Percolation of spatially constraint networks

Daqing Li^1a, Guanliang Li², Kosmas Kosmidis³, H. E. Stanley², Armin Bunde⁴ and Shlomo Havlin¹

¹ Minerva Center and Department of Physics, Bar-Ilan University - Ramat-Gan 52900, Israel
² Center for Polymer Studies, Department of Physics, Boston University - 590 Commonwealth Avenue, Boston, MA 02215, USA
³ Physics Department and Network Operations Center, Aristotle University of Thessaloniki 54124 Thessaloniki, Greece, EU
⁴ Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen - 35392 Giessen, Germany, EU

^a li.daqing.biu@gmail.com

Received: 6 January 2011
Accepted: 27 February 2011

Abstract

We study how spatial constraints are reflected in the percolation properties of networks embedded in one-dimensional chains and two-dimensional lattices. We assume long-range connections between sites on the lattice where two sites at distance r are chosen to be linked with probability p(r) ∼ r^{− δ}. Similar distributions have been found in spatially embedded real networks such as social and airline networks. We find that for networks embedded in two dimensions, with 2 < δ < 4, the percolation properties show new intermediate behavior different from mean field, with critical exponents that depend on δ. For δ < 2, the percolation transition belongs to the universality class of percolation in Erdös-Rényi networks (mean field), while for δ > 4 it belongs to the universality class of percolation in regular lattices. For networks embedded in one dimension, we find that, for δ < 1, the percolation transition is mean field. For 1 < δ <2, the critical exponents depend on δ, while for δ > 2 there is no percolation transition as in regular linear chains.