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Influence Maximization in Temporal Networks with Persistent and Reactive Behaviors

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that its temporal diffusion model cpSI-R, which allows nodes to deactivate, reactivate, and be reinforced by repeated contact, has a monotone and submodular spread function, so greedy seed selection keeps its…

desk verdict The main theoretical claim is a restatement of the conclusion, not a proof, and the model's own tau-window dynamics allow spread to drop when seeds are added; the model idea and sampling heuristic are reasonable, but the paper's core guarantee is unsupported. read the letter →

arxiv 2412.20936 v2 pith:RGGGTCYP submitted 2024-12-30 cs.SI physics.comp-ph

Aaqib Zahoor , Iqra Altaf Gillani , Janib ul Bashir This is my paper · ORCID
classification cs.SIphysics.comp-ph
keywords influencemaximizationtemporalnetworksdiffusionmodelmonotonesubmodularityreactivationandreinforcementsnapshotsamplinggreedyapproximationsocial
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 show that influence maximization on temporal networks does not have to sacrifice optimization guarantees to model realistic behavioral cycles. It introduces cpSI-R, a susceptible-infected model in which nodes spread influence for a limited window, can go inactive, and can be reactivated by contact with active nodes, with each repeated exposure raising infection probability. The paper's main theoretical claim is that expected spread under cpSI-R stays monotone and submodular, so greedy seed selection keeps its approximation bound. If true, this matters because prior temporal models with active-inactive transitions are argued to break those properties, leaving seed selection without quality guarantees. The paper also reports that its sampling-and-selection pipeline beats several temporal baselines in spread and runtime on datasets ranging from school contacts to large online networks.

What carries the argument

The central object is the cpSI-R model: each infected node $u$ can try to infect a susceptible neighbor $v$ at each temporal contact $t^k_{uv}$ with probability $p_{uv}(t^k_{uv}) = p_0(1-e^{-\alpha k})f(t^k_{uv})$, where the temporal factor is $f(t^k_{uv}) = \beta e^{-\gamma(t^k_{uv}-t)}$. A node can spread influence only within a time window $\tau$ after its latest infection attempt, becomes inactive for spreading after that window, and can be reactivated by an active node, while repeated exposures increase the infection probability through the reinforcement factor $\alpha$. The proof of monotonicity and submodularity fixes each stochastic realization $\omega$ and asserts that a larger seed set activates at least as many nodes in every realization, then takes expectations; the optimization pipeline is carried by a Jaccard-Kulczynski similarity score that selects structurally significant snapshots and a lazy forward replacement routine that exploits diminishing returns.

What would settle it

On a three-node temporal path with $a\to v$ at $t=2$, $v\to w$ at $t=3$, $\tau=1$, and infection probability $1$, compare seed sets $\{a\}$ and $\{a,u\}$ where an additional edge $u\to v$ exists at $t=1$. With $\{a\}$, $v$ activates at $t=2$ and still has one time unit to infect $w$ at $t=3$; with $\{a,u\}$, $u$ activates $v$ at $t=1$, $v$'s spreading window expires before $t=3$, and $w$ is never infected. This gives $\sigma_{\mathrm{cpSI\text{-}R}}(\{a,u\}) < \sigma_{\mathrm{cpSI\text{-}R}}(\{a\})$ under a final-activation count, directly violating the claimed monotonicity.

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Extended reading notes

Core claim

The central claim is Theorem 1: for any instance of cpSI-R on a temporal graph with fixed probabilities and a temporal interaction function, the expected spread function $\sigma_{\mathrm{cpSI\text{-}R}}$ is monotone and submodular. The paper argues this makes cpSI-R the first temporal diffusion model with reactivation that retains these properties, so the standard greedy algorithm for influence maximization still applies with the classic $(1-1/e)$ approximation guarantee. The same section contributes a structure-aware snapshot sampling method and a lazy forward seed-selection routine that use submodularity to avoid redundant computations.

