MiS Preprint Repository

This work is a modeling of evolutionary networks embedded in one or two dimensional configuration space. The evolution is based on two attachments depending on degree and spatial distance. The probability for a new node $n$ to connect with a previous node $i$ at distance $r_{ni}$ follows $a \frac{k_i}{\sum_j k_j}+(1-a)\frac{r^{-\alpha}_{ni}}{\sum_j r^{-\alpha}_{nj}}$, where $k_i$ is the degree of node $i$, $\alpha$ and $a$ are tunable parameters. In spatial driven model ($a = 0$), the spatial distance distribution follows the power-law feature. The mean topological distance $l$ and the clustering coefficient $C$ exhibit phase transitions at same critical values of $\alpha$ which change with the dimensionality d of the embedding space. When $a \le 0$, the degree distribution follows the "shifted power law" (SPL) which interpolates between exponential and scale-free distributions depending on the value of $a$.