Toward Scalable Graph Unlearning: A Node Influence Maximization based Approach

Bowen Fan; Guoren Wang; Rong-Hua Li; Xunkai Li; Zhengyu Wu; Zhiyu Li

REVIEW 5 major objections 7 minor 1 cited by

Toward Scalable Graph Unlearning: A Node Influence Maximization based Approach

T0 review · 5 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Graph unlearning works better when the nodes to scrub are chosen by influence scores from the trained model's own propagation, rather than by local neighborhood.

desk verdict A useful plug-and-play HIE selector for graph unlearning with a heuristic core; solid experiments, but the influence measure is asserted and the algorithm/definition mismatch needs fixing. read the letter →

arxiv 2501.11823 v1 pith:4BQG55S2 submitted 2025-01-21 cs.LG cs.AIcs.DBcs.SI

Xunkai Li , Bowen Fan , Zhengyu Wu , Zhiyu Li , Rong-Hua Li , Guoren Wang This is my paper · ORCID

classification cs.LGcs.AIcs.DBcs.SI

keywords graphunlearningnodeinfluencemaximizationhigh-influenceentityselectiondecoupledneuralnetworksfunctionentity-specificfine-tuningforgettingandreasoningtrade-offscalablelearning

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

The paper claims that the nodes a graph-unlearning request must scrub are rarely just the deleted entity's local neighborhood; gradient-driven training entangles many other nodes, often same-class nodes far away. It proposes Node Influence Maximization (NIM), a plug-and-play method that measures how strongly each remaining node was influenced by the unlearning entities during training, treating those entities as seeds in an influence-propagation model. NIM's influence scores come from the already-trained model's propagated features and predictions, so it runs offline before unlearning and can be attached to most existing graph unlearning methods. On top of that, the paper introduces Scalable Graph Unlearning (SGU), which fine-tunes the model only on the deleted entities and the selected high-influence entities with custom forgetting and reasoning losses. On 14 datasets, including the 111-million-node ogbn-papers100M, the claimed result is that NIM improves forgetting for most baseline methods while SGU matches or beats state-of-the-art unlearning quality at scale.

What carries the argument

The carrier is a decoupled influence propagation model (Eq. 4) that treats the GNN's weight-free propagation operator $P$ (SGC-style $k$-step propagation, GBP weighting, GAMLP attention, and similar formulas) as a social-influence diffusion process, with the unlearning entities as the seed set $S$. Topology influence $I_t$ is the $\ell^1$-norm of the expected Jacobian of propagated features, $\partial \tilde{X}^{(k)}_v / \partial \tilde{X}^{(0)}_u$; feature influence $I_f$ is the $\ell^1$-norm of the expected Jacobian of predictions, $\partial \tilde{Y}^{(k)}_v / \partial \tilde{Y}^{(0)}_u$. The two are normalized into a random-walk interpretation (Eq. 7) and summed, and nodes whose maximum influence from $S$ exceeds a threshold $\theta$ are activated as high-influence entities (Eq. 8), with budget $B$ controlling how many are selected. The second mechanism is entity-specific fine-tuning: label-shuffled cross-entropy on unlearning entities, prototype-based embedding forgetting on unlearning entities, contrastive loss on high-influence entities against same-label positives and unlearning-entity negatives, plus an L2/KL memory-based reasoning loss; the scalar $\lambda$ balances forgetting against reasoning.

What would settle it

Train a GNN on a graph, pick an unlearning set, and fully retrain from scratch without those entities; define ground-truth high-influence entities as the nodes whose predictions change most between the original and retrained models. If NIM's selected set overlaps with this ground truth no better than the $L$-hop neighborhood does, the measure fails its core purpose; the paper currently reports membership-inference improvements, not this direct overlap check against retraining.

