Network node immunization: improving Netshield algorithm through random rooted forests

2026-06-02

AI Summary

This work addresses the problem of suppressing viral spread in complex networks by formulating it as the removal of k nodes to minimize the largest eigenvalue of the adjacency matrix. The authors propose K-shield, an algorithm that leverages random walk kernels and random rooted forests to construct a novel submodular objective function paired with a greedy search strategy. While maintaining the same computational complexity as NetShield, K-shield significantly improves spectral optimization performance for multi-node immunization by enhancing the approximation accuracy of the largest eigenvalue. The framework is also generalizable to other submodular optimization tasks. Experimental results demonstrate that K-shield consistently outperforms NetShield across multiple benchmark networks, achieving a more effective reduction in spectral radius without sacrificing computational efficiency.

Abstract

We are interested in the so-called multiple-node immunization problem for complex networks under attack by a viral agent. It consists in identifying and removing a set of nodes of size $k$ in a graph to maximize the impeding of virus spread. A few approaches have been proposed in the literature based on numerical and theoretical insights on how classical models for virus spread evolve on graphs. Based on the analysis of these models, the maximal eigenvalue of the adjacency matrix of the graph has become a classical measure of how resilient the network is. Thus, a clear, well-explored approach for multiple-node immunization consists of identifying a set of $k$ nodes in such a way that the reduced network, obtained by removing these nodes, has a minimal largest eigenvalue. This spectral optimization problem turns out to be a computationally hard problem for which only greedy algorithms offer good solutions at efficient computational time. Among those, the so-called Netshield algorithm represents one of the reference choices. The latter is, in fact, a clearly defined algorithm aiming at optimizing a certain sub-modular functional, called shield-value, which approximates the original optimization problem. We propose here a novel procedure, based on random walk kernels and related random spanning forests, to build a new algorithm, referred to as K-shield, which enhances Netshield searching performance at the same computational complexity. We give theoretical insights behind this novel method, which could also be used for other optimization problems, and then test it via numerical showcase experiments on various benchmarks.

Problem

• network immunization
• virus spread
• eigenvalue optimization
• node removal
• complex networks

Innovation

• random rooted forests
• Netshield
• network immunization
• spectral optimization
• submodular optimization