Estudo de redes complexas sob uma perspectiva combinatória

Abstract

Many problems in several areas of knowledge can be approached through Complex Systems Theory, characterized as problems where a very large number of components interact with each other, promoting the emergence of collective phenomena. The approach to such systems are often made through Complex Networks, which among their most diverse models, can serve in the understanding and treatment of these systems, based on the mapping of these systems through the so-called graphs. One of the information that can be accessed through the Network Theory are the phenomena associated with the species of phase transitions in the system, this through the idea of Percolation, which describes the transition between a regime in which there are several isolated components in the structure and another in which the formation of a giant component occurs. We established a methodology based on combinatorial arguments that leads to the characterization of the connectivity distribution and the behavior of the component size distribution as a function of the average network connectivity up to the percolation threshold for random graphs, by counting and maximizing the ways of building the structure with a given number of vertices. We also use the combinatorial analysis and arrangements ideas to establish the enumeration result for rooted and targeted forests with a certain distribution of connectivity. Keywords: Complex