Locating the eigenshield of a network via perturbation theory

TL;DR

This paper introduces a matrix perturbation framework to identify a small set of vital nodes, called eigenshield nodes, that significantly impact network robustness by reducing spectral radius, considering influence and redundancy.

Contribution

It proposes a novel optimization approach for eigenshield node detection that accounts for influence and redundancy, improving network robustness analysis.

Findings

Eigenshield nodes effectively reduce spectral radius when removed.

The method outperforms existing approaches in network dismantling and spreading tasks.

Analytical insights into influence redundancy explain node importance variations.

Abstract

The functions of complex networks are usually determined by a small set of vital nodes. Finding the best set of vital nodes (eigenshield nodes) is critical to the network's robustness against rumor spreading and cascading failures, which makes it one of the fundamental problems in network science. The problem is challenging as it requires to maximize the influence of nodes in the set while simultaneously minimizing the redundancies between the set's nodes. However, the redundancy mechanism is rarely investigated by previous studies. Here we introduce the matrix perturbation framework to find a small ``eigenshield" set of nodes that, when removed, lead to the largest drop in the network's spectral radius. We show that finding the ``eigenshield" nodes can be translated into the optimization of an objective function that simultaneously accounts for the individual influence of each node and redundancy between different nodes. We analytically quantify the influence redundancy that explains why an important node might play an insignificant role in the ``eigenshield" node set. Extensive experiments under diverse influence maximization problems, ranging from network dismantling to spreading maximization, demonstrate that the eigenshield detection tends to significantly outperforms state-of-the-art methods across most problems. Our findings shed light on the mechanisms that may lie at the core of the function of vital nodes in complex network.