Pan, Gui-JunGui-JunPanDr. NIU Ruiwu2025-08-272025-08-272016Physica A: Statistical Mechanics and its Applications, 2016, vol. 463, pp. 509-515.0378-43711873-2119http://hdl.handle.net/20.500.11861/24725We investigate the problem of the particle or message that travels as a biased random walk toward a target node in Kleinberg’s spatial network which is built from a -dimensional (d=2) regular lattice improved by adding long-range shortcuts with probability P (rij)~rij-a, where rij is the lattice distance between sites i and j, and a is a variable exponent. Bias is represented as a probability of the packet to travel at every hop toward the node which has the smallest Manhattan distance to the target node. We study the mean first passage time (MFPT) for different exponent and the scaling of the MFPT with the size of the network . We find that there exists a threshold probability pth ≈0.5 , for p≥pth the optimal transportation condition is obtained with an optimal transport exponent α= d, while for 0<p<pth, αop ≠ d, the value of αop depends on p, and when p=0, α=0 . Specially, when α=αop, the MFPT may scale logarithmically with L for p>pth , and increases with L less than a power law and get close to logarithmical law for 0<p<pth. Our results indicate that the proper addition of shortcuts to a regular substrate can lead to a formation of complex network with a highly efficient structure for navigation although nodes hold null local information with a relatively large probability, which gives a powerful evidence for the reason why many real networks’ navigability have small world property.enPeer Reviewed Journal Article10.1016/j.physa.2016.07.036