Traffic dynamics of packets generated with non-homogeneously selected sources and destinations in scale-free networks

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by  Rui Jiang

Traffic dynamics of packets generated with non-homogeneously selected sources and
destinations in scale-free networks
Rui Jiang1, Mao-Bin Hu1, Wen-Xu Wang2, Gang Yan3, Qing-Song Wu1, Bing-Hong Wang2
1 School of Engineering Science, University of Science and technology of China, Hefei 230026, China
2 Nonlinear Science Center and Department of Modern Physics,
University of Science and technology of China, Hefei 230026, China and
3 Department of Electronic Science and Technology,
University of Science and technology of China, Hefei 230026, China
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0
0 In this paper, we study traffic dynamics in scale-free networks in which packets are generated
2 withnon-homogeneouslyselectedsourcesanddestinations,andforwardedbasedonthelocalrouting
strategy. We consider two situations of packet generation: (i) packets are more likely generated at
n
high-degreenodes;(ii)packetsaremorelikelygeneratedatlow-degreenodes. Similarly,weconsider
a
two situations of packet destination: (a) packets are more likely to go to high-degree nodes; (b)
J
packetsare more likely togo tolow-degree nodes. Oursimulations show that thenetwork capacity
7 andtheoptimal valueof αcorrespondingtothemaximum networkcapacity greatly dependon the
2
configuration of packets’ sources and destinations. In particular, the capacity is greatly enhanced

when most packets travelfrom low-degree nodes tohigh-degree nodes.
]
h
p PACSnumbers:
-
c
o I. INTRODUCTION delivering capacity or finite queue length of each node.
s Inthesemodels,packetsareforwardedfollowingtheran-
.
s dom walking [6], the shortest path [7], the efficient path
c Complex networks can describe a wide range of sys-
[8], the next-nearest-neighbor search strategy [9], the lo-
i tems in nature and society, therefore there has been a
s cal information[10] or the integrationof local static and
y quickly growing interest in this area [1-3]. Since the
dynamic information [11,12].
h surprising small-worldphenomenon discoveredby Watts
p and Strogatz [4] and scale-free phenomenon with degree Nevertheless, in previous studies, packets are gener-
[ distribution following P(k) ∼ k−γ by Baraba´si and Al- ated with homogeneously selected sources and destina-

