Multifractal analysis for core-periphery structure of complex networks

Revealing the properties of the core-periphery structure of complex networks can help us to deeply understand their organized principle and function. Studies in recent years have indicated that the k-core and peripheral subnetworks of some model and real-world networks exhibit the fractal behavior, but it remains unclear whether they possess the multifractal behavior. In this paper, we study the multifractal property of these networks and their subnetworks. First, we obtain the sequence of k-cores and their corresponding k-peripheries of original networks by using the k-core decomposition method. We find that the multifractal property exists in these k-cores and k-peripheries of the generalized minimal model network and some Brain networks. Furthermore, the multifractality of the k-cores becomes weaker and that of k-peripheries becomes stronger with the increase of k. The result is consistent with the fact that the k-core becomes more homogenous or cohesive with increasing k. Then we decompose each of these original networks into a densely connected core and a sparsely connected periphery. Although the periphery and the original network have almost the same fractal behavior at the tail part of the resulting curves in the log-log plot for some fractal networks, their multifractality are very different. This means that a unique fractal dimension is not enough to characterize the complexity or spatial heterogeneity of these networks when they take a multifractal structure. Our results show that the multifractal analysis is more powerful than the fractal analysis in characterizing the complexity of the core-periphery structure of complex networks.