Load-bearing premise

The proof of Theorem 1 assumes that once a node is activated it stays counted as influenced forever, so adding a seed can never reduce anyone's final activation count, but cpSI-R's $\tau$-window lets an earlier activation expire before later temporal contacts, so the assumption may fail for the implemented objective.

Editorial extensions

If this is right

  • Greedy seed selection under cpSI-R inherits the $(1-1/e)$ approximation ratio, so the model can be optimized with a provable quality bound despite node deactivation and reactivation.
  • The snapshot sampler cuts the runtime from $O(T|E| + k\log|V|\,T)$ to $O(r|E| + k\log|V|\,r)$ with $r \ll T$, because only structurally changed timestamps are simulated.
  • Since reactivation is explicit, seed sets chosen under cpSI-R can sustain spread after nodes disengage, matching repeated-exposure campaigns better than one-shot cascade models.
  • On the six datasets reported, the cpSI-R pipeline achieves higher infection spread and lower runtime than the dynamic degree, reverse-reachable, temporal SIR, entropy, and forward-influence baselines, with the largest margins on the biggest networks.

Reading between the lines

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

  • The paper does not test this, but if the timing objection to monotonicity is real, replacing final-snapshot counts with cumulative ever-active exposure would make the spread function robust to early activation and would change seed rankings under bursty contact patterns.
  • A natural extension is online seeding: the Jaccard-Kulczynski change-point scores could trigger re-seeding whenever a structural shift is detected, instead of committing to one fixed seed set over the whole horizon.
  • The $\tau$-window mechanism implies that seeds should be scored not by static degree but by whether their contacts arrive before the downstream node's window expires; comparing cpSI-R seed sets with temporal-centrality rankings on contact datasets would isolate this effect.
  • If monotonicity does fail under $\tau$ windows, restricting cpSI-R to a persistent-active state in which any contact refreshes the window would restore submodularity without sacrificing reinforcement.
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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 / 6 minor

Summary. The paper introduces cpSI-R, a temporal diffusion model with persistence, reactivation, and reinforcement, and claims that its influence spread function is monotone and submodular, enabling a (1-1/e) greedy approximation for influence maximization. It also proposes an adaptive temporal snapshot sampling method and a lazy-forward seed selection algorithm, reporting experiments on several temporal network datasets. The central theoretical claim is Theorem 1 in Section 5.1, and the main algorithmic contribution is the combination of sampling with lazy forward influence maximization.

Significance. If the central theorem were established, the paper would offer a useful addition to temporal influence maximization by providing an optimization-friendly model with active-inactive transitions. The paper also makes a strong novelty claim: that cpSI-R is the first temporal diffusion model retaining monotonicity and submodularity with recovery/reactivation. However, the proof of Theorem 1 is not a derivation; it assumes the very pointwise inequalities that need to be shown, and the implemented objective in Algorithm 2 is not the expected spread defined in Definition 6. The experimental comparison is also presented without confidence intervals or reproducibility details. The strengths of the paper are its clear motivation and the concrete counterexample in Section 4, which usefully illustrates why standard monotonicity can fail under active-inactive transitions; unfortunately, that same example undermines the proof of Theorem 1 rather than supporting it.

