A characterization of stochastically stable networks

Abstract

Jackson and Watts (J Econ Theory 71: 44–74, 2002) have examined the dynamic formation and stochastic evolution of networks. We provide a refinement of pairwise stability, p-pairwise stability, which allows us to characterize the stochastically stable networks without requiring the “tree construction” and the computation of resistance that may be quite complex. When a $$\frac{1}{2}$$ -pairwise stable network exists, it is unique and it coincides with the unique stochastically stable network. To solve the inexistence problem of p-pairwise stable networks, we define its set-valued extension with the notion of p-pairwise stable set. The $$\frac{1}{2}$$ -pairwise stable set exists and is unique. Any stochastically stable networks is included in the $$\frac{1}{2}$$ -pairwise stable set. Thus, any network outside the $$\frac{1}{2}$$ -pairwise stable set must be considered as a non-robust network. We also show that the $$\frac{1}{2}$$ -pairwise stable set can contain no pairwise stable network and we provide examples where a set of networks is more “stable” than a pairwise stable network.