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REVIEW 4 major objections 5 minor 40 references

Influence Maximization Considering Influence, Cost and Time

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A tri-objective influence maximization problem—maximizing spread while minimizing seed cost and propagation time—is NP-hard, and a variable-length evolutionary algorithm with embedding-aligned crossover finds better Pareto fronts than NSGA-

desk verdict EVEA is a plausible variable-length MOEA variant with a missing experimental protocol and an advertised NP-hardness proof that never appears; the core idea is worth testing once the paper is reproducible. read the letter →

arxiv 2509.07625 v1 pith:HMDKTXQU submitted 2025-09-09 cs.SI cs.NI

Mingyang Feng , Qi Zhao , Shan He , Yuhui Shi This is my paper · ORCID
classification cs.SIcs.NI
keywords InfluenceMaximizationMulti-objectiveOptimizationEvolutionaryAlgorithmVariable-lengthrepresentationGraphembeddingcrossoverPropagationlatencySeedcostNSGA-II
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 introduces IM-ICT, a multi-objective influence maximization formulation that optimizes three conflicting goals at once: maximize the expected number of activated nodes, minimize the total cost of seed nodes, and minimize propagation time under a latency-aware independent cascade model. The authors argue, with a small worked example and Pareto-surface experiments on four networks, that these objectives genuinely conflict, so a single-objective or two-objective treatment misses real decisions. They prove IM-ICT is NP-hard by a polynomial-time reduction from Set Cover, and they propose EVEA, an evolutionary algorithm that lets seed sets grow or shrink during search and uses graph-embedding-based crossover to recombine parents of different lengths. On Facebook, GRQC, Gnutella, and Wiki networks, EVEA reports higher hypervolume (up to 19.3% over the best baseline) and faster convergence (25–40%), plus smoother Pareto fronts. If these results hold, the paper's main contribution is a tractable, expressive formulation and a practical search method for decisions that must weigh reach, budget, and urgency together.

What carries the argument

The load-bearing mechanism is the embedding-aligned crossover: it aligns nodes across two parent seed sets by minimal Euclidean distance in a Node2Vec embedding, then swaps aligned nodes with crossover probability pc, allowing recombination of seed sets of different sizes while keeping topologically similar nodes. This is supported by variable-length mutation (add, delete, replace) and NSGA-II-style non-dominated sorting with crowding distance for environmental selection. The objective functions themselves—expected spread, total seed cost, and longest activation path—are the formal objects the algorithm optimizes; the complexity argument is a polynomial reduction from Set Cover.

What would settle it

Run EVEA and the three NSGA-II variants on the same four SNAP networks with every setting fixed and published (for instance, costs proportional to degree, p_uv=0.1 for all edges, delays sampled from an exponential with mean one time step, 10,000 Monte Carlo simulations, and a common normalized reference point). If EVEA's hypervolume advantage over NSGA-II+Mutation does not pass the Wilcoxon significance threshold on the majority of networks, or its convergence curves do not lead by the claimed 25–40%, the central empirical claim fails.

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Extended reading notes

Core claim

The central claim is that influence maximization should be treated as a three-objective problem—influence spread, seed cost, and propagation time—and that this problem, while NP-hard, is practically solvable by a variable-length evolutionary algorithm. The authors define the objectives formally (expected spread under the IC model; sum of node costs; maximum activation time under a latency-aware IC model), prove NP-hardness via a reduction from Set Cover, and design EVEA, whose key mechanism is an embedding-aligned crossover that pairs nodes from two parents by Euclidean distance in Node2Vec embedding space and swaps them with probability pc. EVEA then applies add/delete/replace mutation so s

Load-bearing premise

The empirical superiority claim rests on experimental settings that are not fully specified in the paper—costs, propagation probabilities, delay distributions, Monte Carlo effort, and hypervolume normalization—so the reported margins are not yet independently reproducible.

