Mitigating Congestion in Complex Transportation Networks via Maximum Entropy

0.12 MB·

by  Yuhang Fan

Mitigating CongestioninComplexTransportation Networks via MaximumEntropy
Yuhang Fan1,2 and Hanyuan Liu3
1StateKeyLab. of IndustrialControl Technology, ZhejiangUniversity, Hangzhou, 310027, China
2School of Mathematics, Zhejiang University, Hangzhou, 310027, China
3CollegeofInformationScienceandElectronicEngineering, ZhejiangUniversity,Hangzhou, 310027, China
In thispaper, wereveal therelationship between entropy rate and thecongestion in complex network and
solveitanalyticallyforspecialcases. Findingmaximizingentropyratewillleadtoanimprovementoftraffic
efficiency,weproposeamethodtomitigatecongestionbyallocatinglimitedtrafficcapacitytothenodesinnet-
workrationally.Differentfromformerstrategies,ourmethodonlyrequireslocalandobservableinformationof
7
network,andislow-costandwidelyapplicableinpractice. Inthesimulationofthephasetransitionforvarious
1
0 network models, our method performs well inmitigatingcongestion both locally and globally. By compari-
2 son, wealsouncover thedeficiencyofformerdegree-biasedapproaches. Owingtotherapiddevelopment of
transportationnetworks,ourmethodmaybehelpfulformodernsociety.
n
a
J
Entropy,asakeyconcept,playsanimportantroleinmany tion.
8
fields such as information theory, statistical physics and so We suppose that we can get each node’s local informa-
1
on [1, 2]. In the study of complex network, the entropy has tion including the frequency of different neighbors it deliv-

] beenusedtocharacterizethepropertiesofnetworktopology erspacketstoandthenumberofpacketsitdeliversintraffic.
h
[3, 4] and diffusion process on network [5]. Via application The frequency and number, as local quantities decided both
p
of maximum entropy method, we can optimize the network bythetrafficdynamicalprocessandbythenetworktopology,
-
c topologyforspecialfunctionandmakepredictionofnetwork is typically easy to be observed and counted in reality. Let
o
properties[6,7]. usanalyzetheroutingofpacketswhichiscertainunderspe-
s
. Trafficflow,modelingtheprocessofmaterialexchangebe- cific strategy from a stochastic standpoint. Mathematically
s
c tween cells, packettransfer on the Internet, municipaltraffic speaking,weregardthetrafficprocessonanetworkcomposed
i and so on [8–10], is one of the significant research fields in from n nodesas a time invariantergodic Markov chain with
s
y complexnetworks. While trafficcongestionis the main hin- atransitionmatrixM=(qij),wheretheentryqij istheaver-
h drancetotransportnetwork’snormalandefficientfunctioning age frequencyof packets to go from node Ni to Nj at a time
p [11], some models, such as random walk [12], shortest path step satisfying (cid:229) n q =1∀i. We introduce entropy rate to
[ j=1 ij
[13] and so on, has been built to analyze this phenomenon characterizethisprocess.AnergodicMarkovchainwithtran-

1 and studies has been focused on optimizing network topol- sition matrix M =(q ) has a unique stationary distribution
ij
v ogyandimprovingthetrafficcapacitytoachievefree-flowfor w=(w ,w ,...,w )T satisfying
1 2 n
4
thesedifferentmodels[14,15].Mostofpreviousstudiescon-
7 centrate on optimizing the topology of network and design- MTw=w, (1)
9
4 ingbetterroutingstrategies[16] toimprovetheefficiencyof
whereMT isthetransposedmatrixofM. Andwcanbecom-
0 transportation. These methods, usually needing high cost to
1. changeinfrastructureorveryconcreteandglobalinformation puteddirectlywithnormalizationcondition(cid:229) nj=1wi=1. The
dynamicalpropertiesofthe networkcanbe evaluatedbyen-
0 ofthetrafficprocess,aredesignedforsomespecialcasesand
7 maybeuneconomicalandunattainableinpractice. tropyratesthatisgivenby
1
Our study is complementary to the above efforts. Here, (cid:229)
: s=− wq lnq . (2)
v basedonmaximumentropymethod,wedevelopanapproach i ij ij
i,j
i to mitigating congestion by allocating limited capacity to
X
nodes reasonably in network with observable information in Theentropyrateisaquantitystudiedinthediffusionprocess
r
a trafficprocess. Werevealthatwecanimprovethewholesys- and measure the time density of the average information in
tem’straffic efficiencybyjustimprovinga little nodecapac- a stochastic process. It hasbeen showed that a highentropy
itywithoutchangingnetworktopology. Ourmethodistested rateindicatesalargerandomness,oreasinessofpropagating
in variousmodelsfor differentnetworks, performingwell in from one node to another [5, 7, 17]. Therefore it can also
raisingtrafficefficiency. berelatedtoanefficientandfreetrafficoverthenetwork. In
Beforepresentingourapproach,wegiveabriefdescription thatpaper,ourmaingoalistomitigatingtrafficcongestionby
ofnetworktraffic. Ateachtimestep,eachnodehasaproba- maximizingtheassociatedentropyratefunction.
