A theory on power in networks

TL;DR

This paper introduces a new 'power equation' as an alternative to eigenvector centrality, emphasizing power dynamics in networks where being connected to less powerful nodes confers more influence.

Contribution

It proposes the power equation as a novel theoretical framework for understanding influence in networks, complementing existing centrality measures.

Findings

Introduces the power equation $x = A x^{\ar{}}$ as a meaningful measure of influence.

Highlights applications in bargaining and influence scenarios.

Explores the theoretical properties of the power equation.

Abstract

The eigenvector centrality equation $\lambda x = A \, x$ is a successful compromise between simplicity and expressivity. It claims that central actors are those connected with central others. For at least 70 years, this equation has been explored in disparate contexts, including econometrics, sociometry, bibliometrics, Web information retrieval, and network science. We propose an equally elegant counterpart: the power equation $x = A x^{\div}$, where $x^{\div}$ is the vector whose entries are the reciprocal of those of $x$. It asserts that power is in the hands of those connected with powerless others. It is meaningful, for instance, in bargaining situations, where it is advantageous to be connected to those who have few options. We tell the parallel, mostly unexplored story of this intriguing equation.