Fractal boundaries of complex networks

¹  Center for Polymer Studies and Department of Physics, Boston University - Boston, MA 02215, USA
²  Department of Physics, Yeshiva University - 500 West 185th Street, New York, NY 10033, USA
³  Department of Mathematics, Bar-Ilan University - 52900 Ramat-Gan, Israel
⁴  Minerva Center and Department of Physics, Bar-Ilan University - 52900 Ramat-Gan, Israel

Corresponding author: jiashao@buphy.bu.edu

Received: 12 May 2008
Accepted: 16 October 2008

Abstract

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.