Phase transitions on heterogeneous random graphs: some case studies

TL;DR

This thesis investigates how heterogeneity in random graphs influences phase transitions and cooperative phenomena across various systems, revealing the impact of network structure on dynamic behaviors.

Contribution

It provides case studies demonstrating the effects of heterogeneity on phase transitions and dynamics in different models, offering new insights into structure-dynamics interplay.

Findings

Inverse phase transitions occur in tri-critical spin systems on heterogeneous graphs.

Transition points scale differently with degree distribution moments for continuous and discontinuous transitions.

Heterogeneity significantly affects the cooperative behavior and phase transition properties.

Abstract

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.