Caminhadas aleatórias sobre redes com distribuição de grau Beta

Abstract

In this work, the study of Random Walks in complex networks with Beta degree distribution was utilised, addressing the interpretations of classical and quantum mechanics. The network was modelled as a scale-free tree. Thus, by applying a walker to the network, the probability of returning to the original position in the degrees and nodes of the network was analysed, using continuous-time walks to describe its dynamics. The methodology involved computational simulations, including the diagonalisation of the connectivity matrix generated by the Beta degree network, associated with the return probabilities in both classical and quantum models, as well as the exact and approximate mean value of the walker, aiming to determine its efficiency. In the classical results, by fixing the adjustment parameter b and varying a of the Beta degree network, the probability curves showed more visible phase differences with higher return probability values as a increased. In contrast, in the quantum approach, high probabilities were associated with lower values of a. By keeping a fixed and varying b, the classical model showed intertwined or linearised behaviours, with higher return probability as b increased, whereas the highest quantum probabilities occurred for lower values of b. Additionally, greater efficiency was observed with an increase in a for fixed b, and for fixed a, efficiency was more pronounced at lower values of b.