Dynamics on spatial networks and the effect of distance coarse graining

Recently, spatial networks have attracted much attention. The spatial network is constructed from a regular lattice by adding long-range edges (shortcuts) with probability P ( r ) ∼ r − δ , where r is the geographical distance between the two ends of the edge. Also, a cost constraint on the total length of the additional edges is introduced ( ∑ r = C ). It has been pointed out that such networks have optimal exponents δ for the average shortest path, traffic dynamics and navigation. However, when δ is large, too many generated long-range edges will be added to the network. In this scenario, the total cost constraint cannot be satisfied. In this paper, we propose a distance coarse graining procedure to solve this problem. We find that the optimal exponents δ for the traffic process, navigation and synchronization indeed result from the trade-off between the probability density function of long-range edges and the total cost constraint, but the optimal exponent δ for percolation is actually due to dissatisfying the total cost constraint. On the other hand, because the distance coarse graining procedure widely exists in the real world, our work is also meaningful in this aspect.