DiffIM: Differentiable Influence Minimization with Surrogate Modeling and Continuous Relaxation

Fanchen Bu; Hyunju Kim; Jihoon Ko; Junghun Lee; Kijung Shin

REVIEW 4 major objections 4 minor 54 references

DiffIM: Differentiable Influence Minimization with Surrogate Modeling and Continuous Relaxation

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that influence minimization—choosing which edges to delete so that a seed set's expected spread drops the most—can be solved far faster with differentiable learning than with discrete combinatorial search, and…

desk verdict A genuinely novel differentiable IMIN pipeline, but the speed and Pareto-optimality claims rest on an unequal hardware comparison and an unvalidated surrogate under edge removal. read the letter →

arxiv 2502.01031 v1 pith:6CLZWDJL submitted 2025-02-03 cs.LG cs.SI

Junghun Lee , Hyunju Kim , Fanchen Bu , Jihoon Ko , Kijung Shin This is my paper · ORCID

classification cs.LGcs.SI MSC 68T0768R1068Q25

keywords influenceminimizationindependentcascademodelgraphneuralnetworksurrogatecontinuousrelaxationgradient-drivenedgeselectionrumorblockingdifferentiableoptimization

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 claims that influence minimization—choosing a small set of edges to delete so that a seed set's expected spread drops the most—can be solved far faster with differentiable learning than with discrete combinatorial search. It proposes DiffIM, which trains a graph neural network to act as a surrogate for Monte Carlo influence estimation, relaxes edge-deletion decisions from binary to continuous, and then uses gradients to pick which edges to remove. On three real-world graphs and several diffusion models, the method is reported to be Pareto-optimal against existing baselines and orders of magnitude faster (up to 15,160×) while matching or improving effectiveness. If the claim holds, time-critical applications like rumor or epidemic blocking gain a practical tool where delays previously made exact or greedy approaches infeasible.

What carries the argument

The load-bearing object is the trained GNN surrogate $\mathrm{GNN}_\theta(v; G, p, S)$, which maps a graph with edge weights $p$ and a one-hot seed indicator to an estimate of each node's influenced probability. Because edge-removal decisions enter through modified weights $\tilde p_{\tilde r}(u,v) = p(u,v)\,\tilde r(u,v)$, the network's output is differentiable with respect to the keep-probabilities $\tilde r$, which is what makes continuous relaxation and gradient-based selection possible. The loss $\mathcal{L}_O = \mathcal{L}_{\mathrm{obj}} + \alpha \mathcal{L}_{\mathrm{budget}} + \beta \mathcal{L}_{\mathrm{certainty}}$ ties the relaxed decisions to the original objective under a budget while pushing them toward binary values.

What would settle it

A reader could test the key assumption by taking a held-out graph, computing the surrogate's per-edge gradient ranking, and comparing it against the true marginal influence reduction from Monte Carlo after each edge removal in sequence; if the ranking correlation degrades sharply after the first few deletions, the gradient-driven and relaxed-optimization variants lose their grounding.

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Extended reading notes

Core claim

The central discovery is that the NP-hard, non-submodular edge-removal problem under the independent cascade model can be approached as a differentiable optimization, contrary to the field's discrete-combinatorial default. DiffIM trains a GNN to predict per-node influenced probabilities given a graph, activation probabilities, and a seed set; replaces each binary keep/remove decision with a continuous keep-probability; and defines a loss whose objective is the estimated influence-reduction ratio, whose budget term enforces exactly b removals, and whose certainty term pushes decisions toward 0/1. A third variant, DiffIM++, skips iterative optimization entirely and removes the edge whose gradient is largest. The authors show empirically that all three schemes accelerate the greedy selection loop, with the full pipeline remaining effective when trained and tested on different graphs.

Load-bearing premise

The entire pipeline rests on the surrogate GNN, trained only on intact graphs, giving trustworthy influence estimates—and more importantly trustworthy gradient rankings—on graphs whose edges have already been deleted or whose probabilities have been rescaled.

