Modeling and Performance Studies of Data Communication Networks using Dynamic Complex Networks

0.23 MB·

by  Suchi Kumari

Modeling and Performance Studies of Data Communication Networks using
Dynamic Complex Networks
SuchiKumari
DepartmentofComputerScienceandEngineering,
NationalInstituteofTechnology,
Delhi,India
email:[email protected]
AnuragSingh
DepartmentofComputerScienceandEngineering,
7 NationalInstituteofTechnology,
1 Delhi,India
0 email:Correspondingauthor:[email protected]
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1 Abstract

] All the existing real world networks are evolving, hence, study of traffic dynamics in these enlarged networks is a
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challengingtask. Thecriticalissueistooptimizethenetworkstructuretoimprovenetworkcapacityandavoidtraffic
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congestion. We are interested in taking user’s routes such that it is least congested with optimal network capacity.
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s Network capacity may be improved either by optimizing network topology or enhancing in routing approach. In
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thiscontext,weproposeanddesignamodelofthetimevaryingdatacommunicationnetworks(TVCN)basedonthe
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dynamicsofin-flowinglinks.Newlyappearednodepreferstoattachwithmostinfluentialnodepresentinthenetwork.

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Inthispaper,influenceistermedasreputationandisappliedforcomputingoverallcongestionatanynode.Userpath
v
withleastbetweennesscentralityandmostreputationispreferredforrouting. Kelly’soptimizationformulationfora
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8 rate allocationproblemisused forobtainingoptimalratesof distinctusersatdifferenttime instants anditis found
5 thattheuser’spathwithlowestbetweennesscentralityandhighestreputationwillalwaysgivemaximumrateatstable
4 point.
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. Keywords:
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0 DataCommunicationNetworks,TimevaryingNetworksmodel,SystemUtility,Routing
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:
v 1. Introduction
i
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Manyrealworldnetworks,suchasdatacommunicationnetwork,WorldWideWeb, transportationnetworketc.,
r follow heavy tailed power law degree distribution results in scale free behavior [1]. These networks are evolving,
a
hence,dynamicsofthenetworkshavebeenstudiedinpastfewyears. Incommunicationnetworks,datapacketsare
transportedfromsomesource(S)todestination(D)viaapath. Thecriticalissueconcernedwiththecommunication
networksistooptimizethenetworkstructuretoimprovesearchingorroutingefficiencyandavoidtrafficcongestion.
The next issue is related to efficient usage of the capacity of the networks. It is observed that few nodes become
congestedandthen,lateritresultsinpropagationofcongestioninthewholenetwork.Ithappensbecausethesenodes
are used most in the communicationpath as they are more central. Network capacity may be improvedusing two
methods:(i)Methodstooptimizenetworktopology,and(ii)Enhancementinroutingapproach.
1.1. Optimizationofnetworktopology
Optimization of network topologycorrespondsto reorganizationof the network for minimizingcongestionand
getting improved network capacity. Zhao et al. [2] have enhanced network capacity by redistributing the load of
heavily loaded nodes to other nodes in the network. Links are sorted in descending order according to their end
PreprintsubmittedtoElsevier January18,2017
nodedegreesandlinksareremovedinthedescendingsortedlist[3]. Forheterogeneousnetworks,Zhangproposed
amethodinwhichnodeshavinghighbetweennesscentralityareremovedfirst[4]. Networksaregrowingduringthe
wholeofthelifespanandnodesareconnectedviapreferentialattachment(BAmodel)[5]. InBAmodel,appearance
ofnewconnectionstotallydependsontheadditionofnewnodestothesystem. But,inrealsystems,linksareadded
andupdatedcontinuously.Tadic[1]hasdevelopedamodelfordynamicsofdirectednetworks.Degreedistributionof
boththeoutgoingandincominglinksfollowpowerlawdegreedistributionandthereisacorrelationbetweenthem.It
suggeststhatlocalstructureofthenetworkisdifferentincomparisonwiththecaseofwithoutupdates.Thereisaneed
toincludetimeparameterwiththenodesandlinksofthebasicnetworkstructureso,manymodelshavebeenproposed
forthesetime-varyinggraphs(TVGs)[6,7,8,9]. ModeldevelopedbyKostakosetal. [10]isbasedontheconceptof
quicktransferanddelayedtransactionofdatabetweendistinctnodesinthecommunicationnetwork. Kimetal. [8]
focustheirworkondiscoveringclosestandmostbetweennesscentralnodeinthedynamicallychangingnetworks.A
seriesofstaticgraphs(i.e.,thesnapshots)areusedtorepresentthenetworkatagiventimeinstant[11]. Thesnapshot
model makes an external assumption that the network must be capable of retaining the link transitivity during the
periodof disconnection. Recently, Wehmuth et al. [6] have proposeda new unifyingmodelfor representingfinite
discreteTVGs. TVGnodesareconsideredasindependententitiesateachtimeanditisanalyzedtheproposedTVGis
isomorphictoastaticgraph. Itisanimportantresultbecauseoftheuseoftheisomorphicdirectedgraphasatoolto
analyzeboththepropertiesofaTVGandthebehaviorofdynamicprocessesoveraTVG.Aframeworktorepresent
mobilenetworksdynamicallyinaSpatio-temporalfashionisdesignedandalgebraicstructuralrepresentationisalso
provided[12].
