Localization of eigenvector centrality in networks with a cut vertex

Area of Science:

• Network Science
• Graph Theory
• Data Analysis

Background:

• Eigenvector centrality is a key metric for identifying influential nodes in networks.
• Network partitioning and node importance are critical in understanding network structures.
• Cut vertices significantly impact network connectivity and node influence.

Purpose of the Study:

• To investigate eigenvector centrality localization phenomena in networks with vertex cut sets.
• To identify and characterize distinct types of eigenvector centrality localization.
• To explore the relationship between eigenvector centrality and Katz centrality for robust measure approximation.

Main Methods:

• Analysis of eigenvector centrality on networks with cut vertices.
• Identification and characterization of localization phenomena.
• Derivation of the relationship between eigenvector centrality and Katz centrality.

Main Results:

• Eigenvector centrality exhibits localization in networks separable by vertex cut sets.
• Three types of localization are identified, including a novel characterization.
• A relationship between eigenvector centrality and Katz centrality is established.

Conclusions:

• Eigenvector centrality localization provides insights into network structure, especially in partitioned networks.
• The principal eigenvector can approximate more robust centrality measures.
• Understanding these phenomena is crucial for network analysis and interpretation.