Traffic dynamics on complex networks are intriguing in recent years due to their practical implications in real communication networks. In this survey, we give a brief review of studies on traffic routing dynamics on complex networks. Strategies for improving transport efficiency, including designing efficient routing strategies and making appropriate adjustments to the underlying network structure, are introduced in this survey. Finally, a few open problems are discussed in this survey.
Large real communication networked systems have become a hot research topic for a rather long time. Typical examples include the Internet, which is an enormous network of many routers connected by physical or wireless links with information packets flowing on them, and high-way network, which is composed of cities and high-ways between cities. The rapid development of society causes the immense increase of traffic amounts in many real networked communication systems. Congestion may firstly occur on some communication units (such as routers on the Internet [
Actually, many empirical studies have revealed that the transport performances are not only relevant to the characteristics of underlying network structure, but also significantly affected by routing strategies. In this light, in order to improve transport performances on real networks, one can either design efficient routing strategies, or make appropriate changes to the underlying network structure. Designing efficient routing strategies is considered to be “soft” strategies, because it does not require any topological changes. Making changes to network structure is considered to be “hard” strategies, because it requires topological changes. One can add a few links to existing networks or delete a few links from existing networks to realize the modification of network structure.
In this survey, we mainly review recent progresses for traffic dynamics on complex networks. We have to mention that this survey is mainly from the perspective of physics. The remainder of this survey is organized as follows. In Section
Processes of random walk on complex networks have been extensively studied recently due to its wide applications in real networks [
The basic model has also been generalized to more realistic models, which incorporate the fact that hub nodes usually have high delivering capacities or can generate more packets at each unit time step than those low-degree nodes can do. For example, degree-dependent delivering capacity was assumed in the form
One of the most important measurements for transport performance of traffic is the traffic capacity
In order to provide a theoretical estimate of traffic capacity, the concept “betweenness” is introduced here. Betweenness centrality [
Then, we can make a theoretical estimate for the traffic capacity
Under the shortest path routing strategy, each packet is always transported along the topological shortest path between the packet’s source and destination. Different from routing strategies with stochastic factors, under the shortest path routing strategy, each packet has a fixed delivering path once the network is constructed. Therefore, the shortest path routing strategy is widely used in real communication systems [
It has been demonstrated that networks cannot handle heavy traffic if packets are always transported along the shortest path from source and destination [
Later on, Zhang et al. in [
The critical value
In [
(a) The traffic capacity rate
In [
The critical
The routing strategies mentioned above, allow each node to have the whole network’s global topological information, which may be practical for small or medium size networks but not for large real communication such as the Internet, WWW [
Critical
The critical generating rate
Actually, the area of information that each router can access can significantly affect the performances of local routing strategy in traffic transport on complex networks. References [
The strategy in [
The network capacity
Average transport time
In [
Evolution in the packet number in the network under different routing strategies [
Figure
(a) The traffic capacity
“Hard” strategies mean network topological structure is appropriately changed so that transport efficiency can be improved. Adding or rewiring links are more costly than soft strategies (i.e., designing efficient routing strategies), because adding or rewiring links usually have to consume much financial, manpower or even energy cost. On the contrary, removing links from networks is usually easy to be implemented at low cost. For example, in a high-way network system, some road ways are usually closed at rush hours to alleviate congestion, especially when crowds of people are rushing to offices in the morning or rushing back home in the afternoon. To realize the closure of roadways, traffic administrators need only to block the entries to the roadways, which is easy to be implemented. Another example is the Internet. Network administrators can easily isolate some connections among computers through computer software.
In fact, the link removal strategy has been extensively studied to enhance or optimize dynamics of different kinds on complex networks. In [
Literatures on enhancing transport capacity by removing links in communication networks have also been reported. The strategies in [
The authors of [
In [
Illustration of uneven (a) and even (b) distributions of neighbors’ degrees of a central node. Node
The VNDR link-removal strategy to enhance traffic capacity in scale-free networks is carried out as follows. For each node
The performance of the VNDR strategy is evaluated by comparing it with the high-degree-first (HDF) strategy [
Illustration of
Link-removal strategy is operationally convenient and economical in real communication networks to enhance transport efficiency. However, with the rapid development of society, the sizes of real communication networks are consistently increasing, which brings new challenges to network administrators in charge of network communications. For instance, the numbers of roadways and cities keep on increasing in highway networks, and the number of computers is also increasing explosively in the Internet. But due to the cost incurred in adding new nodes and links to existing networks, real network designers have to be cautious when preparing to add new nodes and links to a given communication network. Otherwise, if new nodes and links are added to an existing network in an improper way, the new links may not be helpful for enhancing transport efficiency while packets are generated on both existing nodes and new nodes, which may aggravate congestion in the network. More links then need to be added into the networks to alleviate congestion with extra cost. Thus, it is necessary to investigate how to add nodes and links in an efficient way so that the traffic capacity can be enhanced maximally.
In [
In a scale-free network with
The traffic capacity
For the case that both nodes and links are allowed to be added into networks under the shortest path routing strategy, the authors of [
[
It is worth noting that there is an underlying relation between cascading dynamic [
The key purpose of studying routing is to study performances of networks such as delay, admission control, and resource allocation [
The heavy traffic loads in real networked communication systems motivate the intense study of traffic dynamics on complex networks in recent years. The main goal is to enhance transport efficiency so that traffic congestions on complex networks can be alleviated. Generally speaking, the strategies which can enhance the traffic capacity can be categorized into two classes: designing efficient routing strategies and making appropriate adjustments to network structures. Many strategies have been proved to be able to enhance the transport efficiency and are helpful for people in understanding and controlling traffic congestion in real networks.
Although the issues of enhancing transport efficiency have been extensively reported in the literatures, there are still many open questions to be studied. Real communication networks are usually much more complicated than the traffic routing model. Other dynamics such as the virus prevalence and cascading breakdown, may also be mixed with the traffic routing dynamics. A few open questions to be solved in future work are listed as follows. The cascading breakdown dynamics can be incorporated with the traffic routing dynamics. A node or a link may lose its function in delivering packets due to traffic congestion on this node or link. Therefore, the node or the link will be removed from the network. As a result, the traffic capacity may be reduced. Under this circumstance, it is thus interesting to study the relationship between the two kinds of dynamics. It is not always the case in real communication networks that each node in the network can generate packets. Some nodes (i.e., generators) may only generate and send packets to other nodes but do not receive packets from elsewhere. The rest of nodes (i.e., receivers) only receive packets from elsewhere but do not generate and send packets to other nodes. The positions of generators may have impacts on the transport efficiency of packets. It is neccesary, therefore, to study how to place generators and receivers in networks so that the transport efficiency can be maximized. Most current studies about traffic routing dynamics on complex networks is based on traffic routing models. However, due to complexity of real communication networks, traffic routing models cannot fully reflect the characteristics of real communication networks. More attention should be focused on revealing new characteristics of real communication networks rather than only study traffic routing models.
This work was supported by the National Natural Science Foundation of China and Microsoft Research Asia (NSFC-61173096, 60802087), Zhejiang Provincial S&T Department (2010R10006, 2010C33095), and Zhejiang Provincial Natural Science Foundation (R1110679).