1.2. Enhancementinroutingapproach
Networkrestructuringcostismore,hencemostoftheresearchersfocusonfindingbetterroutingstrategies. Tra-
ditionally,packetsaresentalongtheshortestpathsinthetermofhopcountorminimalsumsofweightedlinks[13].
Mostoftheshortestpathsarepassingthroughmostcentralnodesandhence,sendingdataonlythroughshortestpath
isnotefficient. Yanetal. [14]havefocusedtheirworkonsearchingefficientpath,instead ofsearchingofshortest
paths. Incommunicationnetworks,tworatesareassociatedwitheachnode:packetgenerationratel andpacketfor-
wardingratem . Thereisacriticalrateofpacketgeneration(l ),belowwhichnetworkwillbefreefromcongestion.
c
TwomodelsareproposedbyZhaoetal. [2]forfindingpacketdeliveryrateofanode,(i)basedonnode’slinksand
(ii)numberoftheshortestpathpassingthroughthenode.Inlatermodel,criticalrateisindependentofnetwork’ssize
andtopology,unlike,previousmodel. Duetal. [15]havedevelopedamodelinwhichpacketsaregeneratedwitha
ratel betweenrandomsourceanddestinationpairs. Packetsarepropagatedintothenetworksuchthatloadismin-
imizedandwillbedeliveredbyShortestRemainingPacketFirst(SRPF) algorithm. Theyalsostudiedthebehavior
ofcongestednodewiththearrivaloflargenumberofpackets. AheuristicroutingstrategiesareusedbyJiangetal.
[16],inwhichroutingprocessisdividedintoN (sizeofnetwork)steps. Alltheroutesfromthesourcenodearebased
onthebetweennesscentralityandnode’sdegreeinformation. Puetal. [17]haveformedanetworkmodelbasedon
BarabasiAlbert(BA)modelandroutingbetweenpairofnodesaredonesuchthatcostfunctionisminimized.Robust-
nessandnetworkefficiencyarecalculatedagainstcascadingfailure.Networksarechangingrapidlywithtime,hence,
acquiringglobalinformationofthenetworkusingglobalroutingstrategyisnotusefulforlongertime. Variouslocal
routingstrategiesareproposedbyLinetal. [18]. Proposedmethodsincludenodeduplicationavoidance,searching
nextnearestneighborsandrestrictedqueuelength. Alltheproposedstrategiesareusedtoenhancetheperformance
ofthesystem,byincreasingpacketgenerationrateofthenodesandreducingpackettransmissiontime.
1.3. Ratecontrolmechanism
Somenodesaswellaslinksaregettingaddedorremovedbutthereisanetincrementinthesizeofthenetworks
withtime.Asnetworkisexpandingdaybydaysodesignandcontrolofsuchnetworkisachallengingwork.Structure
ofnetworksisdependentondistributionoflinksanddirectlyaffectstheaccessibilityofparticularnode. Topologyis
changingandduetochangeintopology,theperformanceofthenetworkisalsoaffected.Communicationnetworkcan
respondtothe randomlychangingtrafficbutmoderncommunicationnetworksface multiplechallengesatdifferent
layers. Reallifecommunicationnetworksareextremelyvolatileandmakingconnectionsinthevolatileenvironment
withoptimalnetworkutilityisachallengingtask. Modelingtheirratecontrolbehavior[19]withvolatileanddynamic
connectivity helps us in getting optimal utilization of the system. La et al. [20] have extended Kelly’s work by
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introducingasuitablepricingscheme.Analgorithmisproposedforadjustingnetwork’spriceandusers’windowsize
is re-framedto achieve optimalsystem utility by maintainingweighted proportionallyfairness. Priya Ranjan et al.
[21]haveinvestigatedKelly’soptimizationframeworkandusedforfindingstabilityconditionswiththeconsideration
ofarbitrarydelays.Effectofthenonresponsivetrafficonthesystemstabilityisstudied.Itisshownthatubietyofsuch
elementshelpinimprovingthestabilityofthesystem. Recently,progressismadeontheearlierworkofmodelingof
TVCNandanewLyapunovfunctionisbeingformulatedforratecontrolschemeandlongtermstabilityisobtained
[22].
