Modulated Scale-free Network in the 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 $i$-th predecessor of degree $k_i$ with a directed link of length $\mathrm{\ell }$ using a probability proportional to $k_i{\mathrm{\ell }}^{\alpha }$. 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: $D(\mathrm{\ell })\sim {\mathrm{\ell }}^{\delta }$ where $\delta$ is calculated exactly for the whole range of values of $\alpha$.