Editorial extensions

If this is right

If the results hold, influence minimization becomes practical for time-critical settings: the reported speed-ups place decision times well within minutes on graphs where Monte-Carlo-based greedy methods take hours or time out.
The surrogate's once-per-graph training cost is amortized over many seed sets and even across different graphs, since the method retains most of its effectiveness when trained and tested on separate networks.
The same differentiable pipeline transfers to other diffusion models such as linear threshold and G-SIR, so the approach is not tied to independent cascade.
Gradient-driven selection (DIFFIM++) offers near-instant edge recommendations, which could be used for adaptive blocking as an outbreak evolves.

Reading between the lines

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

A natural, untested extension is to use the same surrogate plus continuous relaxation for influence maximization (choosing seed sets), which the authors list as future work but do not evaluate.
The approach implicitly assumes that the gradient ranking on the original graph approximates the true marginal benefit of edge deletion on the progressively sparser graph; if that assumption fails on networks with highly heterogeneous local structure, the speed advantage would come at a larger effectiveness cost.
A quantitative test of gradient quality—comparing surrogate-predicted edge sensitivities with Monte-Carlo marginal reductions after sequential removals—would sharpen the paper's validation and could also serve as a training-time diagnostic.

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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

DiffIM is a genuinely new way to attack influence minimization—treating edge removal as a differentiable learning problem rather than discrete search—but the headline speed and Pareto-optimality claims are not established as stated because the comparison runs on different hardware.

What's new: the three-scheme pipeline (GNN surrogate, continuous relaxation, gradient-driven selection) is, as far as I can tell, the first differentiable formulation of IMIN. The GNN surrogate is adapted from MONSTOR, but the combination is novel. The experiments are thorough: three real datasets, three diffusion models, budgets 3–10, plus large-scale graphs, and the code is public. Lemma 2 gives a reasonable guarantee for the relaxed loss when the surrogate is accurate and the loss weights are large.

Soft spots, in order of importance. First, Appendix F discloses that all DiffIM versions ran on a Xeon Silver with an RTX2080Ti, while every baseline ran on a faster i9-10900KF CPU with no GPU. That means the 30×–15,160× speedups are wall-clock comparisons that bundle together algorithmic gains and hardware gains. The Pareto-optimality claim—'no baseline is faster and more effective'—is therefore not supported as a property of the method; it's a property of these particular runs. A GPU-accelerated MC baseline, or a CPU-only DiffIM, could plausibly change the frontier.

Second, the surrogate is trained only on the intact graph and validated on unmodified validation graphs (Fig. 3, r≥0.999). But DIFFIM+ and DIFFIM++ operate on graphs that have already been modified—edges removed, probabilities rescaled. The paper never checks whether the surrogate's per-node estimates or gradient rankings remain reliable in that regime. That's exactly the regime where the method lives.

Third, a small but real typo: L_certainty in Eq. (1) is written with 'log r(e)' where it should presumably be 'log r̃(e)'. The intended entropy-like loss is clear, but the printed expression is garbled.

Fourth, the gradient-driven selection (DIFFIM++) is a heuristic—the paper doesn't demonstrate that gradient magnitude actually tracks marginal reduction in influence. Its good empirical performance is encouraging, but the reason it works is not fully pinned down.

None of this is fatal. The effectiveness numbers are strong and consistent across datasets and budgets, and the differentiable formulation is worth serious attention. The paper just overreaches in its headline speed claim, and the surrogate-under-removal question deserves a direct experiment before the method is fully trusted.

Who should read it: anyone working on influence minimization, graph neural network surrogates, or learned combinatorial optimization. It deserves a serious referee; a good reviewer should push for equal-hardware baselines and an edge-removal validation study.

