A Complete Characterization of the Inverse Eigenvector Centrality Problem for Undirected Graphs

TL;DR

This paper fully characterizes the inverse eigenvector centrality problem for undirected graphs, revealing complex global constraints involving stable sets and neighborhoods that differ from directed cases.

Contribution

It provides a comprehensive theoretical characterization of the inverse eigenvector centrality problem specifically for undirected graphs.

Findings

Complete characterization in terms of stable sets and neighborhoods.

Undirected case involves nontrivial global constraints.

Differences highlighted between directed and undirected cases.

Abstract

We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets and their neighborhoods, showing that the undirected case requires nontrivial global constraints absent in the directed setting.