Simple binary-state coordination models are widely used to study collective socio-economic phenomena such as the spread of innovations or the adoption of products on social networks. The common trait of these systems is the occurrence of large-scale coordination events taking place abruptly, in the form of a cascade process, as a consequence of small perturbations of an apparently stable state. The conditions for the occurrence of cascade instabilities have been largely analysed in the literature, however for the same coordination models no sufficient attention was given to the relation between structural properties of (Nash) equilibria and possible outcomes of dynamical equilibrium selection. Using methods from the statistical physics of disordered systems, the present work investigates both analytically and numerically, the statistical properties of such Nash equilibria on networks, focusing mostly on random graphs. We provide an accurate description of these properties, which is then exploited to shed light on the mechanisms behind the onset of coordination/miscoordination on large networks. This is done studying the most common processes of dynamical equilibrium selection, such as best response, bounded-rational dynamics and learning processes. In particular, we show that well beyond the instability region, full coordination is still globally stochastically stable, however equilibrium selection processes with low stochasticity (e.g. best response) or strong memory effects (e.g. reinforcement learning) can be prevented from achieving full coordination by being trapped into a large (exponentially in number of agents) set of locally stable Nash equilibria at low/medium coordination (inefficient equilibria). These results should be useful to allow a better understanding of general coordination problems on complex networks.
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