Editorial extensions

If this is right

  • Any campaign that must respect a budget and a deadline—marketing, vaccination, rumor containment—gets a Pareto front of seed sets rather than one fixed k, so decision makers can pick the point matching their budget and deadline.
  • Variable-length representation means seed-set cardinality is not chosen in advance; the search itself reveals which sizes are worth paying for.
  • If the convergence numbers hold, EVEA's embedding-aligned crossover is a generally usable operator for variable-length multi-objective search on graphs, not just IM-ICT.
  • The NP-hardness result formalizes that no polynomial exact algorithm is likely, so evolutionary and heuristic approaches are the relevant class of methods.

Reading between the lines

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

  • An editorially added caution: reproducing Table 2 requires choices the paper does not state—seed cost assignment, edge probabilities, delay sampling distribution, Monte Carlo simulation count, and the reference point or normalization behind hypervolume values near one. Different choices could change the margins.
  • The embedding-alignment idea could be tested on other variable-length combinatorial problems, such as team formation or feature selection, where parent solutions have different sizes and topological structure matters.
  • A natural extension implied by the paper's own future-work remark is coupling the time objective with node geography; the current model treats time as hop-based only.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes IM-ICT, a tri-objective influence maximization problem that simultaneously maximizes influence spread, minimizes seed cost, and minimizes propagation latency, and presents EVEA, an embedding-aligned variable-length evolutionary algorithm. The authors claim that IM-ICT is NP-hard via a reduction from Set Cover, and that EVEA outperforms three NSGA-II variants on four real-world networks, reporting higher hypervolume and faster convergence. The manuscript includes objective definitions, algorithm pseudocode, and experimental tables and figures. The central claims are that the new formulation is well motivated and that EVEA is empirically superior, but both are currently under-supported: the advertised NP-hardness proof is absent, and the experimental protocol is not specified enough to reproduce the reported hypervolume results.

Significance. If the stated results hold, the paper would make a useful contribution to multi-objective influence maximization by adding propagation time as a third objective and by demonstrating a variable-length EA with an embedding-based crossover. The source-code release is a positive element, as are the comparisons against three NSGA-II variants with Wilcoxon significance tests. However, the significance is presently limited by missing technical support: the NP-hardness proof is promised but not delivered, and several load-bearing experimental parameters (seed costs, propagation probabilities, delay distribution, Monte Carlo effort, hypervolume reference and normalization) are not stated. These gaps prevent the reader from validating the main empirical claim and reduce the paper's current value to that of an algorithm proposal with encouraging but incomplete evidence.

major comments (4)
  1. [§1, §3.2; contribution 1] The Introduction and the contribution list state that the paper "rigorously prove[s] the proposed IM-ICT's NP-hardness via a polynomial-time reduction from the Set Cover problem." Section 3, which is the designated complexity section, contains only the objective definitions and formulation (7); no theorem, reduction, or proof appears anywhere in the manuscript. Since NP-hardness is advertised as an original contribution, this is missing support. The authors should either provide the full reduction and proof or remove the claim.
  2. [§5.1, §5.3, Eqs. (4)–(6), Table 2] The central empirical claim that EVEA outperforms all baselines is not reproducible from the information given. The values of c_i in Eq. (5), the edge probabilities p_uv in Eq. (4), the delay distribution P_lat(δt) in Eq. (6), the number of Monte Carlo simulations used to estimate σ(S), and the hypervolume reference point and normalization are all unspecified. For three objectives confined to [0,1] with the origin as reference, HV cannot exceed 1, yet Table 2 reports values from 1.0920 to 1.1659, implying normalization or a different reference that is not disclosed. These choices can change the Pareto fronts, the HV margins, and the Wilcoxon outcomes. The authors must report the full evaluation protocol.
  3. [§5.1, §5.3] The three baseline variants of NSGA-II are not instantiated precisely enough for a fair comparison. The paper cites [35] and [36] for variable-length crossover and mutation, but does not specify how those operators are adapted to IM-ICT, what hyperparameters are used for all algorithms (population size, crossover/mutation rates, embedding settings for EVEA), how many independent runs are averaged, or what exact Wilcoxon procedure is used. These details are necessary to judge whether the reported 5–19% HV improvements are robust or artifacts of asymmetric settings.
  4. [§5.2] The "necessity" of the IM-ICT formulation is argued by showing Pareto surfaces generated by EVEA on the four networks. This is not independent evidence that the three objectives conflict; it only shows that EVEA returns a spread of solutions under its own objective definitions. A stronger demonstration would compare against a set of random or heuristic seed sets, or show that no single solution dominates on all three objectives across a broader sampling of the search space. As written, the empirical motivation is partly circular because the solver that is being evaluated is also used to establish the problem's structure.
minor comments (5)
  1. [§5.1, Table 1] The text says "four benchmark datasets - Facebook, GRQC, and Gnutella," but lists only three, while Table 1 and Table 2 include Wiki as the fourth. Additionally, the abstract mentions Twitter and DBLP, which do not appear in the experiments. These inconsistencies should be fixed.
  2. [§4.1, Algorithm 1] Line 5 reads "Q←Embedding−Aligned_Crossover(Q)" but Q has not been initialized and the two parent arguments are not shown. The algorithm should pass the selected parents P_G (or a mating pool) explicitly to the crossover routine.
  3. [Eq. (6)] The notation for propagation latency is ambiguous: δt_k is sampled from P_lat_u(δt), but the subscript u and the summation range d(v_i,v_j) are not defined clearly, and P_lat_u is never specified as a distribution over hops or nodes. The authors should define the delay model more rigorously.
  4. [§4.4] The time-complexity analysis claims O(N^2) per generation but ignores the cost of evaluating σ(S), which typically requires many Monte Carlo simulations and dominates the runtime in IM problems. The complexity statement should clarify whether evaluation cost is included or excluded.
  5. [References] Reference [27] is incomplete ("Zhu et al." with no full author list or title), and several other references have informal annotations appended inside the bibliography. The reference list needs to be brought to journal style.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: EVEA's central empirical claim is benchmarked against external baselines, and IM-ICT is a stated objective formulation rather than a fit of its own outputs.

