Traffic properties for stochastic routings on scale-free networks

For realistic scale-free networks, we investigate the traffic properties of stochastic routing inspired by a zero-range process known in statistical physics. By parameters $\alpha$ and $\delta$, this model controls degree-dependent hopping of packets and forwarding of packets with higher performance at more busy nodes. Through a theoretical analysis and numerical simulations, we derive the condition for the concentration of packets at a few hubs. In particular, we show that the optimal $\alpha$ and $\delta$ are involved in the trade-off between a detour path for $\alpha < 0$ and long wait at hubs for $\alpha > 0$; In the low-performance regime at a small $\delta$, the wandering path for $\alpha < 0$ better reduces the mean travel time of a packet with high reachability. Although, in the high-performance regime at a large $\delta$, the difference between $\alpha > 0$ and $\alpha < 0$ is small, neither the wandering long path with short wait trapped at nodes ($\alpha = -1$), nor the short hopping path with long wait trapped at hubs ($\alpha = 1$) is advisable. A uniformly random walk ($\alpha = 0$) yields slightly better performance. We also discuss the congestion phenomena in a more complicated situation with packet generation at each time step.