Caracterização de sistemas dinâmicos através da simetria de vértices em redes

Abstract

Symmetry is a concept explored by several areas of science, being always associated with the observation of parts, be it physical or not, that structurally coincide. The simplest problems that deal with symmetry, look for defining if an element is symmetrical or not by the possibility of break it into equal parts. In the most complex problems, the search for patterns, redundancy and regularity takes quantitative terms for the analysis of symmetry. At this juncture, this work proposes a quantitative analysis in order to obtain a measure of symmetry of network nodes by means of the Jensen-Shannon divergence (J), using as probability distribution functions (PDF) as node distance distribution (NDDs) of the networks. Methodologies have been proposed for the analysis and use of the symmetry measure of the network nodes. At first, the behavior of the symmetry of the network nodes of the Barabási-Albert model was observed during the evolution of the network, in order to understand how the measure of symmetry behaves with this evolution. Then, time series of fractional Brownian motion (MBF) were converted into networks using the method proposed by Luque et al. [2009], Horizontal Visibility Graph (HVG), in which it transforms time series into networks and the thus, we propose an application of the measure of symmetry of the nodes of the generated networks for the characterization of the Hurst Exponent (H) of the series. Finally, based on the Pearson correlation, within a time interval between the time series of the stocks that make up the S&P500, networks were built with the purpose of proposing an alternative to replace stocks within a individual stocks portfolio. The yields of the stock portfolios were checked for the replacement of some stocks by others symmetrical to them, and compared to the yield when this replacement is made by stocks of greater correlation and by totally random substitution. The applicability of the symmetry measure of the network nodes, together with the methodologies proposed here, proved to be efficient for the suggested problems.