Routing strategies in traffic network and phase transition in network traffic flow

The dynamics of information traffic over scale-free networks has been investigated systematically. A series of routing strategies of data packets have been proposed, including the local routing strategy, the next-nearest-neighbour routing strategy, and the mixed routing strategy based on local static and dynamic information. The capacity of the network can be quantified by the phase transition from free flow state to congestion state. The optimal parameter values of each model leading to the highest efficiency of scale-free networked traffic systems have been found. Moreover, we have found hysteretic loop in networked traffic systems with finite packets delivering ability. Such hysteretic loop indicates the existence of the bi-stable state in the traffic dynamics over scale-free networks.      

A dynamic routing strategy with limited buffer on scale-free network

In this paper, we propose an integrated routing strategy based on global static topologyinformation and local dynamic data packet queue lengths to improve the transmissionefficiency of scale-free networks. The proposed routing strategy is a combination of aglobal static routing strategy (based on the shortest path algorithm) and local dynamicqueue length management, in which, instead of using an infinite buffer, the queue lengthof each node i in the proposed routing strategy is limited by acritical queue length Q_ic. When the networktraffic is lower and the queue length of each node i is shorter than itscritical queue length Q_ic, it forwardspackets according to the global routing table. With increasing network traffic, when thebuffers of the nodes with higher degree are full, they do not receive packets due to theirlimited buffers and the packets have to be delivered to the nodes with lower degree. Theglobal static routing strategy can shorten the transmission time that it takes a packet toreach its destination, and the local limited queue length can balance the network traffic.The optimal critical queue lengths of nodes have been analysed. Simulation results showthat the proposed routing strategy can get better performance than that of the globalstatic strategy based on topology, and almost the same performance as that of the globaldynamic routing strategy with less complexity.     

Adaptive local routing strategy on a scale-free network

Due to the heterogeneity of the structure on a scale-free network, making the betweennesses of all nodes become homogeneous by reassigning the weights of nodes or edges is very difficult. In order to take advantage of the important effect of high degree nodes on the shortest path communication and preferentially deliver packets by them to increase the probability to destination, an adaptive local routing strategy on a scale-free network is proposed, in which the node adjusts the forwarding probability with the dynamical traffic load (packet queue length) and the degree distribution of neighbouring nodes. The critical queue length of a node is set to be proportional to its degree, and the node with high degree has a larger critical queue length to store and forward more packets. When the queue length of a high degree node is shorter than its critical queue length, it has a higher probability to forward packets. After higher degree nodes are saturated (whose queue lengths are longer than their critical queue lengths), more packets will be delivered by the lower degree nodes around them. The adaptive local routing strategy increases the probability of a packet finding its destination quickly, and improves the transmission capacity on the scale-free network by reducing routing hops. The simulation results show that the transmission capacity of the adaptive local routing strategy is larger than that of three previous local routing strategies.     

Incremental routing strategy on scale-free networks

The most important function of a network is for transporting traffic. Due to the low traffic capacity of network systems under the global shortest path routing, plenty of heuristic routing strategies are emerging. In this paper, we propose a heuristic routing strategy called the incremental routing algorithm to improve the traffic capacity of complex networks. We divide the routing process into N$N$(the network size) steps and, at each step, we heuristically calculate all the routes for one source node considering both the dynamic efficient betweenness centrality and node degree information. We do extensive simulations on scale-free networks to confirm the effectiveness of the proposed incremental routing strategy. The simulation results show that the traffic capacity has been enhanced by a substantial factor at the expense of a slight lengthening in the average path.     

Integrating network structure and dynamic information for better routing strategy on scale-free networks

We study information packet routing processes on scale-free networks by mimicking the Internet traffic delivery strategies. We incorporate both the global network structure information and local queuing information in the dynamic processes. We propose several new routing strategies to guide the packet routing. The performance of the routing strategies is measured by the average transit time of the packets as well as their dependence on the traffic amount. We find that the routing strategies which integrate both global network structure information and local dynamic information perform much better than the traditional shortest-path routing protocol which takes into account only the global topological information. Moreover, from comparative studies of these routing strategies, we observe that some of our proposed methods can decrease the average transit time of packets but the performance is closely dependent on the total amount of traffic while some other proposed methods can have good performance independent of the total amount of traffic with hyper-excellent average transit time of packets. Also, numerical results show that our proposed methods integrating network structure information and local dynamic information can work much better than the methods recently proposed in [S. Sreenivasan, R. Cohen, E. López, Z. Toroczkai, H.E. Stanley, Phys. Rev. E 75 (2007) 036105, Zhi-Xi Wu, Gang Peng, Eric W.M. Wong, Kai-Hau Yeung, J. Stat. Mech. (2008) P11002.], which only considered network structure information.     

