On a Centrality Maximization Game Maria Castaldo ˚ Costanza Catalano ˚˚ Giacomo Como ˚˚ Fabio Fagnani ˚˚ ˚ Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, F-38000 Grenoble, France (e-mail: Maria.Castaldo@gipsa-lab.grenoble-inp.fr). ˚˚ Department of Mathematical Sciences “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (e-mail: {costanza.catalano,giacomo.como,fabio.fagnani}@polito.it). 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. Keywords: Network centrality, network formation, Bonacich centrality, PageRank, game theory, social networks. 1. INTRODUCTION The notion of centrality aims at capturing the importance of a node in a network. This concept arises and finds application in many fields; for example, it selects the nodes in a network that have more chances to lead to cascade effects if hit by a shock (Ballester and Zenou (2006)), or it identifies the nodes that have more influ- ence in the opinion formation and diffusion in a social network (Kempe et al. (2015)), in order to possibly per- form optimal targeting interventions (Galeotti and Goyal (2009), Galeotti et al. (2017)). In the literature different definitions of centrality can be found, such as the degree centrality or the eigenvalue centrality (see for references Latora et al. (2017), Section 2.3); in this paper we focus on the so-called Bonacich centrality measure, introduced in a seminal paper by the American sociologist Bonacich (1987). Formally, the Bonacich centrality π i of a node i in a directed unweighted network is defined as π i “ β ÿ jPN ´ i π j d j `p1 ´ βqη i , (1) where N ´ i is the in-neighborhood of node i in the network, d j is the out-degree of node j , η i can be interpreted as the a-priori centrality of i (possibly the same for all nodes), and β Pp0, 1q is some fixed parameter. Notice that by (1), the centrality of node i depends on the centrality of the ‹ Giacomo Como is also with the Department of Automatic Control, Lund University, Sweden. This work was partially sup- ported by MIUR grant Dipartimenti di Eccellenza 2018–2022 [CUP: E11G18000350001], the Swedish Research Council, and by the Com- pagnia di San Paolo. nodes j linking at i (discounted by the number of their out-links) and on its intrinsic centrality. The centrality of a node is then somewhat inherited by the nodes connected to it: a node is important in the measure that important nodes have a link to it. The Bonacich centrality have found wide applications in many contexts, as in social networks (e.g. representing citations among scientists), in describing Nash equilibria in networked quadratic games (Ballester and Zenou (2006)), in production networks among firms (Acemoglu et al. (2012)), and in opinion dynamics models as the Friedkin- Johnsen model (Friedkin and Johnsen (1990)). A famous instance of the Bonacich centrality is the so-called PageR- ank centrality for web pages, introduced by Brin and Page (1998), which is at the core of modern search engines like Google. Any search query on the web leads indeed to a set of possible related web pages that are sorted and presented according to their centrality ranking by the engine. Due to the relevance of the PageRank centrality for the external visibility of a web page, the problem of understanding how this measure can be efficiently computed and how it can be modified by perturbing the network has recently become very popular; see for example Ishii and Tempo (2014), Como and Fagnani (2015). The effect on the centrality caused by adding or deleting links in the network is not obvious from the recursive definition (1). It is not difficult to see that the addition of a link pi, j q always increases the centrality of the node j ; less clear is how it affects the centrality of node i or, possibly, of all the other nodes in the network. In a context like that of web pages, where each node can decide only where to point its out-links and the arXiv:1911.06737v2 [cs.SI] 6 May 2020