Geographical networks evolving with optimal policy

In this article, we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a new node, having randomly assigned coordinates in a $1 \times 1$ square, is added and connected to a previously existing node $i$, which minimizes the quantity $r_i^2/k_i^\alpha$, where $r_i$ is the geographical distance, $k_i$ the degree, and $\alpha$ a free parameter. The degree distribution obeys a power-law form when $\alpha=1$, and an exponential form when $\alpha=0$. When $\alpha$ is in the interval $(0,1)$, the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge-length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge-length will sharply increase when $\alpha$ exceeds the critical value $\alpha_c=1$, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.