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.
In the typical models, a forwarding node @math is chosen with probability either @math @cite_11 @cite_3 @cite_21 or @math @cite_13 . Here, @math and @math are real parameters, @math and @math denote the degree and the dynamically-occupied queue length by packets at node @math in the connected neighbors @math to the resident node @math of the packet. These methods are not based on a random walk (selecting a forwarding node uniformly at random among the neighbors), but on the extensions (including the uniformly random one at @math ) called preferential and congestion-aware walks, respectively. Note that @math leads to a short path passing through hubs, and that @math and @math lead to the avoidance of hubs and congested nodes with large @math . In stochastic routing methods, instead of using the shortest path, the optimal values @math and @math for maximizing the generation rate of packets in a free-flow regime have been obtained by numerical simulations @cite_3 @cite_21 . A correlation between the congestion at node level and a betweenness centrality measure was suggested @cite_13 .
@cite_21 @cite_13 @cite_3 @cite_11
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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 We present a study of transport on complex networks with routing based on local information. Particles hop from one node of the network to another according to a set of routing rules with different degrees of congestion awareness, ranging from random diffusion to rigid congestion-gradient driven flow. Each node can be either source or destination for particles and all nodes have the same routing capacity, which are features of ad-hoc wireless networks. It is shown that the transport capacity increases when a small amount of congestion awareness is present in the routing rules, and that it then decreases as the routing rules become too rigid when the flow becomes strictly congestion-gradient driven. Therefore, an optimum value of the congestion awareness exists in the routing rules. It is also shown that, in the limit of a large number of nodes, networks using routing based on local information jam at any nonzero load. Finally, we study the correlation between congestion at node level and a betweenness centrality measure. It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V,E), let P(β) = (puvβ)u,v∈V be a transition matrix defined by puvβ = exp [-βU(u, v)] Σw∈N(u) exp [-βU(u, w)] for u∈V, v∈N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(., .) is a potential function defined as U(u, v) = log (max deg(u), deg(v) ) for u∈V, v∈N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2 log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).