Optimal transport in time-varying small-world networks.

Abstract

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)∼r(ij)(-α), where r(ij) is the Manhattan distance. By assigning each shortcut an activity rate subjected to its geometric distance τ(ij)∼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 α(opt) that slightly grows with C as α(opt)∼log(C), while for C ≫ 2 the α(opt)→∞. The unique restriction between C and α unveils an optimization principle in time-varying transportation networks. Empirical studies on British Airways and Austrian Airlines provide consistent evidence with our conclusion.