Percolation of spatially constraint networks

We study how spatial constraints are reflected in the percolation properties of networks embedded in one-dimensional chains and two-dimensional lattices. We assume longrange 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. Copyright c © EPLA, 2011 Complex networks have attracted considerable attention in the last decade [1–12]. It has been realized that networks provide a very useful way to describe and better understand the collective behavior of complex systems composed of a large number of interacting entities. There are two network classes of particular interest: Erdös-Rényi (ER) random graphs [13] and Barabasi-Albert (BA) scalefree networks [14]. In ER networks, the distribution of the degrees k (number of links) of the nodes is a Poissonian (P (k)∼ λ/k!, where λ is the average degree), while in BA scale-free networks, the distribution follows a power law, P (k)∼ k−γ , with γ typically between two and three. Both classes have interesting topological properties, considerably different from those of regular lattices. Both exhibit the “small world” effect meaning that their topological diameter increases slowly, either logarithmically or double logarithmically, with the system size [9,10]. When studying the properties of networks it is usually assumed that spatial constraints can be neglected. (a)E-mail: [email protected] This is probably true for certain networks such as the WorldWideWeb (WWW) or the citation network where the real (Euclidean) distance does not play a role. In contrast, the spatial distance does play a role in the Internet [15,16], airline networks [17,18], human travel networks [12,19], wireless communication networks [20] and social networks [21,22], which are all embedded in two-dimensional space. It has recently been shown that these spatial constraints are important and in certain cases can significantly alter the topological properties of the networks [23–32]. In this letter, we study the robustness of spatially constrained networks embedded in one or two-dimensional space, by analyzing their percolation properties. Percolation is important since it can shed light also on epidemic spreading [33] and immunization strategies [34]. Here, we focus on spatially embedded ER networks where lattice nodes are connected to each other with a probability p(r)∼ r−δ, where r is the Euclidean distance between the nodes. The choice of a power law for the distance distribution is supported from findings in several real