Breaking of ensemble equivalence for complex networks

In equilibrium statistical physics, the micro-canonical ensemble is a distribution on the configuration space of the system where the energy is fixed, while the canonical ensemble is a distribution where the temperature is fixed. In standard textbooks it is claimed that these ensembles become equivalent in the thermodynamic limit, i.e., as the system size tends to infinity the fluctuations of the energy around it average value become negligible. While this is certainly true for many physical systems, it fails in systems that have long-range interactions. In recent work in the physics literature it has become apparent that complex networks are candidates for breaking of ensemble equivalence. The goal is to classify for which classes of complex networks this is indeed the case. The phenomenon is linked to possible non-convexity of large deviation rate functions for the energy.  

Another line of thought is that the breaking of ensemble equivalence may be connected to dynamical Gibbs-non-Gibbs transitions. Here, a system that is prepared in a Gibbs state and is subjected to a stochastic dynamics may no longer be Gibbs at a later time, i.e., it may lose Gibbsianness. At a later time it may or may not recover Gibbsianness. Several scenarios are possible. The Gibbs-non-Gibbs crossover is known to be related to bifurcations in the minima of large deviation rate functions for trajectories of physical quantities like energy. It would be interesting to see whether there is a link with the breaking of ensemble equivalence especially for complex networks.