Modulated scale-free network in Euclidean space

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i\mathrm{th}$ predecessor of degree $k_i$ with a directed link of length $\mathcal{l}$ using a probability proportional to $k_i{\mathcal{l}}^{\alpha }.$ Our numerical study indicates that the network is scale free for all values of $\alpha >{\alpha }_c$ and the degree distribution decays stretched exponentially for the other values of $\alpha .$ The link length distribution follows a power law: $D\left(\mathcal{l}\right)\sim {\mathcal{l}}^{\delta },$ where $\delta$ is calculated exactly for the whole range of values of $\alpha .$