Geographical effects on the path length and the robustness in complex networks

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale‐free(SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical network with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links