Traffic Dynamics on Networks

Tao Zhou

Department of Modern Physics,University of Science and Technology of China

Department of Physics, University of Fribourg, Switzerlandzhutou@ustc.edu

Bing-Hong Wang and Tao Zhou, J. Korean Phys. Soc. 50, 134 (2007)

See an Invited Review Article:

Outline

• Motivation

• Local routing protocol

• Global routing protocol

• Conclusion

• Further discussions *Statistics of real traffic

*Mixing navigation

*Common properties shared by different dynamics

Traffic on complex networks – Motivation & Background

• The increasing importance of large communication networks such as the Internet, upon which our society survives, calls for the need for high efficiency in handling and delivering information.

• To understand the traffic mechanism and find optimal strategies to improve traffic capacity is one of the important issues we have to address.

• Most mentioned simulations are implemented based on Barabasi-Albert scale-free networks (BA model), since

**it is one of the most simplest model that can capture the heterogenous nature of real networks;

**it can avoid the structural bias such as degree-degree correlation;

**it is the most extensively studied model.

• Concepts I:

**Walk: without target, no interaction

**Search: With target, no interaction

**Traffic: With target, interactions can be observed as congestions

• Concepts II:

**Local routing: using only the information from its neighbors

**Global routing (or simply routing): using overall information

Traffic Dynamics Based on Local Routing Protocol

Motivation - Two effect of the preferential choice:

Positive: it leads to the shorter average distance

Negative: it leads to the overload on hub nodes Star-loop

• At each time step, there are R packets generated in the system, with randomly chosen sources and destinations.

• It is assumed that all the routers have the same capability in delivering and handling information packets, that is, at each time step all the nodes can deliver at most C packets

• To navigate packets, each node performs a local search among its neighbors. If a packet’s destination is found within those neighbors, it will be delivered directly to its target.

• Otherwise, it will be forwarded to a neighbor j at a probability proportional to

Particular rules of Model

jk

C.Y. Yin, B.H. Wang, W.X. Wang, T. Zhou*, H.J. Yang, Phys. Lett. A 351, 220 (2006);W.X. Wang, B.H. Wang*, C.Y. Yin, Y.B. Xie, T. Zhou, Phys. Rev. E 73, 026111 (2006).

Bypassing the large-degree nodes will remarkably increase the network throughput*

* A phase transition will take place from free flow to congested traffic at the critical value Rc. For R<Rc, the numbers of created and delivered packets are balanced, leading to a steady free traffic flow. For R>Rc, traffic congestion occurs as the number of accumulated packets increases with time, simply because the capacities of the nodes for delivering packets are limited. Rc := Network Throughput

Phys. Lett. A 351, 220(2006)

N=1000, <k>=6, C=10 - Optimal state: α≈-1

Phys. Rev. E 73, 026111 (2006)

Analytical solution of the optimal α

Denote n(k) the average number of packets on the node with degree k. We obtain the analytical solution of n(k) as:

1( ) an k k

When α= -1, the load distribution is homogeneous. This is the essential reason why system performs best at α= -1.

Traffic Dynamics Based on Global Routing Protocol

The minimal model

• At each time step, there are R packets generated in the system, with randomly chosen sources and destinations.

• Each packet follows one shortest path to the destination.

• It is assumed that all the routers have the same capability in delivering and handling information packets, that is, at each time step all the nodes can deliver at most C packets one step toward their destinations. We set C=1 for simplicity.

• A packet, upon reaching its destination, is removed from the system.

Phase transition

Order parameter: H(R)=lim(t∞) ∆W/R∆t

Homogeneous networks have larger traffic capacities than

heterogeneous ones

Regular lattices Scale-free NetworksT. Zhou et al., Dyn. Contin. Discret. Impuls. Syst. B 13, 463 (2006).

Why scale-free networks perform worse

• This phenomenon can be simply explained by betweenness centralities (BC) distributions. [M. E. J. Newman, Phys. Rev. E 64, 016132(2001)]

• The BC of a node v is defined as

• where is the number of shortest paths going from s to t and is the number of shortest paths going from s to t and passing through v.

• BC gives in transport networks an estimate of the traffic handled by the vertices,

• Define:

where Nk denotes the number of nodes with degree k.

• For most networks, g(k) is strongly correlated with k. In general, the larger the degree, the larger the centrality.