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Extended reading notes

Core claim

The paper's central discovery is that high-influence entity selection is the bottleneck in learning-based graph unlearning: existing methods take the $L$-hop neighborhood of the unlearning entity, which misses same-label nodes reached by similar gradients and includes nearby nodes that carry little signal. The paper claims that a node's influence can be quantified directly from the decoupled propagation operator of the trained GNN: the $\ell^1$-norm of the Jacobian of propagated features (topology influence) plus the $\ell^1$-norm of the Jacobian of model predictions (feature influence). Summing these gives a fine-grained influence score, and nodes whose maximum influence from the unlearning set exceeds a threshold $\theta$ form the high-influence entities. Replacing neighborhood selection with this influence measure lowers membership-inference AUC toward 0.5 for five high-influence-based graph unlearning baselines, and the fine-tuning framework built on it reaches state-of-the-art prediction on non-unlearning entities while remaining efficient on billion-scale graphs.

Load-bearing premise

The load-bearing premise is that the already-trained model's propagation and predictions can be read like an influence map, specifically that the sensitivity of each node's propagated features and predictions to those of the deleted entity reveals which nodes were actually entangled with that entity during gradient-driven training.

Editorial extensions

If this is right

Any high-influence-based graph unlearning method (the paper lists CGU, GIF, D2DGN, GNNDelete, and MEGU) can drop in NIM as a pre-processing step and lower membership-inference leakage, with reported improvements up to about 5.7 percentage points.
Unlearning can scale to graphs with over 100 million nodes because NIM runs offline on precomputed propagated features, and fine-tuning touches only the unlearning entities plus the selected high-influence entities.
The trade-off between forgetting and reasoning is controllable through the budget $B$, threshold $\theta$, and loss weight $\lambda$, so practitioners can steer toward complete removal or preserved prediction.
Same-class influence that extends beyond the $L$-hop neighborhood is explicitly addressed by the influence measure, which is the paper's explanation for why neighborhood-based high-influence selection underperforms.
SGU preserves the reasoning capability of the unlearned model through memory-based supervision, so frequent deletion requests need not degrade predictions on remaining nodes.

Reading between the lines

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

(Editorial inference) If the Jacobian-based influence scores are good proxies for training entanglement, the same scores could serve as a data-attribution tool beyond unlearning, ranking which training nodes most shaped a given prediction.
(Editorial inference) The paper's motivating claims that same-label nodes are more influenced and different-label nodes less so (Sec. 1) are directly testable by inspecting NIM's selected sets; on highly heterophilic graphs, where same-label nodes are structurally distant, NIM's advantage over $L$-hop selection should shrink.
(Editorial inference) Because NIM is computed from the trained model's propagated features and predictions, it inherits the backbone's quality; for sampling-based GNN backbones the added weight-free propagation is an approximation, so the benefit may degrade when that propagation diverges from the actual training distribution.
(Editorial inference) The framework suggests a practical recipe for deletion requests in web-scale graphs: separate the expensive influence computation as a one-time pre-processing step, then let each new deletion request reuse it for fast entity-specific fine-tuning.

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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

Colleague,

Here's my read of the SGU/NIM paper. It deserves a serious referee, but it needs revision before acceptance. The core idea—selecting high-influence entities by a propagation-based score instead of the L-hop neighborhood—is a useful plug-and-play contribution, and the 14-dataset evaluation is substantial.

What is actually new: NIM combines topology influence (gradient of propagated features) with feature influence (gradient of predictions) into a single HIE criterion. That combination is not in the cited literature, and the paper shows it improves forgetting across several existing GU baselines. The SGU fine-tuning losses (label randomization, prototype separation, contrastive learning) are also well-designed for entity-specific updates. Running on ogbn-papers100M is a real strength; many baselines OOM or OOT.

Now the soft spots. The load-bearing premise is that Eqs. (5)-(6) quantify UE-to-node training influence. The paper asserts this in Sec. 3.1.1 without derivation. Reading the equations, they are essentially random-walk reachability plus a classifier Jacobian; they do not involve the labels of the unlearning entity, the loss gradient, or the optimization trajectory. As a heuristic proxy, fine; as a faithful measure of training influence, it is unsupported. The stress-test is right on this. The paper should either derive it under a decoupled linear GNN or explicitly call it a heuristic and justify it empirically.

Second, there is a genuine inconsistency: Definition 3 selects HIE by threshold theta (Eq. 8), while Algorithm 1 selects B nodes by argmax. The experiments tune both B and theta (Fig. 5), so the reader cannot tell which criterion drives the results. That is a fixable but important clarity issue.