major comments (4)
  1. [Section 5.1, Theorem 1] The proof of Theorem 1 is circular and does not engage with the model's own dynamics. It fixes a realization omega and asserts that 'activations are never revoked and additional seeds only increase influence,' but this pointwise monotonicity is exactly what needs to be proved for cpSI-R, and it is false for the active-inactive transitions described earlier in the same section. Lemma 1 in Section 4 provides a concrete temporal counterexample where adding a seed node reduces spread from 6 to 4 because early activation causes a node's tau-window to expire before later edges. The proof never uses p0, alpha, beta, gamma, tau, or the temporal interaction function, so it would apply unchanged to any process satisfying the verbal premise, which indicates that the model-specific claim has not been established. Consequently, the (1-1/e) approximation guarantee and the novelty claim that cpSI-R is the first temporal model to retain these properties rest on an unsupported theorem.
  2. [Section 5.2, Algorithm 2] Algorithm 2 does not compute the expected spread defined in Definition 6. The function returns |S| + sum_{i in N(t)\S(t)} p(i, t_r), which is a deterministic sum of infection probabilities at the final timestamp, not the expected number of nodes that become active during the diffusion process. The model description in Section 5.1 explicitly allows nodes to become inactive for spreading after their tau-window expires, so the final-time infection probability can be non-monotone in the seed set, as Lemma 1 demonstrates. Thus, even if Theorem 1 were correct for a cumulative spread function, the quantity actually optimized by the proposed algorithm is not protected by that theorem.
  3. [Section 5.1 and Algorithm 2] The formal update equations of the cpSI-R model are never given. Section 5.2 refers to 'dynamic update equations' and Algorithm 2 line 7 instructs the reader to compute p(i, t_{j+1}) using those equations, but no recurrence, no state-transition rule, and no reactivation rule are specified. Without these equations, the model is not precisely defined, the proof of monotonicity/submodularity cannot be checked, and the experimental implementation is not reproducible. This is a load-bearing gap because the entire optimization framework depends on the exact stochastic process being optimized.
  4. [Section 6.3] The experimental claims are difficult to assess because no confidence intervals, standard deviations, or repeated-run statistics are reported for the infection spread comparisons in Figures 5, 7, and 9. Additionally, the text states that 'All methods use the cpSI-R model for fair comparison,' which means Figure 9 compares seed selection algorithms under a single model, not the diffusion model itself; the model-vs-SIR comparisons in Figure 5 are not directly comparable to the baseline IM methods. The paper would need a more detailed experimental protocol to support the strong claim of consistent outperformance across all datasets.
minor comments (6)
  1. [Abstract and Introduction] The paper contains two different abstracts: the abstract at the top of the submitted text describes a different emphasis (persistent and reactive behaviors) from the abstract in the introduction, which mentions 'persistent and reactive behaviors' but also repeats the same claims. The duplication and inconsistency should be resolved.
  2. [Section 3, Definition 6] Definition 6 defines expected spread as E_M[|sigma(S0, G)|], but sigma is used both as the random set and as the spread function; this overloading makes the definition ambiguous, especially given the later use of sigma_omega(S) in the proof of Theorem 1.
  3. [Section 5.2.2, toy example] The complexity statement in the toy example, 'selecting fewer snapshots based on eta reduces this to O(eta T)', appears to have the dependence on eta backwards: a higher threshold eta selects fewer snapshots, not more, so the claimed O(eta T) scaling is not justified.
  4. [Section 5.2, Algorithm 1] Algorithm 1 calls SamplingAlgorithm to set tset, but the influence calculations in CalcInfluence also call SamplingAlgorithm, so the interaction between these two calls and the role of tset in the final computation are unclear.
  5. [Section 6, Datasets] The Malawi dataset row in Table 2 states 'Node and edge counts not specified' despite the text giving 142 nodes and 9,436 edges; the table and text should be reconciled.
  6. [General] No code or data release is mentioned, and the parameter settings for p0, alpha, beta, gamma, and tau are not reported in the experiments, which makes it impossible to reproduce the reported spread values.

Circularity Check

2 steps flagged · score 8.0 of 10

Theorem 1's proof assumes the monotonicity and submodularity it claims to prove, while cpSI-R's own tau-window and reactivation dynamics (and the paper's Lemma 1) permit the opposite.

  1. self definitional [Section 5.1, Theorem 1 proof, Monotonicity paragraph]
    "Since activations are never revoked and additional seeds only increase influence, it follows that σω(S ∪ {u}) ≥ σω(S), for all ω ∈ Ω."

