On a Network Centrality Maximization Game

TL;DR

This paper models a network formation game where players strategically link to maximize PageRank centrality, revealing equilibrium structures and network properties through game-theoretic analysis.

Contribution

It introduces a potential game framework for PageRank-based network formation and characterizes equilibrium structures, especially for homogeneous players with specific out-degree constraints.

Findings

The game is a potential game with local best responses.

Equilibria often form undirected, disconnected, or loosely connected networks.

Complete classification of equilibria for d=1 and d=2 cases.

Abstract

We study a network formation game where $n$ players, identified with the nodes of a directed graph to be formed, choose where to wire their outgoing links in order to maximize their PageRank centrality. Specifically, the action of every player $i$ consists in the wiring of a predetermined number $d_i$ of directed out-links, and her utility is her own PageRank centrality in the network resulting from the actions of all players. We show that this is a potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node $i$ to link to other nodes that are at incoming distance more than $d_i $ from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the homogeneous case, where players all have the same number $d$ of out-links, we characterize the structure of the potential maximizing equilibria and, in the special cases $ d=1 $ and $ d=2 $, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks.