Network Prebunking Problem

• Network prebunking problem is a combinatorial optimization challenge that pre-selects a fixed-budget subset of users to reduce misinformation diffusion in directed social networks.
• The approach models prebunking as reducing node susceptibility before diffusion, leveraging stochastic propagation models and addressing ethical intervention versus blocking or corrections.
• The proposed MIA-NPP algorithm uses a greedy strategy based on Maximum Influence Arborescence to achieve near-optimal target selection and scalable intervention performance.

The network prebunking problem is a combinatorial optimization problem for selecting a fixed-budget subset of users in a directed social network to receive prebunking before misinformation diffusion, with the objective of minimizing expected misinformation spread under a stochastic propagation model. In the formulation introduced in "Network Prebunking Problem: Optimizing Prebunking Targets to Suppress the Spread of Misinformation in Social Networks" (Furutani et al., 2 Aug 2025), prebunking is modeled as a pre-diffusion reduction in node-specific susceptibility rather than as node removal, edge blocking, or post-hoc correction. The problem is NP-hard, and the paper proposes an approximation algorithm, MIA-NPP, based on Maximum Influence Arborescence to obtain scalable target selection on large social networks.

1. Formal statement of the problem

Let $G=(V,E)$ be a directed social network with node set $V$ and edge set $E$. Each directed edge $(u,v)\in E$ carries an influence probability $p_{uv}\in[0,1]$, representing the chance that information propagates from $u$ to $v$. Each node $v\in V$ has a susceptibility $q_v\in[0,1]$. Given a set of misinformation seeds $S\subseteq V$, the diffusion process starts at $V$0 with all nodes in $V$1 positively active, while all other nodes are inactive (Furutani et al., 2 Aug 2025).

The propagation model is IC–N, or Independent Cascade with Negative opinion. At each step $V$2, any node that became positive or negative at step $V$3 has one attempt to activate each currently inactive out-neighbor $V$4 with probability $V$5. If the activating node is positive, it transmits misinformation; if it is negative, it transmits corrective information. When an inactive node receives exactly one activation in the next step, two cases are distinguished. If the activation comes from a positive neighbor, the node becomes positive with probability $V$6 and negative with probability $V$7. If the activation comes from a negative neighbor, the node becomes negative with probability $V$8. If both types of activation arrive in the same step, negative precedence applies. Once a node becomes active, whether positive or negative, it remains active.

Prebunking is a pre-diffusion operation on a subset $V$9 with $E$0. For each prebunked node $E$1, susceptibility is reduced according to

$E$2

where $E$3 is the node-specific intervention effect. This formulation makes the intervention probabilistic and heterogeneous. In particular, prebunking does not imply absolute immunity unless $E$4.

2. Probabilistic objective and equivalent formulations

For a given prebunked set $E$5, let

$E$6

The expected misinformation spread is

$E$7

and the corrective spread is

$E$8

The optimization target is the size-$E$9 set

$(u,v)\in E$0

Because $(u,v)\in E$1 for fixed $(u,v)\in E$2, the same problem can be written equivalently as

$(u,v)\in E$3

The paper also gives the gain formulation

$(u,v)\in E$4

which is then maximized subject to $(u,v)\in E$5 (Furutani et al., 2 Aug 2025).

This objective distinguishes the network prebunking problem from formulations that optimize message timing or aggregate coverage. The decision variable is the subset $(u,v)\in E$6 of nodes selected in advance, and the optimization criterion is the network-level expected reduction in misinformation diffusion.

3. Computational hardness

The problem is proved NP-hard by reduction from Set Cover (Furutani et al., 2 Aug 2025). Given a universe $(u,v)\in E$7 and subsets $(u,v)\in E$8, the reduction constructs a graph $(u,v)\in E$9 with a dummy node $p_{uv}\in[0,1]$0, subset nodes $p_{uv}\in[0,1]$1 for $p_{uv}\in[0,1]$2, and element nodes $p_{uv}\in[0,1]$3 for $p_{uv}\in[0,1]$4. The edge set contains $p_{uv}\in[0,1]$5 and $p_{uv}\in[0,1]$6 whenever $p_{uv}\in[0,1]$7. All edge probabilities are set to $p_{uv}\in[0,1]$8, all susceptibilities to $p_{uv}\in[0,1]$9, all intervention effects to $u$0, and the misinformation seed set is $u$1.