bilitytogenerateapacketwhichhasadestinationandchooses Averysimplemethodtomitigatecongestionistoimprove
apathtowardsitsdestinationinnetworkimmediately. Anda thecapacityC ofeachnodeN ashighaspossible. However,
i i
nodeN candeliveratmostC packetstothenextnodeofits thecapacityisusuallylimitedbyvariousconditions. Andan
i i
path at a time step. As the capacity is limited, the packets fundamentalandsignificantproblemisabouthowtoallocate
whicharrivetothenodeearlierhaveprioritytobedelivered. the limited capacity to a given network to make the system
Apacketisremovedfromthetrafficafterreachingitsdestina- efficient. Next we try to solve this problem by maximizing
2
theentropyrate. ThenodeN innetworkcorrespondingtothe BythemethodofLagrangemultipliers,wecanrangenodesin
i
row vector v =(q ,q ,...,q ) of network’s transition ma- networkbythepriorityh (0)/Q forthecaseoflimitedtraffic
i i1 i2 in pi i
trix M=(q ), where q , accordingto the above generation capacitybudgetapproximately,andallocatethebudgettothe
ij ij
methodofmatrixM,canrepresentthepossibilityofapacket nodes with higher priority preferentially within the feasible
at node N going to nodeN at a time step in large scale ap- region.Incidently,withlessinformation,since
i j
proximately. When congestion occurs in node N, we have
i
q >0, thatis, thepacketshaveapossibilityto waitinnode limMTtx=w (7)
ii t→¥
N forbeingdeliveredatalattertimestep.Andthedirectcon-
i
sequence of improving the node capacity Ci is the decrease foranyinitialdistributionx,Qi canbeapproximatedbyQwi,
ofq . We denotetheq asthedetentionprobabilityofnode where Q is the average number of packets in the whole net-
ii ii
N inthispaper,whichplaysanimportantroleinourmodel. workateachtimestep.Thereforewecanalsorangethenodes
i
Suppose the change of qii is p (0≤ pi ≤qii) after being al- by hpi(0)/wi. As the complexityof solving a system of lin-
located new capacity(i.e. the probabilityof a packetstay in earequationswithnvariablesisO(n3),thedistributionstrat-
thenodeN isq −p). Sincethenodesdeliverthepacketsby egycanbegetatmostO(n3)stepsbynumericalcomputation.
i ii i
time-priority which is not associated with the packets’ rout- Furthermore,withascale-free(power-law)node-degreedis-
ing,itisnaturaltogetthatthe jth(j6=i)componentofvec- tribution[20], most of the realnetworks, especiallycommu-
tor v is (1+p/(1−q ))q . Therefor we can view v as a nication networks, are sparse. And there are more efficient
i i ii ij i
vector function of variables p. While M is composed from methodsforthesesparsenetworks[21].
