Modulated Scale-free Network in the Euclidean Space

Abstract

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i$-th predecessor of degree $k_i$ with a directed link of length $\ell$ using a probability proportional to $k_i \ell^α$. Our numerical study indicates that the network is Scale-free for all values of $α> α_c $ and the degree distribution decays stretched exponentially for the other values of $α$. The link length distribution follows a power law: $D(\ell) \sim \ell^δ$ where $δ$ is calculated exactly for the whole range of values of $α$.