512 across datasets
Set per dataset and matched to baselines, but still a free choice affecting quality.
activation function = tanh or exp, depending on variant and dataset
Ablation in Table 7 shows large quality swings (e.g., RFA(H) on Europe: tanh 26.78 vs exp 56.66).
normalization strategy = z-score, l2-norm, or none, per dataset
Selected per dataset and impacts results (e.g., RFA(L) on Youtube without normalization drops to 25.35).

assumptions (5)

standard math Normalized graph Laplacian eigenvalues lie in [0,2] and eigenvectors form an orthonormal basis.
Used in Sections 2-3 to define graph convolution and spectral filtering.
standard math Gershgorin Circle Theorem
Used in Remark 3.3 to interpret how the degree correction term shifts the eigenvalue distribution.
domain assumption High-frequency eigenvectors encode node identities.
The paper's central investigation, supported only by toy examples (Fig. 1, Fig. 8) and a real-graph case study (Fig. 9), not by a theorem.
domain assumption Low-frequency eigenvectors encode community structure.
Standard spectral clustering result, cited to Von Luxburg [46].
ad hoc to paper Random noise input, after one feed-forward pass through a spectral filter with nonlinearity and normalization, preserves and exposes the relevant spectral information.
Empirical property; Section 3.3 gives only a heuristic argument about magnitude control and no formal guarantee.

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Cite this review

Pith. "Pith review of Efficient Identity and Position Graph Embedding via Spectral-Based Random Feature Aggregation." pith.science (2026). https://pith.science/paper/AK3TJZEU

@misc{pith2026250520992,
  author       = {Pith},
  title        = {Pith review of: Efficient Identity and Position Graph Embedding via Spectral-Based Random Feature Aggregation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AK3TJZEU}},
  note         = {Machine review of arXiv:2505.20992}
}

read the original abstract

Graph neural networks (GNNs), which capture graph structures via a feature aggregation mechanism following the graph embedding framework, have demonstrated a powerful ability to support various tasks. According to the topology properties (e.g., structural roles or community memberships of nodes) to be preserved, graph embedding can be categorized into identity and position embedding. However, it is unclear for most GNN-based methods which property they can capture. Some of them may also suffer from low efficiency and scalability caused by several time- and space-consuming procedures (e.g., feature extraction and training). From a perspective of graph signal processing, we find that high- and low-frequency information in the graph spectral domain may characterize node identities and positions, respectively. Based on this investigation, we propose random feature aggregation (RFA) for efficient identity and position embedding, serving as an extreme ablation study regarding GNN feature aggregation. RFA (i) adopts a spectral-based GNN without learnable parameters as its backbone, (ii) only uses random noises as inputs, and (iii) derives embeddings via just one feed-forward propagation (FFP). Inspired by degree-corrected spectral clustering, we further introduce a degree correction mechanism to the GNN backbone. Surprisingly, our experiments demonstrate that two variants of RFA with high- and low-pass filters can respectively derive informative identity and position embeddings via just one FFP (i.e., without any training). As a result, RFA can achieve a better trade-off between quality and efficiency for both identity and position embedding over various baselines.

Figures

Figures reproduced from arXiv: 2505.20992 by the authors.

**Figure 2.** Distributions of frequencies { ˜𝜆𝑟 } w.r.t. different settings of 𝜏 based on the graph in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Intuition of computing NToS for trade-off analysis. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Scalability analysis results of RFA. same inference time. Consistent with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Parameter analysis w.r.t. 𝐾 on PPI, Youtube, Europe, and Actor in terms of micro-F1(%). 0 1 5 10 20 50 τ 19.4 19.6 19.8 20 20.2 Micro-F1(%) (a) RFA(L), PPI 0 1 5 10 20 50 τ 40.5 41 41.5 Micro-F1(%) (b) RFA(L), Youtube 0 1 5 10 20 50 τ 50 55 Micro-F1(%) (c) RFA(H), Euro…

**Figure 7.** Figure 7: Full eigenvectors {u𝑟 } w.r.t. the normalized graph Laplacian L of the example graph in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: A further example of node identities and positions [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: Case study about high-frequency bases [u˜ 𝑁 −2, u˜ 𝑁 −1] for two classes of USA w.r.t. 𝜏 ∈ {0, 10, 20, 50, 100, 500}. frequency). For instance, red nodes {𝑣1, · · · , 𝑣5, 𝑣11, · · · , 𝑣15} have the same value −0.02 and yellow nodes {𝑣6, 𝑣10} have the same value 0.21. T…

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