Fractal boundaries of complex networksJia Shao1, Sergey V. Buldyrev1, 2, Reuven Cohen3, Maksim Kitsak1, Shlomo Havlin4 and H. Eugene Stanley1
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
received 12 May 2008; accepted in final form 16 October 2008; published November 2008
published online 21 November 2008
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 df 2. We present analytical and numerical evidences supporting these results for a broad class of networks.
89.75.Hc - Networks and genealogical trees.
89.75.-k - Complex systems.
64.60.aq - Networks.
© EPLA 2008