    The theorem is supposed to prove monotonicity, but the proof's only argument is the assertion 'activations are never revoked and additional seeds only increase influence' — a restatement of the conclusion. The cpSI-R model itself contradicts that premise: Section 5.1 defines a window tau and states 'If t_k^uv − delta_u > tau, then node u becomes inactive for spreading the infection until reactivated.' The paper's own Lemma 1 (Section 4) exhibits a concrete counterexample: adding V3 to the seed set activates V4 early, V4's window expires before later edges, and spread drops from 6 to 4. The proof uses none of the model parameters (p0, alpha, beta, gamma, tau, f(t)), so it would 'prove' monotonicity for any process satisfying the verbal premise.

  2. self definitional [Section 5.1, Theorem 1 proof, Submodularity paragraph; Algorithm 2]
    "Since S ⊆ S′, the diffusion process starting from S′ activates at least as many nodes as the process from S, possibly more."

    This sentence assumes exactly the monotonicity property being proved; under cpSI-R's time-limited activation, a larger seed set can deplete the tau-window of an early-activated node and end with fewer active nodes (Lemma 1). Moreover, the implemented objective in Algorithm 2 returns '|S| + Σ p(i, tr)', i.e., infection probabilities at the final timestamp, not a permanent ever-active count. Because the quantity optimized is a final-time count, early activation that expires before tr can reduce it. The submodularity conclusion therefore rests on the same unproved monotonicity assumption, and the (1−1/e) greedy guarantee collapses with it.

full rationale

The load-bearing mathematical claim of the paper is Theorem 1, which is used to justify the (1−1/e) greedy approximation and the 'first temporal diffusion model retaining these properties' claim. The proof is circular: for monotonicity it asserts 'activations are never revoked and additional seeds only increase influence,' which is literally the inequality to be established; for submodularity it asserts that a larger seed set 'activates at least as many nodes,' which is monotonicity again. These premises are contradicted by the model's own reactivation/window dynamics and by the paper's Lemma 1 counterexample, so the theorem is unsupported rather than derived. The empirical comparison uses all baselines under the cpSI-R model and is self-contained, but it cannot repair a missing theoretical guarantee. Because the central optimization guarantee reduces by construction to the assumption being proved, the circularity score is high, though the sampling and experimental components themselves are not circular.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the model parameters p0, alpha, beta, gamma, tau, which are never specified in the experiments; on the sampling threshold eta selected post hoc; and on the assumption in Theorem 1 that pointwise monotonicity and submodularity of sigma_omega hold. The model itself is a state machine rather than an invented physical entity, so no independent falsifiable object is introduced. Several axioms are standard (expectation over realizations, Kempe's greedy guarantee conditional on the theorem), but the reactivation non-harm assumption is ad hoc to the paper and unproved.

free parameters (9)
  • p0 (base infection probability) = not specified in paper
    Defines activation probability in puv; all quantitative spread values depend on it, and it is never given in Section 6.
  • alpha (reinforcement factor) = not specified in paper
    Controls the increase in infection probability with exposure count k in p0 * (1 - e^(-alpha*k)); also notationally reused as a similarity weight.
  • beta (temporal scaling factor) = not specified in paper
    Scales the temporal decay function f(t) = beta * e^(-gamma*(t - t_u)); not specified in experiments.
  • gamma (decay rate) = not specified in paper
    Controls the decay of influence over time; not specified in experiments.
  • tau (maximum spread window) = not specified in paper
    Time after which an infected node stops spreading until reactivation; central to the model and omitted from the experimental setup.
  • eta (sampling threshold) = 0.7 (0.6 also mentioned in Section 6.2)
    Selected because it maximized spread and runtime trade-off on the same datasets; the paper fixes eta=0.7 after using Figure 7 to choose it.
  • alpha, beta similarity weights = 0.5, 0.5
    Weights in Score = alpha*L + beta*K, hand-set to equal importance; these names conflict with the model parameters alpha and beta.
  • h (time horizon) = not specified in paper
    Influence horizon used by TemporalInfluenceMaximization; experimental values are never given.
  • MinIter (lazy forward termination) = not specified in paper
    Early-termination threshold in LazyForwardInfluence; no value is reported, so seed quality and runtime cannot be reproduced.
assumptions (6)
  • domain assumption Discrete snapshot representation (Definition 2) preserves the diffusion-relevant structure of the temporal network.
    All algorithms reason over snapshots and never over intra-snapshot edge order; if intra-snapshot timing matters, the modeled spread changes.
  • standard math Expected spread is an expectation over a well-defined sample space of realizations.
    Standard construction in influence maximization, but the paper never specifies the distribution over reactivation and reinforcement events for cpSI-R.
  • ad hoc to paper For every realization, adding a seed cannot reduce the set of activated nodes and cannot change activation timing in a harmful way.
    This is asserted in the monotonicity part of Theorem 1 and is exactly the property that time-limited tau windows put in doubt.
  • ad hoc to paper Pointwise diminishing returns of sigma_omega hold for every realization.
    The submodularity step in Theorem 1 assumes marginal gains of a new seed are no larger for larger seed sets, without modeling reactivation timing effects.
  • ad hoc to paper Jaccard and Kulczynski snapshot similarity is a sufficient proxy for diffusion fidelity.
    The sampling algorithm's validity is never proven or isolated experimentally; the threshold eta is tuned empirically.
  • standard math Kempe et al. greedy approximation guarantee applies once monotonicity and submodularity are established.
    Standard result; it only helps if the cpSI-R theorem is actually proven.