i
{v} ,wecangettheentropyratesasafunctionofvari- Nextwesolveentropyrateasafunctionofdetentionproba-
i 1≤i≤n
able p=(p ,...,p )fromequations1and2. bilityfortwosimpleandbasicsystemsanalyticallyexplicitly:
1 n
It is obvious that a higher node capacityC will lead to a (i) a star-like system and (ii) a chain-like system [22]. For
i
lower detention probability qii. And we need a quantitative simplicity, we suppose the detention probability of node Ni
relationshiptoidentifythefeasibleregionof(p1,...,pn). As isqi,0≤qi<1andq=(q1,...,qn). (i)Forastarlikesystem
mentionedabove,wesupposewecangetthetheaveragenum- composedfromnnodesasfigure1(a)shows,wehavethetran-
berofpacketsthatthenodeNireceivedateachtimestepisQi, sition matrixMs =(qij), whereqii=qi,q1j =(1−q1)/(n−
whichcanbecountedrelativelyeasilyinpracticeorbecom- 1),qj1=1−qj,∀i,j andotherwiseqij =0. Solvetheeigen-
puted analytically for some special cases [11]. Suppose the vectorofMsandwehavethestationarydistribution:w=(wi)
changeofCiisdCi. Approximately,wehavethat wherew1=q /(1−q1),wj=q /(n−1)(1−qn)∀j≥2andthe
normalizationfactorq =1/(1−q )+(cid:229) n 1/(n−1)(1−q ).
dC 1 j=2 j
p =−dq ≈ i, (3) Calculateentropyratefromequations1and2:
i ii
Q
i
that is, the decrease of detention probability approximately q1lnq1 +ln1−q1 +(cid:229) n (1−qi)ln(1−qi)+qilnqi
equaltotheratiooftheincrementofthenumberofpacketsa h(q)= 1−q1 n−1 i=2 (n−1)(1−qi) . (8)
1 +(cid:229) n 1
nodecandeliverpertimestepandthetheaveragenumberof 1−q1 k=2(n−1)(1−qk)
packetsthatthenodeN receivedateachtimestep. Thenfor
i (ii) For a chain-like system composed from n nodes as fig-
agivenmodelandlimitednodecapacitybudgetB. wemodel
ure1(b)shows,we supposethetransitionmatrixM =(q ),
c ij
ourproblemasfollow:for
where q = q ,q = 1−q ,q = 1−q ,q = q and
11 1 12 2 nn−1 n nn n
(cid:229) n (cid:229) n qii−1 = qii+1 = (1−qi)/2,qii = qi,∀2 ≤ i ≤ n−1, other-
pQ = dC ≤B, (4)
i i i wise q = 0. Similar to the above, we get the stationary
ij
i=1 i=1 distribution w=(w), where w = m /(1−q ) for j =1,n,
0≤p ≤q , ∀i, i j j
i ii w =2m /(1−q )for2≤ j≤n,andthenormalizationfactor
j j
maximizethefunctionh(p1,...,pn) m =(cid:229) ni=−212/(1−qi)+(cid:229) j=1,n1/(1−qj). Then the entropy
rateis
maxh(p ,...,p ). (5)
1 n
Fromequations4,thefeasibleregionofoptimizationvariable (cid:229) qjlnqj+(1−qj)ln(1−qj)+(cid:229) n−12qilnqi+(1−qi)ln1−2qi
p is a convexset. And the entropyfunction have some nice h(q)= j=1,n 1−qj i=2 1−qi .
(cid:229) n−1 2 +(cid:229) 1
properties.Manysophisticatedmethods,suchasMonteCarlo i=2 1−qi j=1,n1−qj
(9)
algorithms[18],havebeendevelopedfromsuchproblembe-
For these two systems, we can analyze their entropy
fore[19], andcan be applieddirectly. Supposethath getits
maximumat p∗=(p∗,...,p∗)onthefeasibleregion. Weal- rate functions in the neighborhood of dentation probability
1 n
locatedC =p∗Q capacitytonodeN. Generally, p∗Q isnot (q11,...,qnn)andapplyourmethod.
