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Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the $dk$-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks---the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain---and find that many important local and global structural properties of these networks are closely reproduced by $dk$-random graphs whose degree distributions, degree correlations, and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate $dk$-random graphs.

Projecte final de Màster Oficial reaizat en col.laboració amb Universitat de Barcelona. Departament de Física Fonamental., 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.