Degree heterogeneity in spatial networks with total cost constraint

Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability $P_{ij}\sim r_{ij}^{-\alpha}$, where $r_{ij}$ is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint ($\sum r=C$). These networks have fixed optimal exponent $\alpha$ for transportation (measured by the average shortest-path length). However, we observe that the degree in such spatial networks is homogenously distributed, which is far different from real networks such as airline systems. In this paper, we propose a method to introduce degree heterogeneity in spatial networks with total cost constraint. Results show that with degree heterogeneity the optimal exponent shifts to a smaller value and the average shortest-path length can further decrease. Moreover, we consider the synchronization on the spatial networks and related results are discussed. Our new model may better reproduce the features of many real transportation systems.