Principal eigenvector localization and centrality in networks: revisited

TL;DR

This paper revisits the phenomenon of principal eigenvector localization in networks and its impact on eigenvector centrality, revealing that both localization and delocalization can hinder accurate node importance assessment, and suggesting degree centrality as an alternative.

Contribution

It provides a fundamental analysis of PEV localization and delocalization effects on eigenvector centrality, offering insights for better node importance ranking in complex networks.

Findings

PEV localization causes issues in eigenvector centrality.

PEV delocalization also impairs centrality measures.

Degree centrality is recommended when PEV is delocalized.

Abstract

Complex networks or graphs provide a powerful framework to understand importance of individuals and their interactions in real-world complex systems. Several graph theoretical measures have been introduced to access importance of the individual in systems represented by networks. Particularly, eigenvector centrality (EC) measure has been very popular due to its ability in measuring importance of the nodes based on not only number of interactions they acquire but also particular structural positions they have in the networks. Furthermore, the presence of certain structural features, such as the existence of high degree nodes in a network is recognized to induce localization transition of the principal eigenvector (PEV) of the network's adjacency matrix. Localization of PEV has been shown to cause difficulties in assigning centrality weights to the nodes based on the EC. We revisit PEV localization and its relation with failure of EC problem, and by using simple model networks demonstrate that in addition to the localization of the PEV, the delocalization of PEV may also create difficulties for using EC as a measure to rank the nodes. Our investigation while providing fundamental insight to the relation between PEV localization and centrality of nodes in networks, suggests that for the networks having delocalized PEVs, it is better to use degree centrality measure to rank the nodes.