Controlling centrality: The Inverse ranking problem for spectral centralities of complex networks

TL;DR

This paper investigates the inverse problem of designing unweighted complex networks that achieve a prescribed spectral centrality ranking, including PageRank and eigenvector centrality, for both directed and undirected graphs.

Contribution

It introduces methods to construct networks with a specified centrality ranking, addressing the inverse spectral centrality problem for various network types.

Findings

Networks can be designed to realize any prescribed ranking with ties.

Analytical solutions are provided for different network families.

The approach applies to both directed and undirected graphs, with or without loops.

Abstract

In this paper some results about the controllability of spectral centrality in a complex network are presented. In particular, the inverse problem of designing an unweigthed graph with a prescribed centrality is considered, by showing that for every possible ranking eventually with ties, an unweighted directed/undirected complex network can be found whose PageRank or eigenvector centrality gives the ranking considered. Different families of networks are presented in order to analytically solve this problem either for directed and undirected graphs with and without loops.