Eigenvector localization in real networks and its implications for epidemic spreading

TL;DR

This paper investigates how the principal eigenvector of the adjacency matrix in real networks localizes on specific substructures and how this affects epidemic spreading and immunization strategies.

Contribution

It reveals the localization patterns of the principal eigenvector in real networks and links these patterns to epidemic spreading and immunization effectiveness.

Findings

Principal eigenvector localizes on hubs or dense subgraphs.

Localization correlates with the largest eigenvalue.

Targeted immunization strategies are highly sensitive to eigenvector localization.

Abstract

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum $K$-core in a $K$-core decomposition. The localization of the principal eigenvector is often strongly correlated with the value of the largest eigenvalue, which is given by the local eigenvalue of the corresponding localization subgraph, but different scenarios sometimes occur. We additionally show that simple targeted immunization strategies for epidemic spreading are extremely sensitive to the actual localization set.