Traffic Congestion on Clustered Random

Complex Networks

Thiago Henrique Cupertino and Liang Zhao

Institute of Mathematical Sciences and Computing, Av. Trabalhador Sao-carlense400, Sao Carlos, SP, Brasil

{thiagohc,zhao}@icmc.usp.br

Abstract. In this work we study the traffic-flow on clustered randomcomplex networks. First, we derive a mathematical model to determinethe congestion phase-transition point. This point is defined as the abrupttransition from a free-flow to a congested state. Second, we study the in-fluences of different cluster sizes on the traffic-flow. Our results suggestthat the traffic of centralized cluster network (a network which has a bigcentral cluster surrounded by clusters with significantly smaller sizes) isless congesting than balanced cluster network (a network with clustersof approximately the same size). These results may have practical im-portance in urbanization planning. For example, using the results of thisstudy, the increasing of satellite cities’ sizes surrounding a big city shouldbe well controlled to avoid heavy traffic congestion.

1 Introduction

A rapidly increase on the interest studying networks with complex structureshave been observed since about 12 years ago. The works vary on the areas ofdynamical processes, analytical properties, structural models and so on [1,2,3].Examples of such complex networks on the real world are the Internet, theWWW, P2P networks, social networks, telecommunication systems, biologicalnetworks (e.g. neural networks, protein-protein interaction networks and neuralnetworks), traffic networks, paper author citations networks etc [4,5].

On some kind of networks such as the Internet, transportation networks andpower grids, the free traffic-flow of the characteristic element (e.g., data, goodsand signals, respectively) are crucial for the well-functioning of the whole sys-tem. Furthermore, are also crucial for the society as we depend more and moreon these systems. Motivated by that, several studies have been done consideringtraffic-flow in complex networks [6,7,8,9,10,11,12,13]. Many of the previous workson this area consider networks with regular or homogeneous structures. However,some works revealed that many real networks possesses different kinds of topolo-gies like scale-free, small-world and communities [1,2]. Moreover, a previous work[9] has shown that for two networks with the same average connectivity, nodeprocessing capacity, and total number of nodes, if their topological structuresare different, the traffic phase-transition point can be significantly different, in-dicating that traffic congestion depends sensitively on network structure. On the

L. da F. Costa et al. (Eds.): CompleNet 2010, CCIS 116, pp. 13–21, 2011.c© Springer-Verlag Berlin Heidelberg 2011

14 T.H. Cupertino and L. Zhao

same direction, in this work we study the traffic-flow and congestion on clusteredcomplex networks.

As a specific structure of complex networks, the clustered complex networksor the community structured complex networks are characterized by groups ofnodes densely connected inside the group while the connections between differentgroups are sparse [14]. The structures of communities and between communitiesare important for the demographic identification of network components and thefunction of dynamical processes that operate on these networks (e. g., the spreadof opinions and deceases) [15,16].

Another important feature of many complex networks is that its propertiescan change over time. In a period of time, new nodes or links can be added orredistributed on a given network. As examples of these evolving networks arethe social networks, where opinion formation is mediated by social interactions,consequently taking place on a network of relationships and at the same timeinfluencing the structure of the network and its evolution [17]. Another examplecould be a cellular network composed of wireless mobile devices, where a groupof nodes is defined by their geographical positions (devices next to each otherare connected by the same tower/antenna), and the users are in a continuousmovement which changes the network structure [18].

In [13], it has been shown that the better defined the communities are, themore affected packet transport becomes. Now, we are interested in studyingthe influence of communities sizes in the packet transportation. By combiningthe clustering and the evolving features of complex networks we are able tostudy some properties of evolving clustered complex networks. Specifically, inthis work we are also interested on the traffic-flow on networks where the clustersize changes over time. From the results of our simulations we verified that whena network have clusters of similar sizes the traffic is more congested than whenthere are a central cluster and peripheral smaller ones. An analogy with theseresults can be made using the vehicle traffic between a big city and the satellitecities around it. Our study suggests that the traffic flow can be less congestedif a significant difference in size between the big city and its satellite cities canbe maintained, in other words, the increasing of satellite cities should be wellcontrolled in urbanization process.

