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: $π_i(t) \sim k_i(t)\ell^α$. Our numerical results indicate that the directed scale-free network for $α=0$ belongs to a different universality class compared to the isotropic scale-free network. For $α< α_c$ the degree distribution is stretched exponential in general which takes a pure exponential form in the limit of $α\to -\infty$. The link length distribution is calculated analytically for all values of $α$.

4 pages, 4 figures