Core-periphery disparity in fractal behavior of complex networks

Abstract

We show that there is a disparity in fractal scaling behavior of the core and peripheral parts of empirical small-world scale-free networks. We decompose the network into a core and a periphery and measure the fractal dimension of each part separately using the box-counting method. We find that the core of small-world scale-free networks has a nonfractal structure, whereas the periphery exhibits either fractal or nonfractal scaling. The fractal dimension of the periphery is found to coincide with one for the whole network.

Multifractal analysis for core-periphery structure of complex networks

“…They revealed dierent fractal behaviors between the core and the periphery for some empirical networks, including the WWW network, cellular networks, the protein-protein interaction network of Saccharomyces cerevisiae, and the Internet network at the autonomous system level. More specifically, the resulting curve from the core always displays the exponential behavior in the log-log plot, while the tail part of the curve from the periphery is always consistent with the one from the original network, whether the original network has the exponential or power-law behavior [29]. In addition, they also studied the fractal property of the core and peripheral parts of a fractal model network [20] with some shortcuts, which are added to the model network according to the distance l between nodes with probability p(l) ∼ l −α , where l > 1.…”

“…Thus, a network can be decomposed into a sequence of core-periphery structures. In addition, we adopt the criterion C introduced by Moon et al [29] to separate the core and the periphery from the original network. Before we use the modified sandbox algorithm to perform the multifractal analysis for these subnetworks, we need to calculate the distance between any two nodes of each subnetwork.…”

“…It is known that the fractality, as one of the fundamental properties of complex networks, can be revealed eectively by using some fractal analysis algorithms to calculate their fractal dimensions [19][20][21][22][23][24][25][26][27]. Recently, researchers uncovered the existence of fractality in these subnetworks of some real-world and model networks, and made a detailed comparison with their original networks from the perspective of fractality [28,29]. Kim et al [28] decomposed the world wide web (WWW) network into a sequence of k-cores by the k-core decomposition method.…”

“…Kim et al [28] decomposed the world wide web (WWW) network into a sequence of k-cores by the k-core decomposition method. They performed the fractal analysis for these k-cores and found that there are three dierent fractal behaviours in these k-cores for dierent k. Moon et al [29] decomposed each of some empirical networks possessing the small-world and the scale-free properties with a degree exponent γ (2 < γ < 3) into a core and a periphery, and then studied their fractal properties respectively. They revealed dierent fractal behaviors between the core and the periphery for some empirical networks, including the WWW network, cellular networks, the protein-protein interaction network of Saccharomyces cerevisiae, and the Internet network at the autonomous system level.…”

Modeling the self-similarity in complex networks based on Coulomb’s law

“…Recently, fractal and self-similarity of complex networks have attracted much attention [30,31,32,24,33,34,35,36]. Many researchers have analyzed the fractal property of complex networks and proposed different algorithms to calculate the fractal dimension of complex networks [37,38,39,40,41,42,17]. One of the progress in this field is that Wei et al studied the self-similarity of weighted complex networks [12].…”