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 @math and @math , 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 @math and @math are involved in the trade-off between a detour path for @math ; In the low-performance regime at a small @math , the wandering path for @math and @math is small, neither the wandering long path with short wait trapped at nodes ( @math ), nor the short hopping path with long wait trapped at hubs ( @math ) is advisable. A uniformly random walk ( @math ) yields slightly better performance. We also discuss the congestion phenomena in a more complicated situation with packet generation at each time step.
With a different processing power at each node @cite_3 , it has also been considered that the node capacity @math is proportional to its degree @math , therefore more packets jump out from a node as the degree becomes larger. On the other hand, in the ZRP @cite_9 @cite_8 @cite_5 , the forwarding capacity at a node depends on the number of @math defined as a queue length occupied by packets at node @math . The ZRP is a solvable theoretical model for traffic dynamics. In particular, in the ZRP with a random walk at @math , the phase transition between condensation of packets at hubs and uncondensation on SF networks has been derived @cite_9 @cite_8 . For @math , a similar phase transition has been analyzed in the mean-field approximation @cite_4 .
@cite_4 @cite_8 @cite_9 @cite_3 @cite_5
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We study the condensation phenomenon in a zero range process on weighted scale-free networks in order to show how the weighted transport influences the particle condensation. Instead of the approach of grand canonical ensemble which is generally used in a zero range process, we introduce an alternate approach of the mean-field equations to study the dynamics of particle transport. We find that the condensation on the scale-free network is easier to occur in the case of weighted transport than in the case of weight-free networks. In the weighted transport, especially, a dynamical condensation is even possible for the case of no interaction among particles, which is impossible in the case of weight-free networks. We study the condensation phenomenon in a zero-range process on scale-free networks. We show that the stationary state property depends only on the degree distribution of underlying networks. The model displays a stationary state phase transition between a condensed phase and an uncondensed phase, and the phase diagram is obtained analytically. As for the dynamical property, we find that the relaxation dynamics depends on the global structure of underlying networks. The relaxation time follows the power law @math with the network size @math in the condensed phase. The dynamic exponent @math is found to take a different value depending on whether underlying networks have a tree structure or not. We study a zero range process on scale-free networks in order to investigate how network structure influences particle dynamics. The zero range process is defined with the particle jumping rate function @math . We show analytically that a complete condensation occurs when @math where @math is the degree distribution exponent of the underlying networks. In the complete condensation, those nodes whose degree is higher than a threshold are occupied by macroscopic numbers of particles, while the other nodes are occupied by negligible numbers of particles. We also show numerically that the relaxation time follows a power-law scaling @math with the network size @math and a dynamic exponent @math in the condensed phase. motion of a random walker on heterogeneous complex networks. We find that the random walker is attracted toward nodes with larger degree and that the random walk motion is asymmetric. The asymmetry can be quantified with the random walk centrality, the matrix formulation of which is presented. As an interacting system, we consider the zero-range process on complex networks. We find that a structural heterogeneity can lead to a condensation phenomenon. These studies show that structural heterogeneity plays an important role in understanding the properties of dynamical systems.