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Structure-Aware Optimal Intervention for Rumor Dynamics on Networks: Node-Level, Time-Varying, and Resource-Constrained

Yan Zhu, Qingyang Liu, Chang Guo, Tianlong Fan, Linyuan Lü

TL;DR

The paper addresses rumor propagation on networks under limited intervention resources by formulating a node-level, time-varying optimal-control problem embedded in a networked SIR model. It derives a stability-aware framework where per-node interventions modify transmission and recovery rates via $\beta_i(t)=\beta_0(1 - u w_i(t))$ and $\gamma_i(t)=\gamma_0(1 + u w_i(t))$, and solves the problem with a forward–backward sweep. Existence of an optimal control is established, with Pontryagin's Maximum Principle yielding adjoint dynamics and an explicit projection-based policy $w_i^*(t)$; simulations on synthetic and real networks show substantial reductions in both peak prevalence and total infection area, revealing a robust stage-aware allocation rule (early hub targeting, later peripheral cleanup). The results demonstrate a scalable, interpretable approach for misinformation management and crisis response that leverages network structure to balance effectiveness and resource usage.

Abstract

Rumor propagation in social networks undermines social stability and public trust, calling for interventions that are both effective and resource-efficient. We develop a node-level, time-varying optimal intervention framework that allocates limited resources according to the evolving diffusion state. Unlike static, centrality-based heuristics, our approach derives control weights by solving a resource-constrained optimal control problem tightly coupled to the network structure. Across synthetic and real-world networks, the method consistently lowers both the infection peak and the cumulative infection area relative to uniform and centrality-based static allocations. Moreover, it reveals a stage-aware law: early resources prioritize influential hubs to curb rapid spread, whereas later resources shift to peripheral nodes to eliminate residual transmission. By integrating global efficiency with fine-grained adaptability, the framework offers a scalable and interpretable paradigm for misinformation management and crisis response.

Structure-Aware Optimal Intervention for Rumor Dynamics on Networks: Node-Level, Time-Varying, and Resource-Constrained

TL;DR

The paper addresses rumor propagation on networks under limited intervention resources by formulating a node-level, time-varying optimal-control problem embedded in a networked SIR model. It derives a stability-aware framework where per-node interventions modify transmission and recovery rates via and , and solves the problem with a forward–backward sweep. Existence of an optimal control is established, with Pontryagin's Maximum Principle yielding adjoint dynamics and an explicit projection-based policy ; simulations on synthetic and real networks show substantial reductions in both peak prevalence and total infection area, revealing a robust stage-aware allocation rule (early hub targeting, later peripheral cleanup). The results demonstrate a scalable, interpretable approach for misinformation management and crisis response that leverages network structure to balance effectiveness and resource usage.

Abstract

Rumor propagation in social networks undermines social stability and public trust, calling for interventions that are both effective and resource-efficient. We develop a node-level, time-varying optimal intervention framework that allocates limited resources according to the evolving diffusion state. Unlike static, centrality-based heuristics, our approach derives control weights by solving a resource-constrained optimal control problem tightly coupled to the network structure. Across synthetic and real-world networks, the method consistently lowers both the infection peak and the cumulative infection area relative to uniform and centrality-based static allocations. Moreover, it reveals a stage-aware law: early resources prioritize influential hubs to curb rapid spread, whereas later resources shift to peripheral nodes to eliminate residual transmission. By integrating global efficiency with fine-grained adaptability, the framework offers a scalable and interpretable paradigm for misinformation management and crisis response.

Paper Structure

This paper contains 13 sections, 10 theorems, 37 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Theoretical Background
  4. Problem Definition
  5. Methodology
  6. Controlled rumor intervention model
  7. Stability Analysis Under Control
  8. Research on the optimal control problem
  9. Simulation and Analysis
  10. Correlation Between Node Weights and Centrality
  11. Visualization of Weight Allocation by Centrality
  12. Global Infection I Density Curve Variation
  13. Conclusion

Key Result

Theorem 1

For system eq:SIR-uncontrolled with admissible initial data, solutions exist uniquely on $[0,\infty)$ and satisfy $S_i(t),I_i(t),R_i(t)\in[0,1]$ and $S_i(t)+I_i(t)+R_i(t)=1$ for all $t\ge 0$.

Figures (3)

  • Figure 1: Temporal correlation between control Weights and network static centralities.
  • Figure 2: The weight distribution of nodes with different degrees in different networks varies over time.
  • Figure 3: Different network comparison of infection trajectories: optimal time-varying control vs. centrality-based static policies. These results are captured under SIR model parameters $\gamma_0 = 0.1$, $\beta_0 = 3\beta_c$.

Theorems & Definitions (18)

  • Theorem 1: Positivity And Invariance hethcote2000mathematics
  • proof
  • Theorem 2: Existence Of Equilibria
  • proof
  • Lemma 1: Spectral Bound of Metzler Jacobian horn2012matrixwang2003epidemicvan2008virus
  • Theorem 3: Local Stability van2008viruswang2003epidemic
  • proof
  • Definition 1: Positively Invariant Set
  • Lemma 2: Positivity kermack1927contributionhethcote2000mathematics
  • proof
  • ...and 8 more