In previous researches, various routing strategies are proposed to avoid congestion in the communication net-
works. Here, work is focused on congestion and network utility both and the routing of the users are established
with minimumcongestionand maximumnetworkutility alongthem. Networkutility is calculatedby summingup
all individualuser’s utilities, and it will be maximum when, each user send data to their destinations with optimal
rate. Oncenetworkisdesignedthen, usersselecttheirroutesfordatacommunicationwith theirdestinednodes. In
thispaper, networkis designedwith the considerationof expansion,rewiringand removalof the links. A modelis
proposedforremovinginfrequentlyusedlinksinthenetworkapartfromadditionandrewiringoflinks. Additionand
rewiringoflinksarebasedonpreferentialattachment. Theremovedlinksincreaseunnecessarilymaintenancecost,
hence, the removalof these links will reduce the cost. In the proposed model, the links attached to a node having
lower degree, will be preferredto remove. Optimality of the routes are checked by analyzing Kelly’s optimization
approach[19]forarateallocationproblemincommunicationnetworksatdifferenttimeinstants.
Centralityplaysanimportantroleforcongestioninthenetwork,especiallybetweennesscentrality. Betweenness
centralityofanodevisequaltothenumberofshortestpathsfromallnodepairspassthroughthatnodevandisgiven
by,
g(v)= (cid:229) s S→D(v).
s
S,v,D S→D
Iflargernumberofshortestpathsarepassingthroughanodethen,betweennesscentralityanddegreeofthatnodewill
bemorebut,vice-versaisnottrue[23]. Connectivitydistributionofbothbetweennessanddegreecentralityofthereal
world networksfollow power law distribution with differentbut, correlatedexponents[24]. Nodes with maximum
betweennesscentralityisthemostcongestednodeinthenetwork[2]. Intheproposedroutingstrategy,shortestpaths
withlowesttotalsumofbetweennesscentralityofnodesareselectedforsendingdatapackets.
Somenodesareendorsedbythecongestednodesandcauseincreaseinloadatthatnode.Hence,thesenodesmay
beavoidedforreducingdatatrafficinthenetwork. Eigenvectorcentralityisusedtocheckinfluenceofonenodeto
others. In thispaper,influenceismeasuredintermsof congestioninthe networks. A nodewith highEigenvector
centralityis highly congestedand it is undesirableto connectwith thatnode. We have used a term, “reputation”,
forfindingoverallcongestionatanynode. ReputationofanodeisinverselyrelatedtoEigenvectorcentralityofthe
particularnode,hence,mostEigenvectorcentralnodeisleastreputed. Both,betweennesscentralityandreputation
are collectively used for avoiding congestion in the network. Among all the shortest paths, the path having lowest
betweennesscentralandmostreputedaredesirablefordatacommunicationbetweenuserpairs.Thesedesirablepaths
aretakenasaninputforKelly’soptimizationframeworkandaretheachieversofoptimaldataratesx∗.
Section 2 states background information about the related work and describe the mathematical model used in
the analysis of rate controlbehaviorof the user route. Section 3 introducesmodelfor growthdynamicsof TVCN,
formulationofscalingexponentandtheproposedroutingstrategiesforcommunicationnetwork. Section4presents
theoreticalandsimulationresults,andinSection5,conclusionandfutureresearcharediscussed.
2. BackgroundandRelatedwork
Incommunicationnetwork,eachnodecangenerateandforwardpacketswithdistinctrate,l andm respectively.
Therearethreepossiblerelationshipsbetweenl andm : (i.) l <m impliesthatsysteminfreeflowstate,(ii.) l =m
showsboundarycase forcongestionand(iii.) l >m allowssystem incongestion. Zhaoetal. [2]haveproposeda
modelinwhichpacketforwardingratem ofanodeiformodel:(i)basedondegreeofanode,m =1+⌊b k⌋and(ii)
i i
basedonbetweennesscentralityofanode, m =1+⌊b gi⌋. Here,k =degreeofnodei, g =betweennesscentrality
i N i i
of node i, N = size of the network and b , 0<b <1. By theoretical estimation, the value of critical rate is given
byl = mLmax(N−1). Thenodewithmaximumpacketforwardingcapacityandbetweennesscentralityaredenotedby
c gLmax
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m andg respectively.