The remainder of the paper is organized as follows. In Sec. 2, we describe thetraffic-flow model used in this work. In Sec. 3, we carry out an analysis of theclusterized traffic model to determine the congestion phase-transition point. InSec. 4, we give numerical support for our analysis and study the influence ofdifferent cluster sizes on the traffic-flow. Finally, in Sec. 5 we give some briefconclusions and comments.

2 Traffic-Flow Model

As the model described in [9,10], the study of traffic-flow on clustered complexnetworks in this paper considers two parameters: the information generation rateλ and a control parameter β, that measures the capacity of nodes to processinformation. The capacity of each node for processing and sending information

Traffic Congestion on Clustered Random Complex Networks 15

are proportional to the number of links connected to the node (node’s degree).The quantity of interest is the critical value for λ at which a phase transitionoccurs from free to congested traffic-flow. In particular, for λ ≤ λc, the numbersof created and delivered packets are balanced, resulting in a steady state, orfree traffic-flow. For λ > λc, congestions occur in the sense that the numberof accumulated packets increases in time, due to the fact that the capacities ofnodes for delivering packets are limited. We are interested in determining thephase-transition point, given a clustered complex network, in order to addresswhich kind of clustering is more susceptible to phase transition and thereforetraffic congestion.

We use the same traffic-flow model described in [9,10], which is based on therouting algorithm in computer networks. The capacities for processing informa-tion are different for different nodes, depending on the numbers of links passingthrough them. The routing algorithm consists on the following steps:

(1) At each time step, the probability for node i to generate a packet is λ.(2) At each time step, a node i delivers at most Ci packets one step toward

their destinations, where Ci = (1 + int[βki]) and 0 < β < 1 is a controlparameter, ki is the degree of node i and Bi is its betweenness (calculatedusing Newman’s algorithm [19]).

(3) A new created packet is placed at the end of the queue of its creating node.At the same time, another node on the network is chosen as a destinationnode and the packet is forwarded, on next time steps, along the shortestpath between these two nodes. If there are several shortest paths for onepacket, it is chosen the one having the shortest queue length.

(4) A packet, once reaching its destination node, is removed from the traffic.

Although traffic-flow rules defined in this model are simple, they capture theessential characteristics of traffic-flow in real networks. Our aim is to deter-mine the relationship between the network community structure and the criticalpacket-generating rate λc, which is described in the next Section.

3 Theoretical Analysis

In our work, we use the following random clustered complex network model: Nnodes are divided into M groups, where each group has n = N/M nodes. Insidethe same group, a pair of nodes is connected randomly with probability ρin, andnodes belonging to different groups are connected randomly with probabilityρout. For a clustered complex network, the number of interconnections is typicallyfar less than the number of intra-connections. So, we study the cases in whichρout � ρin, basically ρout ≈ ρin/10.

As stated in [9], the critical point for the packet generation rate on a complexnetwork is defined by:

λc =(1 + int[βkmax])(N − 1)

Bmax, (1)

16 T.H. Cupertino and L. Zhao

where kmax and Bmax are, respectively, the degree and the betweenness of thenode with the largest betweenness on the network.

The problem here is to find these measures analytically for a given networkstructure. In our case, we can promptly find approximated measures for ran-domly clustered complex networks. To easily understand, consider a clusterednetwork with N nodes and 2 communities, each one having n = N/2 nodes.Given that ρout � ρin, there will be just a few nodes connecting both com-munities and, consequently, these few nodes will be loaded with all the trafficbetween the communities. So, considering that n � nρout, i.e., the cluster sizeis much greater than the averaged number of nodes with interconnections, wecan consider that these nodes will have the largest betweennesses of the networkand so they will be the bottleneck for the traffic-flow.

