Scale-free network on a vertical plane

Abstract

A scale-free network is grown in the Euclidean space with a global directional bias. On a vertical plane, nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which are below it using the attachment probability: ${\pi }_i(t)\sim k_i(t){\ell }^{\alpha }$. Our numerical results indicate that the directed scale-free network for $\alpha =0$ belongs to a different universality class compared to the isotropic scale-free network. For $\alpha <{\alpha }_c$ the degree distribution is stretched exponential in general which takes a pure exponential form in the limit of $\alpha \to -\infty$. The link length distribution is calculated analytically for all values of $\alpha$.

4 pages, 4 figures