Efficient routing on small complex networks without buffers

In this paper, we are exploring strategies for the reduction of the congestion in the complex networks. The nodes without buffers are considered, so, if the congestion occurs, the information packets will be dropped. The focus is on the efficient routing. The routing strategies are compared using two generic models, i.e., Barabàsi–Albert scale-free network and scale-free network on lattice, and the academic router networks of the Netherlands and France. We propose a dynamic deflection routing algorithm which automatically extends path of the packet before it arrives at congested node. The simulation results indicate that the dynamic routing strategy can further reduce the number of dropped packets in a combination with the efficient path routing proposed by Yan et al. [5].     

The effect of TTL on the information traffic

In real communication protocols, the information packets have a finite Time-to-Live (TTL) to avoid the waste of network resources, such as infinite loop induced by routing error or too long transferring time. In this paper, we introduce TTL into the information traffic model on Barabási-Albert scale-free networks under local routing strategy and focus on its effect on the network capacity measured by the critical point (R_c) of phase transition from free flow to congestion. Simulations show that the network capacity and the communication velocity are improved. However, some packets are dropped before they arrived at destinations. It is found that the share of successfully arrived packets monotonously increases with the increment of TTL and it is considerably acceptable if TTL is not very small. We also examine the effect of TTL on the positive-feedback preference (PFP) internet model and the results are alike. Our work may be helpful in quantifying the effect of packet lifetime in real communication networks and in routing strategy designing.     

Probability routing strategy for scale-free networks

This study proposes a probability routing strategy for improving traffic capability on scale-free networks. Compared with the shortest path routing strategy depending on central nodes largely and the efficient routing strategy avoiding hub routers as much as possible, the probability routing strategy makes use of hub routers more efficiently, transferring approximate average amount of packs of the whole network. Simulation results indicate that the probability routing strategy has the highest network capacity among the three routing strategies. This strategy provides network capacity that can be more than 30 times higher than that of the shortest path routing strategy and over 50% higher than that of the efficient routing strategy. In addition, the average routing path length of our proposed strategy is over 10% shorter than that of the efficient routing strategy and only about 10% longer than that of the shortest path routing strategy.     

Traffic of packets with non-homogeneously selected destinations in scale-free network

In this paper, we study the information traffic flow in communication networks with scale-free topology. We consider the situation arising when packets are delivered to non-homogeneously selected destinations. It is found that the network capacity R_c increases with the increase of 〈k〉 (average degree of destination nodes) under local routing strategy. In contrast, R_c is essentially independent of 〈k〉 under shortest path strategy. Based on this finding, an integrated routing strategy that can enhance network capacity is proposed by combining the two strategies.     

无标度复杂网络负载传输优化策略

提出了一种能够显著提高无标度复杂网络负载传输性能的优化路由策略.实现了负载在核心节点与边缘节点间的合理分配.分析表明该策略使得网络的负载处理能力正比于网络规模的平方,而与单个节点的度值无关.实验结果显示优化路由策略在保持了最短路由策略小世界效应的同时,成倍地提升了网络的负载传输能力,且随着网络平均节点度的增加其优势越趋显著.此外,与有效路由策略的比较进一步验证了优化路由策略的优异性能. 关键词： 优化路由策略 复杂网络 负载传输 网络阻塞     

Major impact of queue-rule choice on the performance of dynamic networks with limited buffer size

We investigate the similarities and differences among three queue rules, the first-in-first-out(FIFO) rule, last-in-firstout(LIFO) rule and random-in-random-out(RIRO) rule, on dynamical networks with limited buffer size. In our network model, nodes move at each time step. Packets are transmitted by an adaptive routing strategy, combining Euclidean distance and node load by a tunable parameter. Because of this routing strategy, at the initial stage of increasing buffer size, the network density will increase, and the packet loss rate will decrease. Packet loss and traffic congestion occur by these three rules, but nodes keep unblocked and lose no packet in a larger buffer size range on the RIRO rule networks. If packets are lost and traffic congestion occurs, different dynamic characteristics are shown by these three queue rules. Moreover, a phenomenon similar to Braess' paradox is also found by the LIFO rule and the RIRO rule.     

Traffic dynamics in scale-free networks with tunable strength of community structure

In this paper we systematically investigate the impact of community structure on traffic dynamics in scale-free networks based on local routing strategy. A growth model is introduced to construct scale-free networks with tunable strength of community structure, and a packet routing strategy with a parameter α is used to deal with the navigation and transportation of packets simultaneously. Simulations show that the maximal network capacity stands at α=−1 in the case of identical vertex capacity and monotonously decreases with the strength of community structure which suggests that the networks with fuzzy community structure (i.e., community strength is weak) are more efficient in delivering packets than those with pronounced community structure. To explain these results, the distribution of packets of each vertex is carefully studied. Our results indicate that the moderate strength of community structure is more convenient for the information transfer of real complex systems.     

