Modulated Scale-free Network in the Euclidean Space

A random network is grown on the Euclidean space by introducing at unit rate randomly selected points as the nodes of the network. During the growth a node is randomly connected to one of its predecessors with a directed link of length $\ell$ whose probability is jointly proportional to the degree of the node as well as a length dependent factor $\ell^{\alpha}$. Our numerical study indicates that the network is Scale-free for all values of $\alpha > \alpha_c = 1-d$ in $d$ dimensions and the degree distribution decays exponentially for other values of $\alpha$.