Geographical networks evolving with an 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 node, having randomly assigned coordinates in a $1\ifmmode\times\else\texttimes\fi{}1$ square, is added and connected to a previously existing node $i$, which minimizes the quantity ${r}_{i}^{2}∕{k}_{i}^{\ensuremath{\alpha}}$, where ${r}_{i}$ is the geographical distance, ${k}_{i}$ the degree, and $\ensuremath{\alpha}$ a free parameter. The degree distribution obeys a power-law form when $\ensuremath{\alpha}=1$, and an exponential form when $\ensuremath{\alpha}=0$. When $\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}$ exceeds the critical value ${\ensuremath{\alpha}}_{c}=1$, 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.