On a Centrality Maximization Game

TL;DR

This paper models a network formation game where nodes rewire links to maximize Bonacich centrality, analyzing Nash equilibria and showing that such strategies tend to produce specific network structures like 2-cliques and rings.

Contribution

It provides a complete classification of Nash equilibria in a centrality maximization game for m=1 and m=2, revealing the resulting network structures.

Findings

Nash equilibria for m=1 are 2-cliques.

Nash equilibria for m=2 include rings and a Butterfly-shaped graph.

Nodes' local decision-making leads to specific network topologies.

Abstract

The Bonacich centrality is a well-known measure of the relative importance of nodes in a network. This notion is, for example, at the core of Google's PageRank algorithm. In this paper we study a network formation game where each player corresponds to a node in the network to be formed and can decide how to rewire his m out-links aiming at maximizing his own Bonacich centrality, which is his utility function. We study the Nash equilibria (NE) and the best response dynamics of this game and we provide a complete classification of the set of NE when m=1 and a fairly complete classification of the NE when m=2. Our analysis shows that the centrality maximization performed by each node tends to create undirected and disconnected or loosely connected networks, namely 2-cliques for m=1 and rings or a special "Butterfly"-shaped graph when m=2. Our results build on locality property of the best response function in such game that we formalize and prove in the paper.