A Fixed-Mass multifractal approach for unweighted complex networks

Abstract

Complex networks have been studied in recent years due to their relevance in biological, social and technical real systems, such as the world wide web, social networks and biochemical interactions. One of the most current features of complex networks is the presence of (multi-)fractal properties. In spite of the amount of contributions that have been developed in fractal and multifractal analysis, all of them have focused on the adaptation of Fixed size algorithms (FSA) to complex networks, mostly box-counting and sandbox procedures. In this manuscript, a Fixed-mass algorithm (FMA) is adapted to explore the (multi-)fractality of unweighted complex networks is first proposed on the basis on a box-counting procedure. Thus, the aim of this work is to explore the applicability of a FMA to study fractal and multifractal characteristics of unweighted complex networks. After describing the proposed FMA for complex networks, it is tested in several kinds of complex networks such as deterministic, synthetic and real ones. Results suggest that FMA is able to adequately reproduce the fractal and multifractal properties of synthetic complex networks (scale-free, small-world and random). In addition, the algorithm is used for the multifractal description of real complex networks. The main advantage of FMA over FSA is that the minimum configuration of boxes (ideal) for completely covering the network is known. Therefore, results are quite precise when a reasonable covering configuration is obtained, allowing the reconstruction of the Mass exponent, τ , and the Dimension function, D(q), with proper goodness-fits.