Overcoming Non-Submodularity: Towards Constant Approximation for Network Immunization

2024-10-24

AI Summary

Maximizing infection blocking under a fixed immunization budget is an NP-hard problem in network immunization; existing methods rely on surrogate objectives to circumvent the non-submodularity of the true objective, sacrificing theoretical guarantees and empirical performance. Method: We propose a theoretically grounded approach that models stochastic diffusion processes via expectation-based analysis, quantifies non-submodularity deviation, and enhances greedy selection. Contribution/Results: For sparse cascade models (e.g., SIR, IC), we prove the true immunization objective exhibits bounded non-submodular deviation, enabling a guaranteed constant-factor approximation (≥1/4) for the greedy algorithm—its first such theoretical justification. Our method demonstrates robustness across diverse diffusion models and network topologies. Experiments show 18–35% empirical improvement over surrogate-based methods, achieving the first provably approximate optimization of actual immunization efficacy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Given a network with an ongoing epidemic, the network immunization problem seeks to identify a fixed number of nodes to immunize in order to maximize the number of infections prevented. A fundamental computational challenge in network immunization is that the objective function is generally neither submodular nor supermodular. Consequently, no efficient algorithm is known to consistently achieve a constant-factor approximation. Traditionally, this problem is partially addressed using proxy objectives that offer better approximation properties, but these indirect optimizations often introduce losses in effectiveness due to gaps between the proxy and natural objectives. In this paper, we overcome these fundamental barriers by leveraging the underlying stochastic structure of the diffusion process. Similar to the traditional influence objective, the immunization objective is an expectation expressed as a sum over deterministic instances. However, unlike the former, some of these terms are not submodular. Our key step is to prove that this sum has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network. We demonstrate that this approximation holds across various immunization settings and spread models.

Problem

• Network immunization optimization
• Overcoming non-submodularity challenge
• Achieving constant-factor approximation

Innovation

• Leverages stochastic diffusion structure
• Proves bounded submodular deviation
• Enables constant greedy approximation