Referee Report

4 major / 4 minor

Summary. The paper proposes DiffIM, a learning-based method for influence minimization (IMIN) via edge removal under the independent cascade (IC) model. It trains a GNN surrogate on Monte Carlo influence estimates and then selects edges to remove using three variants: DIFFIM (greedy selection with GNN-estimated influence), DIFFIM+ (continuous relaxation of edge-removal decisions optimized by gradient descent with a loss combining influence reduction, budget, and certainty terms), and DIFFIM++ (one-shot selection of edges with largest gradients). Experiments on real-world graphs WC, CL, ET, on LT and G-SIR models, and on large-scale graphs report that all versions are Pareto-optimal against ten baselines, with speedups up to 15,160x. The paper also includes an NP-hardness proof, complexity analysis, ablation studies, and inductive experiments.

Significance. If the empirical claims survive scrutiny, this is a valuable contribution: it is, to my knowledge, the first differentiable-learning treatment of IMIN, it offers a practical way to avoid expensive Monte Carlo simulation at test time, and the three schemes form a clean speed-effectiveness spectrum. The paper ships code and datasets, which aids reproducibility. The central limitation is that the headline speed and Pareto-optimality claims depend on an uneven hardware comparison and on unvalidated surrogate behavior on modified graphs; these need to be addressed before the claims can be accepted as stated.

major comments (4)

[Appendix F; Fig. 1; abstract] The speed comparison is confounded by hardware: all DIFFIM variants ran on a machine with 2.10GHz Intel Xeon Silver 4210R processors and an RTX2080Ti GPU, while all baselines ran on a machine with a faster 3.70GHz Intel Core i9-10900KF CPU and no GPU. Since the claimed speedups and the Pareto-optimality statement are wall-clock comparisons, they do not establish that the method is faster as an algorithm; they only compare a GPU implementation against CPU implementations. Please provide either CPU-only runs of DIFFIM variants, GPU-accelerated MC baselines (e.g., GREEDY or MBPM batched on GPU), or a hardware-normalized analysis, and restate the Pareto claim based on that comparison.
[Sec. 5.2, Alg. 1, Fig. 3] The surrogate is trained only on intact graphs with varied seed sets, but at test time DIFFIM, DIFFIM+, and DIFFIM++ evaluate the surrogate on graphs from which edges have already been removed, and DIFFIM+ also feeds rescaled edge probabilities into the GNN. Fig. 3 validates the surrogate only by correlating total influence estimates with MC on unmodified validation graphs. Please add validation in the operating regime: for example, compare GNN-estimated total influence and per-edge marginal reductions with MC on graphs with 1 to b randomly removed edges and on probability-rescaled graphs. This is needed to support the claim that the selection driven by GNN gradients or greedy scores is reliable.
[Sec. 5.3, Eq. (1), Lemma 2] The displayed L_certainty is not the Shannon entropy and is not minimized at binary r̃. For r̃=0.5 the expression is 0, and for r̃=0.1 (natural log) it is approximately -0.135, so it can be negative and its minimizer on [0,1] is interior. This contradicts the text stating it is inspired by Shannon entropy and is smaller when each r̃(e) is closer to 0 or 1, and Lemma 2's discreteness conclusion does not follow from this formula. If the code implements the intended entropy, please correct the formula and the missing parenthesis in 'logr(e)'; if not, the DIFFIM+ loss and the ablation in Table 6 need to be re-evaluated. Given Table 6 shows L_certainty has only a small effect, the corrected loss is unlikely to change the main empirical conclusions, but the inconsistency must be fixed.
[Sec. 5.3, Alg. 1, Sec. 6.1] The budget handling in DIFFIM+ is not fully specified. EdgeSelection+ optimizes L_budget = (|E| - Σ_e r̃(e) - b)^2, but after each removal the graph shrinks while Algorithm 1 line 7 keeps b fixed; the initialization r̃(e)=1-b/|E| makes L_budget zero only on the initial graph. Please clarify whether r̃ is reinitialized in each round, whether b is decremented to reflect the remaining budget, and how the budget term behaves after the first removal. This is necessary for reproducibility of DIFFIM+.