Under this construction, without prebunking all nodes become positive for sure. Prebunking a subset node $u$2 turns $u$3 and all its downstream element nodes $u$4 negative. Selecting $u$5 subset nodes covers the universe $u$6 if and only if one can cause all element nodes $u$7 to go negative. Hence deciding whether $u$8 is equivalent to Set Cover.

A plausible implication is that exact optimization on large graphs is not expected to be tractable in general, which motivates approximation algorithms that exploit local structure in the propagation process.

4. MIA-NPP approximation algorithm

MIA-NPP is based on the Maximum Influence Arborescence approach (Furutani et al., 2 Aug 2025). For any path $u$9, define its path probability by

$v$0

The Maximum Influence Path from $v$1 to $v$2 is

$v$3

It is computed by Dijkstra on edge weights $v$4.

Fix a threshold $v$5. The Maximum Influence In-Arborescence of $v$6, denoted $v$7, is the union of all maximum influence paths $v$8 with $v$9. The Maximum Influence Out-Arborescence of $v\in V$0, denoted $v\in V$1, is the union of all maximum influence paths $v\in V$2 with $v\in V$3. These arborescences restrict the influence computation around each node to a local directed tree rooted at that node.

Local influence estimation is performed on $v\in V$4 by a time-indexed dynamic program. Let $v\in V$5 and $v\in V$6 be the probability that $v\in V$7 first becomes positive or negative at step $v\in V$8, and let

$v\in V$9

Then, for each child $q_v\in[0,1]$0 of $q_v\in[0,1]$1 in $q_v\in[0,1]$2,

$q_v\in[0,1]$3

$q_v\in[0,1]$4

where

$q_v\in[0,1]$5

Target selection is greedy with local updates. For each candidate $q_v\in[0,1]$6, the marginal gain estimate is maintained as

$q_v\in[0,1]$7

The algorithm repeats $q_v\in[0,1]$8 times: it selects

$q_v\in[0,1]$9

adds $S\subseteq V$0 to $S\subseteq V$1, and updates only those $S\subseteq V$2 and their ancestors in $S\subseteq V$3. This avoids recomputing the full spread after each iteration.

The complexity is

$S\subseteq V$4

where $S\subseteq V$5 and $S\subseteq V$6 is the maximum size of any $S\subseteq V$7 or $S\subseteq V$8. No closed-form approximation ratio is proved. The paper states that, by analogy to MIA in influence maximization, the greedy scheme is efficient and, in practice, near-optimal.

5. Experimental evaluation

The experimental study uses two misinformation-oriented social networks and eight additional social networks (Furutani et al., 2 Aug 2025).

Network	Construction	Size
PolitiFact	merge 314 diffusion trees	30,813 nodes, 33,488 edges
GossipCop	merge 5,464 diffusion trees	75,915 nodes, 85,308 edges

Additional experiments are reported on Facebook, WikiVote, LastFM, ca-HepTh, Deezer, Enron, Epinions, and Twitter. Parameter settings are specified as follows: $S\subseteq V$9 with $V$00 so that approximately $V$01 activation occurs from the root; $V$02 with additive smoothing; $V$03; $V$04; and the prebunk budget $V$05 ranges from $V$06 to $V$07 in steps of $V$08. For AdvancedGreedy, $V$09 graph samples are used. For MIA-NPP and CMIA-O, $V$10.

The baselines are Random, Gullible, Degree, Distance, AdvancedGreedy, and CMIA-O. Performance is measured by relative misinformation spread $V$11, averaged over $V$12 Monte Carlo runs, with lower values indicating better suppression.