Third, MIA numbers in Tables 1 and 5 come without error bars, and hyperparameters are tuned per dataset to report the best performance (App. A.7). That is a selection-bias risk. The large-scale comparisons have many OOM/OOT gaps, so "comprehensive SOTA" is stronger than what the data show.

One more thing: some forgetting is by construction—the losses directly randomize UE labels and separate UE embeddings, so MIA gains over baselines are partly baked in. That is standard in this literature, but "complete knowledge removal" should be tempered; MIA AUC remains above 0.5.

Who is this for: the graph unlearning community and anyone building scalable unlearning pipelines. It is a solid empirical paper with a valuable plug-and-play component. I would accept it for peer review, with a strong expectation of major revision. A serious referee should push on the influence measure's justification, the algorithm/definition mismatch, and error bars.

Referee Report

5 major / 7 minor

Summary. The paper proposes a graph-unlearning framework built on a new high-influence entity (HIE) selection module called Node Influence Maximization (NIM). NIM treats unlearning entities as seeds in an influence-propagation model and scores each non-UE node by the L1 norm of Jacobians of propagated features and predictions (Eqs. (5)-(6)). SGU then fine-tunes the trained GNN with entity-specific losses: random-label cross-entropy, prototype perturbation, contrastive learning, and a memory-based KL term (Eqs. (9)-(12)). The authors claim that NIM is plug-and-play and improves the forgetting capability of existing graph-unlearning methods, and that SGU achieves state-of-the-art forgetting/reasoning while scaling to ogbn-papers100M. The experiments cover 14 datasets, multiple backbones, and transductive, inductive, and link-level settings.

Significance. If the central premise of NIM were established, the paper would make a useful conceptual and practical contribution: it draws a concrete connection between graph unlearning and influence maximization, and the plug-and-play HIE selector is simple enough to retrofit into existing GU pipelines. The experimental breadth is a genuine strength: 14 datasets, transductive/inductive/link settings, and several decoupled and sampling backbones. The paper also reports consistent gains of NIM across multiple HIE-based GU methods (Table 1), which is the most direct evidence for the plug-and-play claim. The main caveat is that the influence measure itself is asserted rather than derived, and the forgetting losses directly target the MIA signal, so the current experiments do not fully separate the effect of NIM from the effect of the fine-tuning objective. With a derivation or validation study and more careful reporting, the contribution would be solid.

major comments (5)

[Sec. 3.1.1, Eqs. (5)-(6)] The influence measures in Eqs. (5)-(6) are asserted rather than derived. For a decoupled linear GNN of the form in Eq. (2), the effect of removing UE u on the trained weights is mediated by the loss term at u and by the curvature of the training objective; it is not determined by the L1 norm of dX_v/dX_u or dY_v/dY_u alone, because those Jacobians are evaluated at the already-trained model and do not involve the labels of u, the loss gradient at u, or the optimization trajectory. Since NIM's selected HIE are the input to all SGU losses (Eqs. (9)-(12)), this is a load-bearing assumption. Please either provide a derivation linking Eqs. (5)-(6) to leave-one-out influence for a linear GNN, or add a validation experiment on small graphs (e.g., compare NIM-selected HIE against actual retraining-based influence). Without this, the claim that NIM identifies nodes that carry UE knowledge is not established.
[Sec. 3.1.2 (Def. 3) and Algorithm 1] Definition 3 (Eq. (8)) defines HIE as the set of nodes whose maximum influence from the UE seed set exceeds threshold theta. Algorithm 1, however, repeatedly selects the single node with the largest influence score until B nodes are collected (lines 11-17) and never uses theta. Thresholding and top-B selection are not equivalent rules, and App. A.7 reports searching theta in [0.5,1] while the algorithm uses B=3|DeltaV|. The reader cannot tell which criterion produced Tables 1-5 and Fig. 5. Please reconcile the formal definition with the implementation, and specify how theta and B interact (e.g., threshold then cap, or top-B with theta used only for sensitivity analysis).
[Sec. 3.2.1, Tables 1 and 5] The forgetting losses directly remove the signals measured by the MIA: L1_f randomly shuffles UE labels, L2_f pushes UE embeddings away from class prototypes, and L3_f uses UE as negative samples. Low MIA AUC is therefore partially a consequence of the loss design, not necessarily of NIM's HIE selection. The paper should add a control experiment in which the SGU fine-tuning losses are held fixed and only the HIE source is changed (NIM vs L-hop neighborhood vs random nodes), and should report an additional membership-evaluation metric or a held-out UE set. This would separate the contribution of NIM from the contribution of the objective itself.
[Tables 1, 5 and App. A.7] MIA results in Tables 1, 5, and 9 are reported without standard deviations, even though App. A.7 states that each experiment is repeated 10 times. The differences are small (e.g., Table 1 improvements of 4-5% on AUC values around 0.53-0.61), so without variance or significance testing the improvements may be within noise. In addition, App. A.7 says hyperparameters (B, theta, lambda, k) are searched per dataset/backbone to report the best performance; this selection protocol should be applied identically to baselines, and the chosen values should be reported in the main text or appendix. Please add error bars or per-seed values and a fixed validation-based hyperparameter protocol.
[Table 3] Table 3 contains many OOM, OOT, or missing entries for baselines on arxiv, products, papers100M, Flickr, and Reddit. As a result, the comprehensive SOTA and scalability claim on large graphs rests on a small number of completed comparisons (notably SGU vs ScaleGUN, GNNDelete, and MEGU on papers100M). Please report runtime and peak memory for all completed cells, state the resource limits that caused OOM/OOT, and clarify whether the missing entries mean the baselines could not run at all under the reported environment.

