Optimal transport in time-varying small-world networks

The time-order of interactions, which is regulated by some intrinsic activity, surely plays a crucial role regarding the transport efficiency of transportation systems. Here we study the optimal transport structure by measure of the length of time-respecting paths. Our network is built from a two-dimensional regular lattice, and long-range connections are allocated with probability P-ij similar to r(ij)(-alpha), where r(ij) is the Manhattan distance. By assigning each shortcut an activity rate subjected to its geometric distance tau(ij) similar to r(ij)(-C), long-range links become active intermittently, leading to the time-varying dynamics. We show that for 0 < C < 2, the network behaves as a small world with an optimal structural exponent alpha(opt) that slightly grows with C as alpha(opt) similar to log(C), while for C >> 2 the alpha(opt) -> infinity. The unique restriction between C and alpha unveils an optimization principle in time-varying transportation networks. Empirical studies on British Airways and Austrian Airlines provide consistent evidence with our conclusion.