Traffic resource allocation for complex networks

In this paper, an optimal resource allocation strategy is proposed to enhance traffic dynamics in complex networks. The network resources are the total node packet-delivering capacity and the total link bandwidth. An analytical method is developed to estimate the overall network capacity by using the concept of efficient betweenness (ratio of algorithmic betweenness and local processing capacity). Three network structures (scale-free, small-world, and random networks) and two typical routing protocols (shortest path protocol and efficient routing protocol) are adopted to demonstrate the performance of the proposed strategy. Our results show that the network capacity is reversely proportional to the average path length for a particular routing protocol and the shortest path protocol can achieve the largest network capacity when the proposed resource allocation strategy is adopted.     

Traffic dynamics based on a traffic awareness routing strategy on scale-free networks

By incorporating local traffic information into the shortest path routing strategy, we numerically investigate the effectiveness of the traffic awareness routing strategy for scale-free networks with different clustering. In order to characterize the efficiency of the packet-delivery process, we introduce an order parameter and an average transmission time that allow us to measure the network capacity by the critical value of phase transition from free flow to congestion. Compared with the shortest path routing protocol, the network capacity is greatly enhanced by the traffic awareness routing strategy. We also find that there exists an optimum value for the tunable parameter in the congestion awareness strategy. Moreover, simulation results show that the more clustered the network, the less efficient the packet-delivery process.     

Routing in scale-free networks based on expanding betweenness centrality

In this paper, an improved routing strategy is proposed for enhancing the traffic capacity of scale-free networks. Instead of using the information of degree and betweenness centrality, the new algorithm is derived on the basis of the expanding betweenness centrality of nodes, which gives an estimate of the traffic handled by the vertex for a certain route set. Since the nodes with large betweenness centrality are more susceptible to traffic congestion, the traffic can be improved by redistributing traffic loads from nodes with large betweenness centrality to nodes with small betweenness centrality in the process of computing the collective routing table. Comparing with results of previous routing strategies, it is shown that the present improved routing performs more effectively.     

Optimal routing strategy based on the minimum information path

In this paper, we propose a new routing strategy based on the minimum information path, named the optimal routing (OR) strategy, to improve the transportation capacity of scale-free networks. We define the average routing centrality degree of the node to analyze the traffic load on nodes of different degree. We analyze the transportation capacity by using the critical values of R_c, the average packet travel time, and the average path length. Both theoretical and experimental results show that the capacity of the network under our strategy will be maximized when the packet-delivery rate of the node is directly proportional to the degree.     

一种应用于含权无标度网络的全局路由算法

针对含权无标度网络提出了一种全局路由算法.该算法利用网络路径上的节点强度信息构建了一种全局路由代价函数,选择使该代价函数最小的路径来传输信息包,有效避开了网络中易发生拥塞的核心节点.实验结果表明,与最短路径算法相比,该算法以较小的平均路径长度的增加为代价,将网络容量提高了十多倍.     

A packet routing strategy using neural networks on scale-free networks

We propose a dynamic packet routing strategy by using neural networks on scale-free networks. In this strategy, in order to determine the nodes to which the packets should be transmitted, we use path lengths to the destinations of the packets, and adjust the connection weights of the neural networks attached to the nodes from local information and the path lengths. The performances of this strategy on scale-free networks which have the same degree distribution and different degree correlations are compared to one another. Our numerical simulations confirm that this routing strategy is more effective than the shortest path based strategy on scale-free networks with any degree correlations and that the performance of our strategy on assortative scale-free networks is better than that on disassortative and uncorrelated scale-free networks.     

Robustness analysis of static routing on networks

Robustness is one of the crucial properties that needs to be considered in the design of routing strategies on networks. We study the robustness of three typical routing strategies, which are the SP (shortest path), EP (efficient path), and OP (optimal path) strategies, by simulating several different kinds of attacks including random attacks, target attacks and cascading failures on scale-free networks. Results of the average path length, betweenness centrality, network capacity, etc., demonstrate that the EP strategy is more robust than the other two, and the OP strategy is more reliable than the SP strategy in general. However, on the power-grid network, the OP strategy is more resistant against cascading failures than the EP and SP strategies.     

面向成本-收益好的无标度耦合网络构建方法

较大平均路径长度的网络会带来较大的网络延迟, 难以支持时间敏感业务与应用. 通过增加连接可以降低源和目的节点之间的跳数, 进而降低网络平均延迟, 使得更加快速地传播信息, 但是增加连接的同时也增加了网络构建成本. 分层网络是研究网络耦合的一个有效方法, 但目前网络构建过程中将每层网络分别处理并认为每层网络之间没有强相关性. 本文提出了一种面向成本-收益的无标度网络动态构建方法. 此方法将网络分为多层, 基于连续论在高层网络中添加连接, 使得网络演化为无标度网络. 此连续过程包括节点度增加过程和局部网络半径增长两个连续过程, 在增加连接的过程中引入表征网络构建成本和收益的成本-收益指标. 模拟结果表明引入成本-收益指标的无标度耦合网络构建方法能够在合理范围内有效降低网络平均路径长度, 提升网络性能, 并且本文给出了耦合网络的动态业务性能, 通过调整高层网络避免网络拥塞.