The Fractal Dimensions of Complex Networks

It is shown that many real complex networks share distinctive features, such as the small-world effect and the heterogeneous property of connectivity of vertices, which are different from random networks and regular lattices. Although these features capture the important characteristics of complex networks, their applicability depends on the style of networks. To unravel the universal characteristics many complex networks have in common, we study the fractal dimensions of complex networks using the method introduced by Shanker. We find that the average `density' <ρ(r)> of complex networks follows a better power-law function as a function of distance r with the exponent d_f, which is defined as the fractal dimension, in some real complex networks. Furthermore, we study the relation between d_f and the shortcuts N_add in small-world networks and the size N in regular lattices. Our present work provides a new perspective to understand the dependence of the fractal dimension d_f on the complex network structure.