i i i i i i
Wetestourmethodforthecongestionmodelbasedonref-
integerandweneedtoadjustourdistributionascloseaspos-
erence[11], wherethe nodecapacityC =1+int[b k] atthe
sibletoit. Forthecasethefunctionhisanalyticat0, i i
beginning,0≤b ≤1isacontrolparameterandk isthede-
i
(cid:229) n ¶ h greeofnodeN. Thetotalnumberofnodesinthenetworkis
h(p)=h(0)+ | p +o(kpk). (6) i
i=1¶ pi p=0 i n. As l , the possibility for a node to generate a package at
3
eachtimestep,isincreasedfromzero,twophasescanbeob- 0.14
0.147
served: freeflowforsmalll andacongestedphaseforlarge 0.12
0.146
l . Andthereisaphasetransitionatl . Tosimulatetheprac- 0.1
c
ticalcongestionprocess,suchasthetrafficcongestionduring V0.145 V0.08
peakhoursandthecongestionofcommunicationduringma- 0.144 0.06
jorholiday,wecreatel npacketsateachtimestepwithsome 0.04
0.143
probabilitydistributionforatimeperiod,andstopcreatingaf- 0.02
1 1.02 1.04 1.06 1.08 1.1 1 1.2 1.4 1.6 1.8
tercongestionoccurringforawhile,thenthecongestioncease
(a) (b)
after a time period. As what we have mentionedbefore, we
0.7 0.26
count the frequency in the process of congestion and apply
0.65
ourmethodtoallocatelimitedcapacitybudgettothenodesin 0.24
0.6
network. It is worth mentionedthat, differentfromprevious 0.55 0.22
V V
work[11],ourmethodandcalculation,basedonobservation, 0.5 0.2
donotrequireaknownprobabilitydistributionofpacketgen- 0.45
0.18
eration and the packets’ concrete routing strategy. For com- 0.4
parison, we allocate the same capacitybudgetrandomlyand 0.351 1.05 1.1 1.15 1.2 1.25 1.3 0.161 1.1 1.2 1.3
byadegree-biasedstrategy,wherethedegree-biasedstrategy,
(c) (d)
similartoreference[11], istoincreasethecontrolparameter
b toallocatecapacitybudgettonodes. Thenwecreatepack- FIG.1. (Colour On-line) hVi−a graph, the average speed hVi is
ets in the same way again and compare the traffic efficiency shownasafunctionofa for(a)Cayleytree,z=3,l=6,hki=2,thus
betweenthesemethods. HereweusequantityhVitocharac- N=1093,b =30(b)regularnetwork,N=1000,hki=4,b =0.4,
terize the traffic efficiency. HereV is the velocity of a node (c)randomnetwork,N=1000,hki=4,b =0.3(d)scale-freenet-
beingnavigatinginnetwork. work, N =1000, b =0.2, withdifferent capacity allocation meth-
ods,wheretheargumenta >1istheratiobetweeninitialcapacity
L sumandimprovedcapacitysumofthewholenetwork,andtriangle,
V = , (10)
T plusandcirclecurvescorrespondtothesimulationsofourmethod,
degree-biasedmethodandrandommethod. Theinitialcapacitydis-
where L is the length (i.e. the number of nodes a packets tributionfollowsCi=1+int[b ki]atthebeginning. Wesimulatethe
passedfromstartpositiontodestination)andT issumoftime process of congestion by keeping l =0.8 for 500 time steps and
steps it takes. The h...i indicates the average over all pack- choosingthegeneratingpackages’destinationsrandomly.
ets. Wepresentoursimulationfor(i)Cayleytree,(ii)regular
network, (iii) randomnetworkand (iv)scale-freenetworkin
networkwithdifferentl . Similartotheabove,werepeatthe
Fig. 1. As it shows, we can see that, for different node ca-
sameprocessforrandomstrategyanddegree-biasedstrategy
pacitybudgets,correspondinghVibeinglowerthanothersin
asacomparison.Ithasbeenshowedthat[8],aftercongestion
general,ourmethodgenerallyperformswellinimprovingthe
occurring,thenumberofpacketsinthenetworkQwillgrow
trafficefficiencywithlimitedcapacitybudget.