minor comments (7)

[Eq. (7)] The product notation in the random-walk expression lacks explicit limits and appears inconsistent with the definition of P_{v->u}^k; please rewrite with i=1..k.
[Definitions 1 and 2] The phrase 'expected Jacobin matrix' should be 'Jacobian', and the expectation and the exact L1 matrix norm should be specified.
[Eq. (4)] The notation overloads X_tilde for both the aggregated propagated features and per-layer quantities, and Y_tilde^{(k)} in Eq. (6) is never defined; please align the notation.
[Algorithm 1] Line 3 names S the 'seed set (HIE)', although HIE is the activated set; line 16 also uses u without a definition. Please rename and clarify.
[Table 6] The table uses '!' and '%' symbols without a legend; please define them in the caption.
[App. A.6 / Table 6] The table lists 'GCU [9]', but reference [9] is CGU; please correct the citation.
[Reproducibility] No code or data link is provided; please include a reproducibility artifact.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: NIM's influence scores are a modeling choice and SGU's reported gains are empirical, not derived from the evaluation metrics.

full rationale

The derivation chain is: NIM defines influence via Jacobians of propagated features and predictions (Eqs. 5-6), selects HIE by threshold or budget (Eq. 8 and Algorithm 1), and fine-tunes UE/HIE with entity-specific losses (Eqs. 9-12). None of these steps is defined in terms of the reported metrics (MIA AUC, F1, link-prediction HR/MRR), and no equation reduces to its own input by construction. The claim that Eqs. (5)-(6) faithfully capture training-time UE influence is an asserted modeling assumption rather than a tautology; the paper itself leaves 'GU-tailored influence propagation model and quantification functions' as future work, which is a limitation but not circularity. The paper does cite prior work by overlapping authors (e.g., MEGU [29], Rethinking Node-wise Propagation [28]), but these citations are baselines and background support, not a load-bearing uniqueness theorem or the source of the main result. The random-label forgetting loss L1_f directly randomizes UE labels, so MIA on UE is aligned with the unlearning objective; this is the standard definition of the task, and the comparative claims (NIM vs L-hop HIE selection, SGU vs baselines) are empirical findings evaluated against external benchmarks rather than definitional equivalences. The inconsistency between threshold-based Definition 3 and argmax-based Algorithm 1 is a clarity issue, not circularity. Overall, the central derivation is self-contained and the reported improvements are empirical, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method is primarily empirical; its central mechanism depends on unstated modeling assumptions about what influence means. The free parameters theta, B, lambda, and k are fitted to data, which limits the strength of any causal claim about the method's components.

free parameters (4)

