We consider a general set of stochastic models of social network formation and evolution. In these models, agents form and sever links in order to increase their payoff from interacting with others, which depends on the centrality of their position in the network. Since the centrality of an agent in a dynamic network is an important determinant in many economic phenomena, the possible applications of this work range from communication networks to business and R&D networks. In the economic and sociology literature, there are different characterizations of centrality, which are designed to highlight the differences between important and non-important agents (Ballester 2006). With this work, we provide a tractable model that is able to unify and compare the impact of different centrality measures on heterogeneous agents maximizing their centrality embedded in a dynamic social network.

In particular, we consider three different measures of centrality, based on the degree of the agents, the eigenvector- (Wasserman 1994) or eigenvalue-centrality (Koenig et al. 2008) associated with the adjacency matrix representing the interactions between agents. The model allows to include unintentional mistakes (noise) made by the agents when selecting the best possible acquaintance. This means that agents are bounded-rational and they are exposed to perturbations in a network which is permanently out-of-equilibrium. We show that, when the level of noise vanishes, all three measures of centrality can be unified in a basic model in which, agents are simultaneously optimizing their degree, eigenvector- and eigenvalue-centrality.

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