linearlywithtime. Andthereisaparameter:
Formerapproachesof controllingtraffic congestion, espe-
ciallyforheterogenousnetworkssuchasscale-freenetworks hD Qi
[14,23],concentratedonoptimizingthenodeswithhighde- h = lim (11)
t→¥ l D t
gree. However,asFig. 2shows, wefindthattheimportance
of nodesin the traffic processis not totaly dependenton the toqualifythetransition,whereD Q=Q(t+1)−Q(t)andh...i
theirdegree,thatis,thenodeswithhighdegreemaynotmore indicates the average over time windows of D t. We use this
importantand need to be allocated morecapacity than those parametertoevaluatetheglobaleffectofouroptimizationfor
withlowdegree. Thenodeswithhigherpriorityarefairlyin- two heterogenous networks: Cayley tree and scale-free net-
dependentof their degree. For a fixed degreerange, there is work. As Fig. 3 shows, the corresponding h of method is
a wide spread of priority. Therefore our method might be a lowerandthethresholdofphasetransitionishigherthanoth-
modificationoftheformerdegree-biasedstudy. ers. Our method has a global optimal effect on mitigating
Furthermore,werevealthatourmethodcanmitigatingthe congestion.
congestiongloballywithlocalinformation.Inmoredetail,the Introducingmaximumentropyratemethodintotrafficnet-
transitionfrequencyofeachnodewhatwegetisthephaseof work may also help us assess a traffic network globally and
networkinaspecialphasewithal . However,asthestation- evaluatethegapbetweenitandtheoptimalstatebycompar-
arydistributionandentropyrateareglobalforthedynamical ing it to the maximal-entropy random walk [7, 24], which
processoncomplexnetwork,ourmethodmayalso be effec- can be relatively localized easily. Suppose that the practi-
tual on a much larger scale. By simulation, we find that we cal transition matrix is M =(p ), and the transition matrix
ij
can optimize a wide range of phases with only the informa- corresponding to maximal-entropy random walk is P =p .
ij
tion got in one phase. We allocate capacity budget by our Regardingthe transition matrix as randomvariables, we can
methodfor a properphase with a largel . And then test the useKullback-Leiblerdivergencetoidentifythedifferencebe-
4
10
16 divergenceismoreefficientgenerally.
14 8 Recently, it has been showed that many real networks are
temporal networks which has many advantages[25]. As our
12
6 method,basedonthe dynamicalstate of network,isless de-
10
k pendentonthefixednetworktopology.Itmaybesuitableand
8 4 helpful for the optimization of the communication on tem-
6 poral network since the priority can be computed for every
2
4 timestep. Furthermore,asthecapacityusuallychangesover
timeintemporalnetworkstoo,ourmethodmayenlightenre-
2 0
200 400 600 800 searcherstostudyhowtoallocatetrafficcapacitydynamically
priority high low
ontemporalnetworkstoimprovingitstrafficefficiency.
(a)
Summarizing,inthispaper,weanalyzethequantitativere-
60
lationshipbetweentrafficcapacityincongestionandentropy
25
50 rate in complex network and solve it explicitly for star-like
20 system and chain-like system. Based on the principle that
40 a higher entropy rate leads to the easiness of propagatingin
15
k complexnetwork,wedevelopamethodtomitigatingconges-
30
10 tion by allocating the limited traffic capacity to nodes ratio-
20 nally. We analyze the complexity of our method, get that it
5 canbecompletedinatmostO(n3)steps,thereforisfeasible.
10
Whatismore,asitonlyrequiresthelocalandobservablein-
0
200 400 600 800 formationanddoesnotchangethe topologyofnetwork,this
priority high low
methodis shown to be low-costand generally practical. We
(b)
testourmethodandcompareittoothermethodsfor(i)Cayley
tree,(ii)regularnetwork,(iii)randomnetworkand(iv)scale-
FIG.2. (ColourOn-line)Ourmethodidentifiestheimportantnodes
for limited capacity budget more reliably than the degree-biased free network amd get that our method is more functional in
method. Thenodesarepermutatedwiththeirpriorityp=hpi(0)/Qi general. The congestion is considerably mitigated after ap-
computed from the equation 6 in the order from highest to lowest plyingourstrategy. Freeflowsonnetworksarecriticalinour
onthehorizontalaxisandtheordinateaxiscorrespondstotheirde- worldandhowtomitigatecongestionwithlimitedbudgetisa
greehki. Thedistributionof(p,hki)isshownviaheatgraphfor(a)
fundamentalandsignificantproblem. Ourproposedaheuris-
random networkand (b)scale-freenetwork. Theconditions where
tic method for this problem and may be useful for modern
wegettheresultsfromarethesameastheconditionsinFig. 1for
society.
randomnetworkandscale-freenetwork.
600 600
500 500
400 400
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