• In scale-free networks, traffic congestion generally occurs at nodes with the largest degree or BC, and immediately spreads over all the nodes. When all the packets follow their shortest paths, it will easily lead to overload of a heavily linked router, which is precisely the cause of traffic congestion.

st ( )st v

** To alleviate the congestion, a feasible and effective way is to bypass such high-degree nodes in the traffic-routing design. This leads us to question the commonly used shortest-path routing mechanism.** Nodes with larger connections are more likely to bear traffic congestion, thus a packet will in average spend more waiting time to pass through a high-BC node.** Often, bypassing those high-BC nodes, a packet will reach its destination quicker than taking the shortest path.

Motivation Efficient path & efficient path length

For a path from i to j:

Define its efficient path length:

The efficient path between i and j is the route that makes the sum minimum.

β Red path

Green path

β= 0 3 2

β= 1 7 7

β= 2 17 29

G. Yan, T. Zhou*, B. Hu, Z.Q. Fu, B.H. Wang, Phys. Rev. E 73, 046108 (2006)

Efficient routing protocol

Minimal Conclusions

• High-BC (or generally high-degree) nodes play the major role in the occurrence of traffic congestions.

• Bypassing these hub nodes rather than following shortest path will generally

enhance the traffic capacity.

Statistics of real network traffic• Some previously studies indicated that the real traffic time series has long-range correlation and displays

the 1/f-type scaling. Moreover, this scaling behavior can only be observed near the critical state.

• In the previous studies, the autocorrelation function and power spectrum are widely used to analyse the self-similar scaling behaviors of both real and artificial traffic data. However, it is shown that all the above methods do not work very well for the effect of non-stationary and are less accurate than the detrended fluctuation analysis (DFA) proposed by Peng et al., which has now been accepted as an important time series analysis approach and widely used especially for financial and biological data.

• Data from an Ethernet of a University - http://www.doc.ic.ac.uk/Buh/QUAINT/data/

Real Traffic Heng Seng Index DFA results, an evidence of the existence of 1/f-type scaling

P.L. Zhou, S.M. Cai, T. Zhou*, Z.Q. Fu, Proc. ICCCAS’06 (IEEE Press 2006, pp. 2687-2691)

The time series of traffic load in network level of an artificial traffic model on BA networks

The scaling exponents calculated with DFA are approximately H = 1, which implies the existence of 1/f-type scaling in the power spectrum and the long-range correlated behavior. Correspondingly, the traffic in free-flow state displays almost no long-range correlations.

Traffic time series of Yan model

Agree with the empirical study that the 1/f noise can only be detected near the critical point!!

S.M. Cai, G. Yan, T. Zhou*, P.L. Zhou, Z.Q. Fu, B.H. Wang, Phys. Lett. A 366, 14 (2007).T. Zhou, Int. J. Mod. Phys. B 21: 4071-4075 (2007).

Mixing Navigation

Black: routers; White: forwarders

Targeted strategy (in BA networks):

Clustering strategy in geographical networks:

T. Zhou, Physica A 387, 3025 (2008)

A latent and maybe oversimplified assumption is that all the nodes are functionally equivalent. Here we raise a partially centralized navigation system where a tiny fraction of nodes, called routers, know where the shortest path is, while most nodes, called forwarders, can only randomly forward the message to one of their neighbors. This mixing navigation mechanism is practicable and necessary in some significant self-driven systems, such as the wireless sensor networks and the autonomous robotic system.

Common features among traffic, synchronization and epidemic spreading

Soul of network-based dynamics: FLOW!! Here we show some previous works indicating some common properties shared by different dynamics.

The network synchronizability is also mainly determined by the maximal betweenness, see Hong, Kim, Choi & Park, Phys. Rev. E 69, 067105(2004) and Nishikawa, et al., Phys. Rev. Lett. 91, 014101(2003). See also a counterexample in Physica A 371, 773(2006).

Some strategies used to enhance network synchronizability can also be used to improve traffic conditions and accelerate the epidemic spreading.

Motter, Zhou & Kurths, Phys. Rev. E 71, 016116 (2005)Chavez et al., Phys. Rev. Lett. 94, 218701 (2005)And also some works by usPhys. Rev. E 72, 057102(2005)Phys. Rev. E 73, 037101(2006)Eur. Phys. J. B 53, 375 (2006)Phys. Rev. E 74, 056109(2006)

Optimal synchronizability is obtained when the coupling strength is inversely proportional to degree, see Motter, Zhou and Kurths, PRE 71, 016116 (2005).

A variant contact process where the contact probability to a neighboring node is determined by its degree. A variational method proves that this generic function has a unique optimal form, inversely proportional to degree,subject to the maximum of the average density of occupied nodes in the steady state. Collaborated with Y.-C. Lai’s group.