To find Bmax, we divide the analysis in two parts. The first concerns thebetweenness inter-clusters and the second, intra-clusters. Considering that thereare just a few nodes with interconnections, the averaged betweenness for themis found by dividing the number of shortest paths inter-clusters by the averagednumber of that kind of nodes, i.e.:

Binter =n(N − n)

nρout=

N − n

ρout(2)

As we are studying random clusters, the intra-cluster betweenness can befound with good approximation by averaging the total betweenness inside acluster, i.e.:

Bintra =Dn(n − 1)

n= D(n − 1), (3)

where D is the averaged shortest path inside the clusters. This approximationis reasonable because each bottleneck node is randomly selected and thus ithas high probability that it degree is near the average degree of the cluster.Consequently, its betweenness should be close to the average betweenness of thecluster.

So, the maximum betweenness is given by:

Bmax = Binter + Bintra =N − n

ρout+ D(n − 1), (4)

and the critical packet generating rate (Eq. 1) becomes:

λc =(1 + int[β < k >])(N − 1)

N−nρout

+ D(n − 1), (5)

where we used the averaged degree < k > of the network.Now, consider a clustered random network where initially all clusters have the

same size n. After a while, one cluster evolves by expanding its size n while theother clusters become smaller. This process can be understood by creating an-other network with the same parameters, but with different communities’ sizes.So, applying Eq. 5 to the biggest cluster, it can be noted that λc depends on

Traffic Congestion on Clustered Random Complex Networks 17

n through two measures: Binter ((N − n)/(ρout)) and Bintra (D(n − 1)). As nincreases, Binter decreases linearly (and so λc increases and traffic becomes lesscongested) until the limit n = N when it vanishes. On the other hand, as nincreases Bintra increases logarithmically (D ∼ ln(n) [21]) (and so λc decreasesand traffic becomes more congested). Also, from the network structure we haveBinter � Bintra and, therefore, from these two opposite consequences we can ex-pect more influence from Binter on increasing λc (attenuating traffic congestion)when a cluster is expanding.

4 Numerical Support

In order to characterize the transition from free traffic-flow to congestion, weuse the order parameter η = limt−→∞ < ΔΘ > /λΔt [20] where < ΔΘ >=Θ(t + Δt) − Θ(t), Θ(t) is the total number of packets in the network at time t,and < ... > indicates average over a time window of length Δt. For λ < λc thenetwork is in a free flow state. In this case, we have < ΔΘ >≈ 0 and η ≈ 0. Forλ > λc, traffic congestion occurs so that ΔΘ increases with Δt.

Figure 1 shows λ versus η for various clustered complex networks. For net-works with N = 1000 nodes with clusters of n = 200 nodes, various probabil-ity connection rates are studied. In these results, we simulated the probabilityof interconnections ρout ranging from 0.01 to 0.13, while the probabilities forintra-connections are ρin = 1 − ρout. As expected from Eq. 5, the network ismore congested when the connections between clusters are sparse. As ρout in-creases, the clusters of the network become more connected and the traffic is lesscongested.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

100

200

300

400

500

600

700

800

900

1000

λ

η

pout

= 0.01

pout

= 0.03

pout

= 0.05

pout

= 0.07

pout

= 0.09

pout

= 0.11

pout

= 0.13

Fig. 1. Order parameter η vs the packet-generation rate λ for five-cluster networks ofdifferent connections probabilities. For each data point the network size is N = 1000,n = 200, < k >= 8 and β = 0.1. 30 different scenarios were averaged.

18 T.H. Cupertino and L. Zhao

0 0.01 0.03 0.05 0.07 0.09 0.11 0.130

0.05

0.1

0.15

0.2

0.25

0.3

0.35

pout

λ c

Simulation

Theoretical

Fig. 2. Comparison of the critical generating rate λc from theoretical prediction(squares) and from numerical simulations (circles). For each data point the networksize is N = 1000, n = 200, < k >= 8 and β = 0.1. 30 different scenarios were averaged.