full rationale

The paper's central claims are (i) IM-ICT is a new tri-objective IM problem, (ii) EVEA outperforms NSGA-II and variants in hypervolume, and (iii) IM-ICT is NP-hard. No circularity is present in the derivation chain: the problem formulation in Eqs. (4)-(7) is a definition, not a self-referential reduction; the experimental comparison (Table 2) uses independent baseline algorithms and external datasets; and no fitted parameter is relabeled as a prediction. The only quasi-self-referential passage is Section 5.2, where Pareto fronts generated by EVEA are cited as evidence of the 'necessity' of Pareto-based approaches. That is an empirical demonstration rather than a formal derivation: the Pareto front is a property of the objectives, so the passage is weak justification but not a circular reduction. Two non-circular problems should be weighed separately: the Introduction and Contribution 1 advertise a rigorous NP-hardness proof via Set Cover reduction, but no such reduction appears anywhere in Section 3, an omitted proof; and the experimental protocol in Section 5.1 leaves unspecified the seed costs c_i, propagation probabilities p_uv, delay distribution P_lat, number of Monte Carlo runs, and HV reference point/normalization (the reported HV values exceed 1.0 in a three-objective space), making Table 2 non-reproducible. Neither issue is circularity, so the circularity score is 0.

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

The paper's empirical core depends on numerical choices it never states: per-node costs, edge probabilities, delay distribution, Monte Carlo counts, and the hypervolume reference point. These are free parameters of the evaluation rather than derived from data or theory. The axiomatic content is standard IM background plus two domain assumptions about the diffusion model and the embedding-guided crossover.

