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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 $α$ and $δ$, 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 $α$ and $δ$ are involved in the trade-off between a detour path for $α< 0$ and long wait at hubs for $α> 0$; In the low-performance regime at a small $δ$, the wandering path for $α< 0$ better reduces the mean travel time of a packet with high reachability. Although, in the high-performance regime at a large $δ$, the difference between $α> 0$ and $α< 0$ is small, neither the wandering long path with short wait trapped at nodes ($α= -1$), nor the short hopping path with long wait trapped at hubs ($α= 1$) is advisable. A uniformly random walk ($α= 0$) yields slightly better performance. We also discuss the congestion phenomena in a more complicated situation with packet generation at each time step.