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Cite this review

Pith. "Pith review of Influence Maximization in Temporal Networks with Persistent and Reactive Behaviors." pith.science (2026). https://pith.science/paper/RGGGTCYP

@misc{pith2026241220936,
  author       = {Pith},
  title        = {Pith review of: Influence Maximization in Temporal Networks with Persistent and Reactive Behaviors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGGGTCYP}},
  note         = {Machine review of arXiv:2412.20936}
}
read the original abstract

Influence maximization in temporal social networks presents unique challenges due to the dynamic interactions that evolve over time. Traditional diffusion models often fall short in capturing the real-world complexities of active-inactive transitions among nodes, obscuring the true behavior of influence spread. In dynamic networks, nodes do not simply transition to an active state once; rather, they can oscillate between active and inactive states, with the potential for reactivation and reinforcement over time. This reactivation allows previously influenced nodes to regain influence potency, enhancing their ability to spread influence to others and amplifying the overall diffusion process. Ignoring these transitions can thus conceal the cumulative impact of influence, making it essential to account for them in any effective diffusion model. To address these challenges, we introduce the Continuous Persistent Susceptible-Infected Model with Reinforcement and Re-activation (cpSI-R), which explicitly incorporates active-inactive transitions, capturing the progressive reinforcement that makes nodes more potent spreaders upon reactivation. This model naturally leads to a submodular and monotone objective function, which supports efficient optimization for seed selection in influence maximization tasks. Alongside cpSI-R, we propose an efficient temporal snapshot sampling method, simplifying the analysis of evolving networks. We then adapt the prior algorithms of seed selection to our model and sampling strategy, resulting in reduced computational costs and enhanced seed selection efficiency. Experimental evaluations on diverse datasets demonstrate substantial improvements in performance over baseline methods, underscoring the effectiveness of cpSI-R for real-world temporal networks

Figures

Figures reproduced from arXiv: 2412.20936 by the authors.

Figure 1
Figure 1. Counter example showing the violation of monotonicity under active-inactive transition (a): Seed set [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Counter example showing the violation of submodularity under active-inactive transition (a): Seed [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Example illustrating the working of cpSI-R model (a): Toy temporal network represented with initial [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Five snapshots of a Primary School temporal network sampled at 1.1K timestamps in each snapshot. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Advantage of cpSI-R model over SIR in terms of infection spread on Primary school dataset across [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Toy temporal network illustrating the advantage of cpSI-R model over SIR in densely connected [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Comparative infection spread across multiple datasets for different values of [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Comparative execution time (sec) across multiple datasets for our influence maximization method [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Comparative infection spread across multiple datasets for different influence maximization methods. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Execution time in seconds across multiple datasets for different influence maximization methods. [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

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Forward citations

Cited by 1 Pith paper

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.