free parameters (6)
  • Seed cost model c_i = not stated; degree-proportional only in toy example
    Objective f2 (Eq. 5) needs a cost per node on Facebook, GRQC, Gnutella, and Wiki; Section 5.1 never defines it, so the cost objective in Table 2 is unspecified.
  • Edge propagation probabilities p_uv = not stated
    Objective f1 (Eq. 4) requires p_uv for the IC process; the values used on the four datasets are not reported.
  • Hop-delay distribution P_lat_u(delta t) = not stated
    Objective f3 (Eq. 6) samples per-hop delays from this distribution; its family and parameters are absent from the paper.
  • Hypervolume reference point and objective normalization = not stated; HV ~1.0 implies normalization
    Table 2 HV values are uninterpretable without the reference point and the normalization applied to f1, f2, f3.
  • EA hyperparameters (population size, p_c, p_m, embedding settings) = not stated
    Algorithms 2 and 3 require p_c and p_m; only the generation count (1000) is given. Node2Vec dimensions and walk parameters are also unstated.
  • Monte Carlo simulation count for sigma(S) = not stated
    Eq. 4 is an expectation; the number of simulations per evaluation determines the stability of every reported HV and significance test.
assumptions (4)
  • standard math IM is NP-hard under IC and LT models, so IM-ICT (a superset) is NP-hard
    Introduction claims a Set Cover reduction but no proof is shown; the argument implicitly relies on Kempe et al.'s cited NP-hardness (Section 2.1).
  • domain assumption The LAIC per-hop random delay model is the right formalization of 'time'
    Eq. 6 defines time as the longest sampled path; whether this matches real urgency (deadlines, campaigns) is assumed, and alternative time models are not discussed.
  • domain assumption Euclidean distance in Node2Vec embedding space captures exchangeable topological semantics for crossover
    Algorithm 2 pairs nodes by embedding distance and swaps them; no evidence or ablation shows this preserves solution quality better than random pairing.
  • domain assumption The three objectives are inherently conflicting in the stated directions
    Section 3.1 asserts the conflicts; Section 5.2 supports them with Pareto fronts generated by the authors' own EVEA, which is evidence but not independent proof.

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Pith. "Pith review of Influence Maximization Considering Influence, Cost and Time." pith.science (2026). https://pith.science/paper/HMDKTXQU

@misc{pith2026250907625,
  author       = {Pith},
  title        = {Pith review of: Influence Maximization Considering Influence, Cost and Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMDKTXQU}},
  note         = {Machine review of arXiv:2509.07625}
}
read the original abstract

Influence maximization has been studied for social network analysis, such as viral marketing (advertising), rumor prevention, and opinion leader identification. However, most studies neglect the interplay between influence spread, cost efficiency, and temporal urgency. In practical scenarios such as viral marketing and information campaigns, jointly optimizing Influence, Cost, and Time is essential, yet remaining largely unaddressed in current literature. To bridge the gap, this paper proposes a new multi-objective influence maximization problem that simultaneously optimizes influence, cost, and time. We show the intuitive and empirical evidence to prove the feasibility and necessity of this multi-objective problem. We also develop an evolutionary variable-length search algorithm that can effectively search for optimal node combinations. The proposed EVEA algorithm outperforms all baselines, achieving up to 19.3% higher hypervolume and 25 to 40% faster convergence across four real-world networks, while maintaining a diverse and balanced Pareto front among influence, cost, and time objectives.

Figures

Figures reproduced from arXiv: 2509.07625 by the authors.

Figure 1
Figure 1. supermarket ing containment costs while maximizing malware erad￾ication speed and scope). The optimization of the proposed IM-ICT problem incurs two algorithmic challenges. First, we rigor￾ously prove the proposed IM-ICT’s NP-hardness via a polynomial-time reduction from the Set Cover prob￾lem. In addition to the NP-hard challenge, the length of solutions requires variability to enable the algorithm to naturally evo… view at source ↗
Figure 2
Figure 2. Embedded crossover 4.3. Variable Length Mutation To further promote diversity and adaptability, EVEA incorporates a variable-length mutation operator (Algo￾rithm 3). At each mutation step, one of the following strategies is randomly selected with equal probability: Algorithm 2: Embedding Aligned Crossover Input: Parents S 1,S 2,Embedding EM, Crossover Rate pc Output: Children C1,C2 1 Align nodes in S 1, S 2 by embed… view at source ↗
Figure 3
Figure 3. The Pareto-front of EVEA calculations to maintain a well-distributed Pareto front. This selection step has a time complexity of O(N 2 ). Additionally, the embedding-aligned crossover oper￾ator involves at most min(k1, k2) pairwise distance com￾putations between seed nodes in the parent solutions, where k1 and k2 are the lengths of the respective parent solutions. Since k1, k2 ≪ |V| (with |V| being the number of node… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Convergence curves (HV vs. generations) of four algorithms across four networks. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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