Figure 2 shows the comparison of the critical generating rate from theoreticalprediction (calculted by using Eq. 5) and from numerical simulations. Thesesimulations are the same as the Fig. 1 and so we use the same parameters. Onecan note here that the theoretical model given by Eq. 5 can get a good estimationof the critical packet generating rate for the clustered complex networks understudy.

Two different simulations are performed to study the influence of the changingof the network cluster structure on the traffic-flow. The first simulation considersa network of size N = 400 with two clusters. Initially, both clusters have the samenumber of nodes n = 200. As the time passes, one cluster starts growing whilethe other one becomes smaller, until the limit of n1 = 360 and n2 = 40 nodes,respectively. This situation is shown by Fig. 3: when a network maintains thesame number of nodes while its communities are changing over time, the phase-transition point from the free flow state to the congested state changes. In thesecases, the most congested network is the one which has the most similar clustersizes. The second simulation considers an environment where both clusters areexpanding over time, but one increases quicker than the other. Here, the networkstarts with two clusters: one with n1 = 200 nodes and the other with n2 = 40nodes. At each iteration, the former expands with a rate α1 = 5% while thelater, with α2 = 20%. At the last iteration, the maximum sizes for the firstand the second cluster are n1 = 353 and n2 = 357, respectively. As shown byFig. 4, two conclusions can be made. First, the values of η naturally increasethrough the iterations as the total network size N increases and more packetsare being created. But here the second observation is more important: as theprevious simulation, when the clusters have similar sizes the network is morecongested. So, from the two scenarios simulated, the first with fixed N and the

Traffic Congestion on Clustered Random Complex Networks 19

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

50

100

150

200

250

λ

η

200−200

240−160

280−120

320−80

360−40

Fig. 3. Order parameter η vs the packet-generation rate λ for two-cluster networks withdifferent cluster sizes. Initially, both clusters have the same number of nodes n = 200.On the last scenario one cluster has n = 360 nodes while the other one has only n = 40nodes. The connection probabilities are pout = 0.05 and pin = 0.95. For each datapoint the network size is N = 400, < k >= 8 and β = 0.1. 30 different scenarios wereaveraged.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

50

100

150

200

250

300

350

400

450

500

λ

η

200 − 40

210 − 48

220 − 58

231 − 69

242 − 83

254 − 100

266 − 119

279 − 143

292 − 172

306 − 206

321 − 248

337 − 297

353 − 357

Fig. 4. Order parameter η vs the packet-generation rate λ for two-cluster networkswith different cluster sizes. Initially, one cluster has n1 = 200 and the other, n2 = 40nodes. The first cluster increases with rate α1 = 5% while the other, with α2 = 20%.The connection probabilities are pout = 0.05 and pin = 0.95. For each data point theaverage degree is < k >= 8 and β = 0.1. 30 different scenarios were averaged.

20 T.H. Cupertino and L. Zhao

second with an increasing N , we observed the same influence of the cluster sizeson the network traffic-flow.

5 Conclusions

In this work, we studied the traffic-flow in complex networks. Motivated bypractical applications and previous works [9,10], our primary interest was toinvestigate the congestion process on a clustered structure. We first derived amathematical analysis to determine the phase-transition congestion point andthen we shown numerical results. The simulation results suggest that when anetwork has clusters with different sizes, e. g., a central cluster surrounded byclusters with significantly smaller sizes, the traffic is less congested than whenthe clusters have approximately the same size. To our knowledge, this is thefirst time a study on how sub-network sizes influence the traffic-flow in complexnetworks. We believe that these results have practical importance for controlingflow and designing networks in which there is some kind of traffic and thatpresent a clustered structure. For example, in order to decrease traffic congestionin a metropolitan region of a big city, the satellite cities or communities shouldnot increase rapidly, in other words, it should have more small satellite citiesrather than few big satellite cities. As future works, it should be addressed morecomplex network structures, i.e., having a more number of clusters and differenttopologies as scale-free and small-world.

Acknowledgments. T.H.C. is supported by FAPESP under Grant No. 2009/02036-7. L.Z. is supported by CNPq under Grant No. 302783/2008-3.

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