Analysis of the relationship between multifractality and robustness in complex networks

Abstract

Complex networks are a strategy to study different real systems through graph-based representation, which allows their observation with different graph measures, such as degree distribution, clustering coefficient, among others. However, these measures are based on the whole of the network, thus, it hides the macro-structures that composes the network.To study the macro-structures in the networks, an approach is the application of the Multifractal Analysis (MFA), which consists in the measure of fractal dimensions in different scales of the network, allowing the observation of different structures into the network. Nevertheless, by using MFA, the observation of the dynamics of the network is unable when it loses nodes or arcs due to a perturbation. On the other hand, the Robustness Analysis (RA) provides a useful tool to study the network dynamics because It gives measures of the network when it loses nodes or arcs.For the above, the combination of the MFA and RA (MFA-RA) can be a strategy to study the dynamics of the macro-structures into the networks. Our experiments applying MFA and RA in different scale-free networks, small-world networks and random networks presented evidence that the measure applying MFA-RA can be used to categorize different types of networks. We can conclude that the study of the dynamic of macro-structures into the networks can provide a most-completed measure to categorize them.