minor comments (4)

[Fig. 1 caption] The caption says the running time of the best baseline is compared with 'one of our methods with the most similar reduced ratio'; this wording is ambiguous. Please specify exactly which baseline and which DIFFIM version are used for each reported speed-up, and give the underlying numbers in the appendix or table.
[Table 7] There are formatting typos in the WC column: '0.1641 (2315)' and '0.3545 (2243)' should presumably read '0.1641 (0.2315)' and '0.3545 (0.2243)'.
[Appendix G.3, Fig. 7 caption] The text says the together variant of DIFFIM++ has lower effectiveness than 'the original DIFFIM+'; this should presumably say 'the original DIFFIM++'.
[Appendix A.1, proof of Theorem 1] The reduction would be easier to verify if the budget were stated explicitly as b = m - k and the seed set as S = {v_init}; currently the construction is described only in prose.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the surrogate is trained on MC ground truth and final effectiveness is evaluated against the same external MC metric, not against the surrogate's own predictions.

full rationale

The paper's derivation chain is not circular. The surrogate GNN is trained by supervised regression on Monte Carlo influence estimates (Alg. 3: 'Obtain π(v; G, Si, p) by MC... Update θ w.r.t. L(GNNθ; π̃, S, G, p)'), and the final effectiveness metric is the reduced ratio Rr defined in Eq. (2) in terms of σ(S; G, p), which is computed by Monte Carlo evaluation on the actually removed-edge graphs. Thus the edge-selection outcome is judged by an external stochastic simulation benchmark, not by the surrogate. The surrogate could be inaccurate under edge removal, and Appendix F discloses that DIFFIM ran on a GPU machine while baselines ran on faster CPUs without GPUs; both are experimental/validity concerns, not cases where a prediction reduces by construction to its input. The MONSTOR citation (Ko et al. 2020) is a self-citation with overlapping authors, but it is not load-bearing: the paper independently trains and validates its own GCN surrogate (Fig. 3, Pearson r >= 0.999 on validation), and the differentiable IMIN optimization is new relative to MONSTOR's influence-estimation/maximization objective. No uniqueness theorem, ansatz, or renaming is imported from the authors' prior work. The continuous relaxation and gradient-driven selection are algorithm designs whose reported effectiveness is still measured against MC, so they do not constitute fitted-input-called-prediction circularity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The core contribution rests on three empirical premises: the IC model, the transferability of a trained GNN to edge-modified graphs, and the validity of surrogate gradients as edge-importance scores. The first is standard; the last two are asserted rather than demonstrated.

free parameters (6)

alpha (loss coefficient for budget term) = 0.1
Chosen hyperparameter in DIFFIM+ loss (Eq. 1); no sensitivity analysis across values shown.
beta (loss coefficient for certainty term) = 1.0
Chosen hyperparameter in DIFFIM+ loss (Eq. 1); ablation shows marginal effect.
nep (gradient epochs per removal) = 100
Number of gradient descent steps per edge removal in DIFFIM+ (Sec. 6.1).
MC simulations for ground truth = 10,000
Number of Monte Carlo runs used as ground-truth influence for training and evaluation; standard but approximate.
GNN hyperparameters = tuned via Optuna
Learning rate, decay, hidden dimensions, and six-layer GCN architecture chosen by hyperparameter search, not derived.
time split threshold t_th = not reported
Determines which edges form training vs test graphs; exact threshold omitted.

assumptions (4)

domain assumption Independent cascade model with edge activation probabilities p correctly models the influence process.
Problem formulation and all experiments rely on this model (Sec. 2, Definition 1).
ad hoc to paper The GNN surrogate, trained on intact graphs, generalizes to graphs with removed edges or rescaled probabilities.
All three DiffIM variants evaluate edge removals through the surrogate, but validation only covers unmodified graphs (Sec. 5.2, Fig. 3).
ad hoc to paper The gradient of the surrogate's total influence with respect to an edge's keep-probability ranks the true marginal benefit of removing that edge.
DIFFIM++ selects edges by largest gradient (Sec. 5.4) without theoretical or direct empirical justification.
domain assumption 10,000 Monte Carlo simulations approximate the exact expected influence well enough to serve as ground truth.
Used for training labels and evaluation metric (Sec. 6.1), following standard practice.

