Characterization of the clustering phase transition of a complex network embedded in a hyperbolic plane

Abstract

English: If we distribute nodes homogeneously in an hyperbolic plane and connect each possible pair of nodes with a probability that depends on the hyperbolic distance among them, heterogeneous degree distributions and strong clustering emerge naturally. Both metrics are key properties observed in real complex networks but are rarely seen together in standard network models. Our model considers edges in a network as noninteracting fermions whose energies are equal to the hyperbolic distances between nodes. This interpretation allows us to use statistical mechanics methods, like the Metropolis Hastings algorithm, in order to perform numerical simulations and to get precise measurements of the network properties. In this master thesis, we focus on the study of clustering, which undergoes a phase transition at a certain critical temperature. We develop an analytical framework to obtain the critical exponents of this phase transition and compare them with numerical simulations. Finally, we check whether the Finite Size Scaling (FSS) assumption holds in this case or not.