bert[5], the evolution mechanism of the structure and tions, i.e., sources and destinations are randomly cho-
1
sen without preference. However, in the real networked
v the dynamics on the networks have recently received a
traffic, packets are more likely to be generated at some
9 lot of interests among physics community. Due to the
0 importanceoflargecommunicationnetworkssuchasthe nodes than at others and are more likely to go to some
3 Internet and WWW with scale-free properties in mod- nodes than to others. Therefore, in this paper, we study
1 traffic dynamics with considering packets are generated
ern society, processes of dynamics taking place upon the
0 with non-homogeneously selected sources and destina-
underlying structure such as traffic congestion of infor-
7
tions, and delivered based on the local routing strategy,
0 mation flow have drawn more and more attention from
which is favoredin cases where there is a heavy commu-
/ physical and engineering fields.
s nication cost to searching the network.
c The ultimate goalof studying these large communica-
The paper is organized as follows. In section 2, the
i tion networks is to control the increasing traffic conges-
s traffic model is introduced. In section 3, the simulations
y tionandimprovetheefficiencyofinformationtransporta-
results are presented and discussed. The conclusion is
h tion. Many recent studies have focused on the efficiency
p improvement of communication networks which is usu- given in section 4.
v: ally considered from two aspects: modifying underlying
i network structures or developing better routing strate-
X gies. In view of the high cost of changing the underlying II. MODEL AND RULES
r structure, the latter is comparatively preferable.
a
Recentworksproposedsomemodelstomimicthetraf- Baraba´si-Albert model is the simplest and a well
fic routing on complex networks by introducing packets known model which can generate networks with power-
generatingrateaswellashomogeneouslyselectedsources law degreedistribution P(k)∼k−γ, where γ =3. With-
and destinations of each packet [6-12]. These kinds of outlosing generality,we constructthe networkstructure
models alsodefine the capacity of networksmeasuredby by following the same method used in Ref. [5]: Starting
a critical generating rate. At this critical rate, a contin- fromm0fullyconnectednodes,anewnodewithm0edges
uous phase transition from free flow state to congested is added to the existing graph at each time step accord-
state occurs. In the free state, the numbers of created ingto preferentialattachment,i.e., the probabilityQ of
i
and delivered packets are balanced, leading to a steady being connected to the existing node i is proportionalto
state. While in the jammed state, the number of accu- the degree k .
i
mulated packets increases with time due to the limited Then we model the traffic of packets on the given
2
graph. At each time step, there are R packets gener- III. RESULTS
ated in the system. Each packet is generated at node i
with probability Pkiβk1β1, where ki is the degree of node muInmtnheetwsporekcicaalpcaacsietyofisβR1 =≈β420=,w0hi(cchaisser1e)a,chtheedamtatxhie-
i and the sum runs over all nodes. Furthermore, the c
kβ2 optimal value αc ≈ −1. This can be explained by not-
packet goes to the node j with probability Pjkβ2, where ing two facts that the degree-degree correlation is zero
the sumalsorunsoverallnodes. Hereβ1 andβ2 aretwo in BA networks and the average number of packets on
parameters. Inthespecialcaseofβ1 =β2 =0,the pack- nodes does not depend on degree k when αc ≈−1 [10].
ets are generated with homogeneously selected sources Next we investigate the case of β1 = −5.0 and β2 =
and destinations. When β1 > 0 (β1 < 0), packets are −5.0 (case 2), i.e., most packets travel from low-degree
more likely generatedat high-degree(low-degree)nodes. nodes to low-degree nodes. Fig.1 compares the network
When β2 > 0 (β2 < 0), packets are more likely to go to capacityRc incases1and2. Onecanseethatatagiven
high-degree (low-degree) nodes. α, the network capacity decreases. This is easily to be
We treat all the nodes as both hosts and routers and understood because a low-degreenode has less links and
assume that each node can deliver at most C packets therefore more difficult to be found by packets than a
per time step towards their destinations. All the nodes high-degree node.
perform a parallel local search among their immediate Furthermore, the optimal value αc is essentially the
neighbors. If a packet’s destination is found within the same in cases 1 and 2, which is explained as follows. Let
searched area of node l, i.e. the immediate neighbors nidenotethenumberofpacketsofnodeiattimet. Then
of l, the packets will be delivered from l directly to its we have
target and then removed from the system. Otherwise, dn
the probabilityofa neighbornodei,to whichthe packet i =−ndeliver+nreceive+ngenerate−nremove, (3)
dt
will be delivered is as follows:
wherendeliver,nreceive,ngenerate,andnremove denote
kα the number of packets delivered from node i to other
Pl→i = Pikα, (1) nodes, received from other nodes, generated at node i
j j andremovedat node i. In case 1, ngenerate =nremove,
thus
where the sum runs over the immediate neighbors of the
dn
nodel. αisanintroducedtunableparameter. Duringthe dti =−ndeliver+nreceive. (4)
evolution of the system, FIFO (first- in first-out) rule is
applied. Allthesimulationsareperformedwithchoosing From Eq.(4), Wang et al. show that n(k) ∼ k1+α [10].
C =10. Therefore, when α=−1, the average number of packets
on nodes is independent of degree k and the maximum
In order to characterize the network capacity, we use
capacity is reached.
the order parameter presented in Ref. [13]:
In case 2, we have ngenerate = nremove > 0 for low-
<△Load> degree nodes and ngenerate = nremove ≈ 0 for high-
η(R)= lim , (2) degree nodes. Thus, Eq.(4) is valid for both low-degree
t→∞ R△t
nodes and high-degree nodes. Therefore, the optimal
value α ≈−1 does not change.
c
whereLoad(t)isdefinedasthenumberofpacketswithin Fig.2 compares the network capacity R in case 1 and
c
the networkat time t. △Load=Load(t+△t)−Load(t) in case 3 (where β1 = −2.0 and β2 = 2.0) and case 4
with < ··· > indicates average over time windows of (whereβ1 =−5.0andβ2 =5.0),i.e.,mostpacketstravel
width △t. The order parameter represents the ratio be- fromlow-degreenodestohigh-degreenodes. Onecansee
tween the outflow and the inflow of packets calculated that the network capacity is greatly enhanced in cases 3
over long enough period. In the free flow state, due to and 4. The maximum network capacity increases from
the balance of created and removed packets, the load is 40 in case 1 to 119 in case 3 and to 720 in case 4. This
independent of time, which brings a steady state. Thus is because a high-degree node has much more links and
whentimetendstobeunlimited,η isaboutzero. Other- thereforemucheasiertobe foundbypacketsthanalow-
wise,whenRexceedsacriticalvalueR ,thepacketswill degree node. Based on this result, we suggest that the
c
continuously pile up within the network, which destroys local routing strategy is very suitable if the packets are
the stead state. Thus, the quantities of packets within more likely to go from low-degree nodes to high-degree
the system will be a function of time, which makes η a nodes.
constantmore than zero. Therefore,a sudden increment Moreover, the optimal value of α corresponding to
c
of η from zero to nonzero characterizes the onset of the the maximum capacity increases from −1.0 in case 1 to
phase transition from free flow state to congestion, and −0.9 in case 3 and to −0.8 in case 4. This is also ex-
the network capacity can be measured by the maximal plained from Eq.(3). In case 1, α > −1 means high-
c
generating rate R at the phase transition point. degree nodes have more packets. In cases 3 and 4, we
c
3

