Structural holes in social networks: A remark

In a recent article, [S. Goyal, F. Vega-Redondo, Structural holes in social networks, J. Econ. Theory 137 (1) (2007) 460–492] the authors (GVR) showed the importance of stars and cycles in a given network formation game. Implicitly, in their article, a network is called an equilibrium if it is generated by an equilibrium strategy. We extend the results of GVR to the case of a stronger requirement: namely, that a network can be called an equilibrium only if all the strategies generating it are equilibria. We also show, in a dynamic framework, that both definitions differ in crucial ways.