Tesis doctoral de Michele Catanzaro
Many natural, technological, and social systems are made of a large amount of elements connected by an irregular pattern of interactions (for example, ecosystems, the internet, social relations, etc.). In the last few years, some advances in understanding these systems have been made by mapping them into a simpler geometric structure, a graph: a collection of dots connected by lines. Statistical measures performed on graphs representing real-world systems have revealed features (like a great heterogeneity in the degree. I.E. The number of connections attached to a node) that are common to widely different systems and deviate both from a regular pattern of interactions and from a completely random one. real-world networks are the scenario of a wide variety of dynamical processes. These dynamics are strongly influenced by the topology of the underlying networks. In order to study them from a numerical point of view, it is necessary to model the networked substrate where the dynamics takes place. We focus our attention on the static model, that allows to generate networks with the desired degree heterogeneity. We carry out its analytic solution and check it by numerical simulations, revealing the presence of degree correlations in the model. When using this model for the study of dynamical processes, it is thus impossible to disentangle the respective effects of degree heterogeneity and correlations. We show that this problem is shared by most customarily used network models. In order to overcome it, we propose the uncorrelated configuration model (ucm), that allows maximal freedom in the manipulation of the degree heterogeneity and still avoids all correlations. a framework that allows the modeling a wide class of dynamical processes is the theory of reaction-diffusion processes, where particles diffuse and react with each other on a substrate. In lattices, these processes can be analyzed by means of renormalization group techniques. A successful approach to analyze them in networks is the heterogeneous mean field theory, that takes into account the degree heterogeneity. We develop this theory for large classes of dynamical processes. First of all, we focus on dynamics with an exclusion principle (i.E. With fermionic particles). We carry out the analytical solution of the diffusion-annihilation process (a dynamics decaying towards an absorbing state) and of the branching-annihilating random walk (a process displaying a non-equilibrium phase transition), and check it by numerical simulations, using the ucm. We reveal important differences with the results obtained for lattices: for example, the first dynamics is remakably faster, and the phase diagram of the second dynamics has a different shape. then, we focus on dynamics without exclusion principles (i.E. With bosonic particles). In this case, we can carry out the analysis of the whole classes to which the previously studied processes belong (respectively, that of decay processes and that of steady-state processes). We develop the general solution of these classes, that encompasses a wide range of dynamical systems and check it by means of numerical simulations (using the ucm) with a recipe especially designed for bosonic processes. When possible, we perform a comparison between fermionic and bosonic processes, revealing both differences and similarities. finally, we focus on the effect on dynamics of imposing non-local constraints to networked substrates, like the absence of loops between network nodes. We find that even a simple random walk is strongly affected by these constraints. We use fermionic diffusion-annihilation process to sample the effects on more complex dynamics, revealing important deviations from the general theoretical results. We explain them in terms of a slowing-down effect that sets on in the dynamics that are the building blocks of the diffusion-annihilation, namely the diffusive trapping and the diffusive caputure processes.
Datos académicos de la tesis doctoral «Dynamical processes in complex networks«
- Título de la tesis: Dynamical processes in complex networks
- Autor: Michele Catanzaro
- Universidad: Politécnica de catalunya
- Fecha de lectura de la tesis: 10/07/2008
Dirección y tribunal
- Director de la tesis
- Romualdo Pastor Satorras
- Tribunal
- Presidente del tribunal: alessandro Vespignani
- María de los ángeles Serrano moral (vocal)
- alain Barrat (vocal)
- albert Díaz guilera (vocal)