On a Centrality Maximization Game

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.Comment: 10 pages, 11 figure

Endogenous efforts on communication networks under strategic complementarity

This article explores individual incentives to produce information on communication networks. In our setting, efforts are strategic complements along communication paths with convex decay. We analyze Nash equilibria on a set of networks which are unambiguous in terms of centrality. We first characterize both dominant and dominated equilibria. Second, we examine the issue of social coordination in order to reduce the social dilemma.Communication Network, Endogenous Efforts, Strategic Complements

Social Networks

We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. We expose, in particular, the two main ways of modeling network formation. While the physics/applied mathematics approach is capable of reproducing most observed networks, it does not explain why they emerge. On the contrary, the economics approach is very precise in explaining why networks emerge but does a poor job in matching real-world networks. We also analyze behaviors on networks, which take networks as given and focus on the impact of their structure on individuals’ outcomes. Using a game-theoretical framework, we then compare the results with those obtained in sociology.random graph, game theory, centrality measures, network formation, weak and strong ties

Social Networks

We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. We expose, in particular, the two main ways of modeling network formation. While the physics/applied mathematics approach is capable of reproducing most observed networks, it does not explain why they emerge. On the contrary, the economics approach is very precise in explaining why networks emerge but does a poor job in matching real-world networks. We also analyze behaviors on networks, which take networks as given and focus on the impact of their structure on individuals’ outcomes. Using a game-theoretical framework, we then compare the results with those obtained in sociology.Random Graph; Game Theory; Centrality Measures; Network Formation; Weak

Delinquent Networks

Delinquents are embedded in a network of relationships. Social ties among delinquents are modelled by means of a graph where delinquents compete for a booty and benefit from local interactions with their neighbors. Each delinquent decides in a non cooperative way how much delinquency effort he will exert. Using the network model developed by Ballester et al. (2006), we characterize the Nash equilibrium and derive an optimal enforcement policy, called the key-player policy, which targets the delinquent who, once removed, leads to the highest aggregate delinquency reduction. We then extend our characterization of optimal single player network removal for delinquency reduction, the key player, to optimal group removal, the key group. We also characterize and derive a policy that targets links rather than players. Finally, we endogenize the network connecting delinquents by allowing players to join the labor market instead of committing delinquent offenses. The key-player policy turns out to be much more complex since it depends on wages and on the structure of the network.Social networks, delinquency decision, key group, NP-hard problem, crime policies

Strategic Interaction and Networks

This paper brings a general network analysis to a wide class of economic games. A network, or interaction matrix, tells who directly interacts with whom. A major challenge is determining how network structure shapes overall outcomes. We have a striking result. Equilibrium conditions depend on a single number: the lowest eigenvalue of a network matrix. Combining tools from potential games, optimization, and spectral graph theory, we study games with linear best replies and characterize the Nash and stable equilibria for any graph and for any impact of players’ actions. When the graph is sufficiently absorptive (as measured by this eigenvalue), there is a unique equilibrium. When it is less absorptive, stable equilibria always involve extreme play where some agents take no actions at all. This paper is the first to show the importance of this measure to social and economic outcomes, and we relate it to different network link patterns.Networks, potential games, lowest eigenvalue, stable equilibria, asymmetric equilibria

Communication, coordination and networks

We study experimentally how the network structure and length of pre-play communication affect behavior and outcome in a multi-player coordination game with conflicting preferences. Network structure matters but the interaction between network and time effects is more subtle. Under each time treatment, substantial variations are observed in both the rate of coordination and distribution of coordinated outcomes across networks. But, increasing the communication length improves both efficiency and equity of coordination. In all treatments, coordination is mostly explained by convergence in communication. We also identify behaviors that explain variations in the distribution of coordinated outcomes both within and across networks

Small World Networks with Segregation Patterns and Brokers

Many social networks have the following properties: (i) a short average distance between any two individuals; (ii) a high clustering coefficient; (iii) segregation patterns; the presence of (iv) brokers and (v) hubs. (i) and (ii) define a small world network. This paper develops a strategic network formation model where agents have heterogeneous knowledge of the network: cognizant agents know the whole network, while ignorant ones are less knowledgeable. For a broad range of parameters, all pairwise Nash (PN) networks have properties (i)-(iv). There are some PN networks with one hub. Cognizant agents have higher betweenness centrality: they are the brokers who connect different parts of the network. Ignorant agents cause the emergence of segregation patterns. The results are robust to varying the number of cognizant agents and to increasing the knowledge level of ignorant ones. An application shows the relevance of the results to assessing the welfare impact of an increase in network knowledge due to, e.g., improved access to social networking tools.Network, Cognitive Network, Small World, Broker, Segregation

On a Network Centrality Maximization Game

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.Comment: 42 pages, 11 figure