Eigenvector Centrality and Uniform Dominant Eigenvalue of Graph Components

TL;DR

This paper investigates the calculation of eigenvector centrality in graph components, comparing methods for dominant eigenvalues and analyzing cases with shared dominant eigenvalues across components.

Contribution

It introduces a new approach for computing eigenvector centrality in partitioned graphs and explores theoretical and numerical aspects of shared dominant eigenvalues.

Findings

Compared the new method with the power method for single dominant eigenvalue

Analyzed theoretical properties of shared dominant eigenvalues in graph components

Provided numerical results illustrating the behavior of eigenvector centrality in these cases

Abstract

Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results. Keywords: Eigenvector centrality, power iteration, graph, strongly connected component.