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Cite this review

Pith. "Pith review of DiffIM: Differentiable Influence Minimization with Surrogate Modeling and Continuous Relaxation." pith.science (2026). https://pith.science/paper/6CLZWDJL

@misc{pith2026250201031,
  author       = {Pith},
  title        = {Pith review of: DiffIM: Differentiable Influence Minimization with Surrogate Modeling and Continuous Relaxation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6CLZWDJL}},
  note         = {Machine review of arXiv:2502.01031}
}

read the original abstract

In social networks, people influence each other through social links, which can be represented as propagation among nodes in graphs. Influence minimization (IMIN) is the problem of manipulating the structures of an input graph (e.g., removing edges) to reduce the propagation among nodes. IMIN can represent time-critical real-world applications, such as rumor blocking, but IMIN is theoretically difficult and computationally expensive. Moreover, the discrete nature of IMIN hinders the usage of powerful machine learning techniques, which requires differentiable computation. In this work, we propose DiffIM, a novel method for IMIN with two differentiable schemes for acceleration: (1) surrogate modeling for efficient influence estimation, which avoids time-consuming simulations (e.g., Monte Carlo), and (2) the continuous relaxation of decisions, which avoids the evaluation of individual discrete decisions (e.g., removing an edge). We further propose a third accelerating scheme, gradient-driven selection, that chooses edges instantly based on gradients without optimization (spec., gradient descent iterations) on each test instance. Through extensive experiments on real-world graphs, we show that each proposed scheme significantly improves speed with little (or even no) IMIN performance degradation. Our method is Pareto-optimal (i.e., no baseline is faster and more effective than it) and typically several orders of magnitude (spec., up to 15,160X) faster than the most effective baseline while being more effective.

Figures

Figures reproduced from arXiv: 2502.01031 by the authors.

**Figure 2.** The running time of each DIFFIM version when budget b increases from 1 to 10. The running time of each version grew linearly with b, showing good scalability [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The Pearson correlation coefficients between the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The effectiveness (the reduced ratio of influence) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: The effectiveness (the reduced ratio of influence) and running time of each method, with budget [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 8.** Figure 8: The average validation estimation errors in estimat [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 6.** Figure 6: The effectiveness of the variant (selecting edges [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: The effectiveness of the variant (selecting edges [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Akiba, T.; Sano, S.; Yanase, T.; Ohta, T.; and Koyama, M. 2019. Optuna: A Next-generation Hyperparameter Optimization Framework. In KDD

work page 2019

[4] [4]

Blakely, D.; Lanchantin, J.; and Qi, Y. 2021. Time and space complexity of graph convolutional networks. Technical report, University of Virginia

work page 2021

[5] [5]

Borgs, C.; Brautbar, M.; Chayes, J.; and Lucier, B. 2014. Maximizing social influence in nearly optimal time. In SODA

work page 2014

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Budak, C.; Agrawal, D.; and El Abbadi, A. 2011. Limiting the spread of misinformation in social networks. In WWW

work page 2011

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Chang, C.-W.; Yeh, M.-Y.; and Chuang, K.-T. 2016. On the guarantee of containment probability in influence minimization. In ASONAM

work page 2016

[8] [8]

Chen, W.; Wang, C.; and Wang, Y. 2010. Scalable influence maximization for prevalent viral marketing in large-scale social networks. In KDD

work page 2010