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The project started with the analysis of a previously proposed model of hierarchical deposition of blocks. We extended the study of the behavior of the resulting surfaces to a more detailed analysis which includes the level sets. It was shown that, depending on the deposition probabilities, they display a rich variety of geometries which go all the way from Euclidean objects to classical fractals passing through the special case of logarithmic fractals. Applications to biofilms were also presented, based on a microscopic model that, with very elementary assumptions, leads to geometries similar to the ones imposed a priori in a previous model. The main part of the thesis considers the subject of scale-free networks with connectivity-dependent interactions. A fundamental relation which links topology and interactions on such networks was proven analytically and verified by Monte-Carlo simulations. The relation was initially proven for the Ising model but it was later shown that, sometimes in a slightly modified form, it applies to a variety of phenomena, including sand-pile models and contact processes. An extremely simple model for distribution networks was also proposed, which allows the modeling of failure-propagation through a network. The third part of the thesis concerns the possibility of a significant increase of the critical temperature in the bulk of a superconductor, in which artificial enhancement centers have been deposited. Within the limits of the Ginzburg-Landau theory, our calculations provide a simple analytical approximation, which is very close to the full numerical solution. The results are in agreement with the significant increase of the critical temperature observed in Ga-sponges. The thesis consists of three distinct but interrelated parts. In each of them it is shown that when peculiar structures or geometries are involved, previously known phenomena might display very different properties. When pieces of different sizes are deposited on a surface, the big ones first, the resulting surfaces have a peculiar fractal structure, as was shown in a previously proposed hierarchical deposition model. Here, the “coastlines” of this model are studied and it is shown that they display a rich variety of geometries, which go all the way from Euclidean objects to classical fractals passing through the special case of logarithmic fractals. The model might have applications in cluster deposition in applied surface physics or in the development of bacterial biofilms. The main part of the thesis considers the subject of scale-free networks with connectivity-dependent interactions. For a long-time the structures encountered in sociology, lexicology, biology, etc., were out of the reach for the usual techniques applied in exact sciences on regular lattices. This changed very recently with the introduction of the concepts of small-world networks and scale-free networks. The simplest form of interaction among the nodes (1 if two nodes are connected, 0 if not) was enough to reveal an astonishingly rich behavior, depending on the topology of the network. Connectivity-dependent interactions provide a new insight into the properties of scale-free networks, with possible applications in fields ranging from power-grid collapses to opinion formation. Using a wide range of techniques the conclusion was reached that the behavior of scale-free networks depends on the form of the connectivity-dependent interactions in a similar way as on the topology. More than that, topology and interactions can be interchanged, “traded” by a simple rule. Real-life networks that usually were considered not to have critical phase transitions according to their topology may display them by virtue of non-trivial interactions. After a detailed discussion at a theoretical level a simple application is presented that should be considered when analyzing power grid failures. The third part of the thesis concerns the possibility of a significant increase of the critical temperature in a superconductor. Provided artificial enhancement centers have been deposited inside the bulk, one can observe a phenomenon analogous to wetting, in which a layer of superconductivity appears at the border of these centers at a temperature higher than the usual critical temperature. The conditions under which these layers can percolate leading to a superconducting phase extended in the bulk are studied with a simple analytical approximation. The results are in agreement with the numerical solution and agree with the increase of the critical temperature in Ga-sponges observed experimentally.