Phase transitions on heterogeneous random graphs: some case studies

The focus of this thesis is about statistical mechanics on heterogeneous random graphs, i.e. how this heterogeneity affects the cooperative behavior of model systems. It is not intended as a review on it, rather it is showed how this question emerges naturally and can give useful insights to specific instances. The first chapter is about the statistical mechanics of congestion in queuing networks. The second is devoted to the study of the glassy dynamics of facilitated spin models on disordered structures. In the third chapter, the presence of inverse phase transitions in tri-critical spin systems on heterogeneous random graphs is pointed out. Finally, the last chapter is on the role of volatility in the evolution of social networks. In the conclusions, a general insight about the interplay between structure and dynamics on heterogeneous random graphs is given. It is based on the different scaling of the transition point with the moments of the degree distribution for continuous and discontinuous transitions, respectively.