Fractal boundaries of complex networks

We introduce the concept of the boundary of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundary nodes seen from a given node of complex networks. We find that for both Erdős-Rényi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes B follows a power law probability density function which scales as B -2 . The clusters formed by the boundary nodes seen from a given node are fractals with a fractal dimension d f ≈2. We present analytical and numerical evidences supporting these results for a broad class of networks.