40 1=0, 2=0 40 1=0, 2=0
1=-5, 2=-5 1=5, 2=-5
30 30
Rc20 Rc20
10 10
0 0 =-1.5
c=-1 c c=-1
-4 -2 0 2 -4 -2 0 2
FIG. 1: (color online) The network capacity Rc against α in FIG. 3: (color online) The network capacity Rc against α in
cases 1 and 2. cases 1 and 5.
packets generated in high-degree nodes and removed in
low-degree nodes enable the average number of packets
on nodes to be independent of degree k. Consequently,
the maximum capacity is reached at α <−1.
c
100 1=-5, 2=5 Fig.4 compares the network capacity Rc in case 1 and
in case 6 (where β1 = 3.0 and β2 = 3.0) and case 7
Rc (where β1 = 5.0 and β2 = 5.0), i.e., most packets travel
from high-degree nodes to high-degree nodes. In case 6,
10 the network capacity is essentially independent of α for
α < −1 and decreases with the increase of α. In case 7,
1=0, 2=0 c=-1 thenetworkcapacityisessentiallyindependentofαwhen
=-0.9 α is in the range studied. This is explained as follows.
=-2, =2 c
1 1 2 c=-0.8 Incase6,the probabilitythat the highest-degreenode
-4 -2 0 2 is chosen as origin is 0.33 and it is 0.6 in case 7. There-
fore, in case 6, when R > 31, the number of particles
c
generated in the highest-degree node exceeds the capac-
FIG. 2: (color online) The network capacity Rc against α in ityofthenode. Thisleadstothecongestion. Asaresult,
cases 1, 3 and 4. the constant network capacity for α < −1 occurs. Sim-
ilarly, in case 7, when R > 17, the number of particles
c
generated in the node exceeds the capacity of the node.
have ngenerate > 0,nremove ≈ 0 for low-degree nodes Therefore, the constant network capacity in the stud-
and ngenerate ≈ 0,nremove > 0 for high-degree nodes. ied range emerges. To avoid the constant small network
The packets generated in low-degree nodes and removed capacity, it is necessary to enhance the capacity of the
inhigh-degreenodes enablethe averagenumber ofpack- nodes of high degrees.
ets on nodes to be independent of degree k. As a result,
the maximum capacity is achieved.
Fig.3 compares the network capacity R in case 1 IV. DISCUSSION AND CONCLUSION
c
and in case 5, where β1 = 5.0 and β2 = −5.0, i.e.,
mostpacketstravelfromhigh-degreenodestolow-degree In this paper, we have investigated the network ca-
nodes. One can see that the network capacity becomes pacity in the scale-free networks, in which packets are
smaller and the optimal value of αc decreases. The rea- generated with non-homogeneously selected sources and
son of capacity decrease is the same as in case 2, i.e., a destinations, based on the local routing strategy.
low-degree node has less links and therefore more dif- Generally speaking, when most packets travel to low-
ficult to be found by packets. The decrease of αc is degreenodes,thenetworkcapacitywilldecrease. Incon-
explained as follows. In case 1, αc < −1 means high- trast, when most packets travel to high-degree nodes,
degree nodes have less packets. In case 5, we have whether the network capacity decreases or increases de-
ngenerate > 0,nremove ≈ 0 for high-degree nodes and pends on the selection of origins. When β2 is large, i.e.,
ngenerate ≈ 0,nremove > 0 for low-degree nodes. The most packets are generated from high-degree nodes, the
4
routing strategy is very suitable if the packets are more

likely to go from low-degree nodes to high-degree nodes.
50
Inaddition,α ,i.e.,theoptimalvalueofαcorrespond-
c
1=0, 2=0 ing to the maximum network capacity also depends on
40
the distribution of packets’ origins and destinations. We
=3, =3
1 2
have explained the reasonwhy α changes when the dis-
c
30 1=5, 2=5 tribution of packets’ origins and destinations changes.
Rc Finally,we wouldlike to mentionthatourresults may
be used to design a new local routing strategy, in which
20
theparameterαispacket-related. Concretely,αdepends
ontheoriginanddestinationofthepacketα=α(k ,k ),
o d
10 wherek andk denotethe degreeofthenodewherethe
o d
=-1 packetisgeneratedandthatofthenodewherethepacket
c
0 goes to. A suitable choice of α(ko,kd) may enhance the
-2 0 2 network capacity. Further investigations will be carried
out in future work.
FIG. 4: (color online) The network capacity Rc against α in
cases 1, 6 and 7.
Acknowledgements
highest-degree node is easily congested, which leads to We acknowledge the support of National Basic Re-
the congestion of the whole network. To avoid this, it is search Programof China (2006CB705500),the National
necessary to enhance the capacity of high-degree nodes. Natural Science Foundation of China (NNSFC) under
Whenmostpacketsaregeneratedfromlow-degreenodes, Key Project No. 10532060 and Project Nos. 10404025,
the network is greatly enhanced. Therefore, the local 10672160,70601026,and the CAS special Foundation.
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