  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.
