Novo modelo hierárquico para decomposição do backbone de percolação revela novas leis de escala da distribuição de correntes

Abstract

In this work, we conducted two studies. First, we applied the Boltzmann Machine Learning lgorithm to analyze eye movement data during the reading of different types of texts. We found that we can describe the complexity of texts through the average magnetization, and that the distance between the reading temperature and critical temperature (To − Tc) is capable of reflecting their coherence. Second, we studied the properties of the percolation aggregate, more specifically, the application of the percolation model in the random resistors network, where each bond in the network has a resistance. When a current is introduced into the network, it is distributed through each bond according to Kirchhoff’s laws. Many of the properties of the random resistor system are obtained from the probability distribution P(i). Previously, it was believed that the distribution of currents in the backbone at the critical percolation point followed a log-normal distribution, like a simple hierarchical model. However, later it was observed that the distribution of currents in the backbone of the critical percolation aggregate did not follow a log-normal distribution. Due to the self-similarity of the critical percolation backbone, we created a hierarchical model by decomposing the backbone into triconnected components. This allowed us to discover that the distribution of currents is formed by two distributions: one corresponds to the distribution of the number of connections at each level, and the other corresponds to the distributions of the multiplicative factors that make up the currents at each level. Our decomposition methodology also allowed us to accurately find small currents up to 10−35 for systems up to L = 8192.