Activation threshold theta = searched in [0.5, 1] per dataset
Determines which Non-UE nodes are selected as HIE in Eq. (8); higher theta means fewer nodes. Tuned to report best performance (App. A.7).
Unlearning budget B = default 3 x |Delta V|
Controls the number of HIE; set to three times the UE set size, with per-dataset tuning implied by sensitivity analysis.
Loss balance lambda = searched in [0, 1]
Balances forgetting loss and reasoning loss in L = lambda L_f + (1-lambda) L_p; tuned per dataset (App. A.7, A.11).
Propagation steps k = searched in [3, 10]
Number of graph propagation steps in Eq. (4); searched per dataset (App. A.7).

assumptions (4)

domain assumption The L1 norm of the expected Jacobian of propagated features (Eq. 5) is a valid measure of the topology influence of a node.
Stated in Sec. 3.1.1 without derivation or empirical validation that this Jacobian captures training-time influence.
domain assumption The L1 norm of the expected Jacobian of model predictions (Eq. 6) is a valid measure of feature influence, i.e., gradient-driven label supervision.
Stated in Sec. 3.1.1; no proof that this quantity corresponds to the influence of UE on optimization.
domain assumption Nodes with influence above threshold theta are exactly the high-influence entities whose knowledge must be removed.
Definition 3 in Sec. 3.1.2; this equivalence is assumed, not proven.
domain assumption Decoupled weight-free graph propagation (Eq. 4) adequately represents the message-passing of the GNN for the purpose of influence quantification.
Invoked for all backbones; for non-decoupled GNNs the paper adds a propagation pass, assuming the learned weights do not change the influence structure.

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Cite this review

Pith. "Pith review of Toward Scalable Graph Unlearning: A Node Influence Maximization based Approach." pith.science (2026). https://pith.science/paper/4BQG55S2

@misc{pith2026250111823,
  author       = {Pith},
  title        = {Pith review of: Toward Scalable Graph Unlearning: A Node Influence Maximization based Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BQG55S2}},
  note         = {Machine review of arXiv:2501.11823}
}

read the original abstract

Machine unlearning, as a pivotal technology for enhancing model robustness and data privacy, has garnered significant attention in prevalent web mining applications, especially in thriving graph-based scenarios. However, most existing graph unlearning (GU) approaches face significant challenges due to the intricate interactions among web-scale graph elements during the model training: (1) The gradient-driven node entanglement hinders the complete knowledge removal in response to unlearning requests; (2) The billion-level graph elements in the web scenarios present inevitable scalability issues. To break the above limitations, we open up a new perspective by drawing a connection between GU and conventional social influence maximization. To this end, we propose Node Influence Maximization (NIM) through the decoupled influence propagation model and fine-grained influence function in a scalable manner, which is crafted to be a plug-and-play strategy to identify potential nodes affected by unlearning entities. This approach enables offline execution independent of GU, allowing it to be seamlessly integrated into most GU methods to improve their unlearning performance. Based on this, we introduce Scalable Graph Unlearning (SGU) as a new fine-tuned framework, which balances the forgetting and reasoning capability of the unlearned model by entity-specific optimizations. Extensive experiments on 14 datasets, including large-scale ogbn-papers100M, have demonstrated the effectiveness of our approach. Specifically, NIM enhances the forgetting capability of most GU methods, while SGU achieves comprehensive SOTA performance and maintains scalability.

Figures

Figures reproduced from arXiv: 2501.11823 by the authors.

**Figure 2.** Overview of our proposed SGU framework. 3.1 Node Influence Maximization The message-passing mechanism in most GNNs creates intricate interactions among graph entities during training, which results in a complex and unquantifiable gradient flow. This poses a significant challenge in completely removing the model knowledge contributed by UE. Recent studies have observed that GNNs primarily derive their benefits from… view at source ↗

**Figure 3.** Node-level performance on arxiv within Edge Attack. The x-axis is the ratio of noisy edges to the existing edges. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Predictive performance of node unlearning within different ratios on PPI. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Sensitive analysis within GAMLP and node removal. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Running efficiency within S2GC and node removal. 4.4 Robustness Analysis To answer Q4, we first evaluate SGU and the two most competitive baselines across varying unlearning scales in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Predictive performance of feature unlearning within different ratios on Cora. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Predictive performance of edge unlearning within different ratios on Photo. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: Sensitivity analysis on entity-specify optimization objectives. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

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