Under perfect knowledge of $V$13 and $V$14, MIA-NPP achieves the greatest reduction in $V$15 on both PolitiFact and GossipCop, outperforming all baselines. In the two-dimensional scatter of the first $V$16 selected nodes, MIA-NPP concentrates on nodes with both high susceptibility $V$17 and high $V$18. Under parameter uncertainty, where $V$19 is noisy and $V$20 is replaced by its mean, MIA-NPP degrades gracefully and remains better than the second-best method, AdvancedGreedy, for moderate noise. On the eight other social networks with synthetic $V$21, MIA-NPP again outperforms all heuristics and baselines. Sensitivity tests show that varying $V$22 over $V$23 has negligible effect on spread, while higher $V$24 yields faster runtime.

These results indicate that the method’s empirical advantage is not limited to a single graph family or to perfectly known parameters. The consistent pattern is that target quality depends jointly on susceptibility and on influence path structure, rather than on degree or distance alone.

6. Relation to timing-based and compartmental prebunking models

The network prebunking problem addresses one axis of intervention design: whom to prebunk. Other arXiv formulations address different axes. "Prebunking Design as a Defense Mechanism Against Misinformation Propagation on Social Networks" formalizes optimal prebunking as optimizing the timing of delivering accurate information, ensuring users encounter it before receiving misinformation while minimizing the disruption to user experience. That work uses a susceptible-infected epidemiological process to model misinformation propagation, frames optimal prebunking as a policy synthesis problem with safety constraints, proposes a policy that approximates the optimal solution to a relaxed problem, and reports that the policy cuts the user experience cost of repeated information delivery in half compared to delivering accurate information immediately after identifying a misinformation propagation (Bayiz et al., 2023).

A different abstraction appears in the IPSR model, "IPSR Model: Misinformation Intervention through Prebunking in Social Systems" (Rai et al., 18 Feb 2025). There, the population is partitioned into $V$25 for Ignorant, $V$26 for Prebunked, $V$27 for Spreader, and $V$28 for Stifler, with

$V$29

Before misinformation introduction at time $V$30, the model has only $V$31 transitions: $V$32 After $V$33, the dynamics become a four-compartment mean-field system on a homogeneous network of average degree $V$34, with transition parameters $V$35. The model derives a misinformation reproduction threshold and a minimal prebunk-coverage condition,

$V$36

together with an equilibrium condition under which $V$37 is asymptotically stable: $V$38 Its simulation findings are that raising $V$39 lowers both the peak and final size of $V$40 and $V$41, lowering $V$42 sharply reduces misinformation spread, and increasing the ratio $V$43 keeps more people in $V$44 longer and further curtails $V$45.

Taken together, these formulations separate three distinct but related design questions. The network prebunking problem formalizes target selection on an explicit graph. The 2023 timing paper formalizes when accurate information should be delivered. The IPSR model formalizes how advance coverage, prebunk potency, and forgetting shape outbreak conditions at the population level. A plausible implication is that practical prebunking systems may need to coordinate all three dimensions simultaneously: budgeted target selection, delivery timing, and maintenance of prebunking through sufficiently strong or repeated interventions.

7. Practical interpretation and scope

Within the formulation of (Furutani et al., 2 Aug 2025), prebunking is less intrusive than node or edge blocking and less delayed than post-hoc clarification, which is why the paper describes it as ethically and operationally attractive. The deployment requirements listed for practice are an estimate of each user’s susceptibility $V$46, a measure of prebunking effectiveness $V$47 or at least its average, and real-time construction of $V$48 and $V$49 for candidate seeds.

The model also clarifies the scope of what is being optimized. The intervention is neither universal exposure nor indiscriminate warning; it is a budget-constrained choice of $V$50 users, applied before diffusion, under uncertainty in both propagation and susceptibility. The empirical results indicate that effective targets are not identified solely by centrality or gullibility. Instead, MIA-NPP prioritizes nodes that jointly combine high susceptibility with high-probability influence paths from the misinformation seed.

This framing places the network prebunking problem at the intersection of misinformation diffusion, influence blocking, and intervention design. Its main technical contribution is to turn prebunking from a generic recommendation into a graph-structured optimization problem with explicit diffusion semantics, a hardness result, and a scalable approximation procedure (Furutani et al., 2 Aug 2025).