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.
Other routing schemes @cite_20 @cite_18 @cite_17 have also been considered, taking into account lengths of both the routing path and of the queue. In a deterministic model @cite_18 , a forwarding node @math is chosen among neighbors @math by minimizing the quantity @math with a weight @math , @math denoting the distance from @math to the terminal node. Since we must solve the optimization problems, these models @cite_21 @cite_18 are not suitable for wireless or ad hoc communication networks. Thus, stochastic routing methods using only local information are potentially promising. In a stochastic model @cite_17 , @math is chosen at random, and a packet at the top of its queue is sent with probability @math or refused with probability @math as a nondecreasing function of the queue length @math . This model is simplified by the assumption of a constant arrival rate of packets, for analyzing the critical point of traffic congestion in a mean-field equation @cite_17 .
@cite_18 @cite_21 @cite_20 @cite_17
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We numerically investigate jamming transitions in complex heterogeneous networks. Inspired by Internet routing protocols, we study a general model that incorporates local traffic information through a tunable parameter. The results show that whether the transition from a low-traffic regime to a congested phase is of first- or second-order type is determined by the protocol at work. The microscopic dynamics reveals that these two radically different behaviors are due to the way in which traffic jams propagate through the network. Our results are discussed in the context of Internet dynamics and other transport processes that take place on complex networks and provide insights for the design of routing policies based on traffic awareness in communication systems. We propose a routing strategy to improve the transportation efficiency on complex networks. Instead of using the routing strategy for shortest path, we give a generalized routing algorithm to find the so-called efficient path, which considers the possible congestion in the nodes along actual paths. Since the nodes with the largest degree are very susceptible to traffic congestion, an effective way to improve traffic and control congestion, as our strategy, can be redistributing traffic load in central nodes to other noncentral nodes. Simulation results indicate that the network capability in processing traffic is improved more than 10 times by optimizing the efficient path, which is in good agreement with the analysis. DOI: 10.1103 PhysRevE.73.046108 PACS numbers: 89.75.Hc Since the seminal work on scale-free networks by Barabasi and Albert BA model1 and on the small-world phenomenon by Watts and Strogatz 2, the structure and dynamics of complex networks have recently attracted a tremendous amount of interest and attention from the physics community see the review papers 3‐5 and references therein. The increasing importance of large communication networks such as the Internet 6, upon which our society survives, calls for the need for high efficiency in handling and delivering information. In this light, to find optimal strategies for traffic routing is one of the important issues we have to address. There have been many previous studies to understand and control traffic congestion on networks, with a basic assumption that the network has a homogeneous structure 7‐11. However, many real networks display both scale-free and small-world features, and thus it is of great interest to study the effect of network topology on traffic flow and the effect of traffic on network evolution. present a formalism that can cope simultaneously with the searching and traffic dynamics in parallel transportation systems 12. This formalism can be used to optimize network structure under a local search algorithm, while to obtain the formalism one should know the global information of the whole networks. Holme and Kim provide an in-depth analysis on the vertex edge overload cascading breakdowns based on evolving networks, and suggest a method to avoid networks. The strategy is governed by a single parameter. Simulation results show that maximizing the network capacity and reducing the packet travel time can generate an optimal parameter value. Compared with the strategy of adopting exclusive local static information, the new strategy shows its advantages in improving the efficiency of the system. The detailed analysis of the mixing strategy is provided for explaining its effects on traffic routing. The work indicates that effectively utilizing the larger degree nodes plays a key role in scalefree traffic systems. We define a minimal model of traffic flows in complex networks in order to study the trade-off between topological-based and traffic-based routing strategies. The resulting collective behavior is obtained analytically for an ensemble of uncorrelated networks and summarized in a rich phase diagram presenting second-order as well as first-order phase transitions between a free-flow phase and a congested phase. We find that traffic control improves global performance, enlarging the free-flow region in parameter space only in heterogeneous networks. Traffic control introduces nonlinear effects and, beyond a critical strength, may trigger the appearance of a congested phase in a discontinuous manner. The model also reproduces the crossover in the scaling of traffic fluctuations empirically observed on the Internet.