Lmax Lmax
Anorderparameterz (l )[25],isusedtodescribethetrafficandgivenby,
m hD Num i
z (l )=limt→¥ l D t p
D Num =Num (t+D t)−Num (t),Num (t)isnumberofpacketsattimetandh.ishowsthataveragevalueoverthe
p p p p
timewindowD t.hD Num i=0showsthatnopacketinthenetworkandwhenh =0,systemisinfreeflowstate. Yan
p
etal. [14]havefoundaveragebetweennesscentralityg(k)forthenodeswithsamedegreek:
1 (cid:229)
g(k)= g(v)
N(k)
v:kv=k
a
where, N(k) is number of nodes with degree k. For Scale Free networks, g(k)∼k . An efficient path P(i→ j)
between node i and j is calculated such that the sum L(P(i→ j :a )=(cid:229) N−1k(x)a should be minimized for an
i=0 i
optimal value of a . Critical rate, l also depends on m and hence, l (b )= N(N−1). Liu et al. [3] have studied
c c b
g
Lmax
networktrafficbyconsideringlocalandglobalroutingstrategies. BetweennessCentrality(globallycalculated)ofa
linkwhoseendnodesaremwithdegreek andnwithdegreek ,islinearlycorrelatedwithmultiplicationofdegree
m n
ofendnodes,k k (locallyevaluated). Toavoidcongestioninthenetwork,linkshavingmaximumk k valuesare
m n m n
closed. Afterclosingfewlinks,l ,averagepathlength(L )andaveragetraveltime(hTi)arestudied. Jiangetal.
c avg
[16]havefoundheuristicmethodforroutingandusedanincrementalapproachforaddingnewpathsinthenetwork.
Asadditionofnewpathscausechangeinbetweennessanddegreeofeachnode.
v=N−1
Ls(P(i→ j))= (cid:229) fS(n )
v
v=0
where, fS(n )=(gS DegS )a isdynamicweightofnodev,n forsourcenodeS,gS isdynamicbetweennesscentrality
v nv nv v xv
ofn andDegS is dynamicdegreeof nodev. Hence, weightof each link(i,j) is (gSDegS)a +(gSDegS)a . All the
v nv i i j j
previousworksdiscussedtillnowarefocusedonroutingandoptimizingnetworkstructure. Tadic[1]hasdiscussed
aboutrestructuringofdirectednetworksbyconsideringgrowthandrearrangementatuniquetimescale. Ateachtime
unitt a new nodeisaddedtothe network(growth)anda numberX(t)andfractionb areusedto findnewlyadded
andrearrangedlinksinthenetwork. (i.) Aportion f (t)=b X(t)ofnewlinksareout-flowinglinksfromthenew
add
added node i=t and (ii.) the remaining part f (t)=(1−b )X(t) are the updated links at other nodes in the
update
network. Optimization of networktopologyand developingefficient routingstrategies help us to avoid congestion
andtoimproveoverallnetworkperformance. Incommunicationnetwork,varioususerswanttoestablishconnection
betweenspeciousnodepairsviaanoptimalroute. Routesareselectedsuchthatusercansenddatawithoptimalrate
to their destinations. Kelly [19] has formulated the rate allocation problem into optimization problem for a static
network. We have used the formulationof rate controlalgorithmand proposeda mathematicalmodelfor dynamic
communicationnetworks.
2.1. Mathematicalmodel
Abriefdescriptionispresentedaboutmathematicalrepresentationofthedynamicnetworks. Networkconsistsof
N nodesandE linksandlifespanofthenetworkisT. AsetofRusersarewillingtosenddatainthenetwork. Any
linke connectsnodemwithnodencansendmaximumC unitsofdatathroughit,where,C iscapacityofthe
mn emn emn
linke ande ∈E. InTVCN,eachuserrisassignedarouter∈Rforaparticulartimeinstantt ∈T. Attheend
mn mn i
oftth time,azero-onematrixAofthesize(N×N) isdefinedwhere,A =1,ifnoden andn areconnectedat
i ti ni,nj,ti i j
timet otherwisezero. Whentheuserrisassignedaratex then,utilityofuserratratex isgivenasU (x )is
i r,ti r,ti r,ti r,ti
increasing,strictlyconcavefunctionofx overtherangex ≥0. Aggregateutilityiscalculatedbysummingupall
r,ti r,ti
utilitiesofuserr withratex andisdenotedas(cid:229) U (x ). Rateallocationproblemcanbeformulatedasthe
r,ti r∈R,ti r,ti r,ti
followingoptimizationproblem.
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SYSTEM(U ,A )
ti ti
(cid:229)
maximize U (x ) (1)
r,ti r,ti
r∈R,ti
ATx ≤C andx ≥0
ti ti ti ti
A is a matrix at time intervalt tot. The given constraint states that a link can not send data more than its
ti i−1 i
capacity[19]. Itisdifficultandunmanageabletoallocateasuitableutilityfunctionandoptimalratetodistinctusers
in complex networks. Hence, Kelly has divided this problem into two simpler problems named as user’s optimal
problemand network’soptimal problem [19]. Let user r decide an amount P (t) to pay per unit time, while r is
r i
demandedapriceh forunitflow. Hence,userwillacquireaflow,x (t)=P (t)/h attimet. User’soptimalprice
r r i r i r i
forsendingx (t)attimet canbeobtainedusingtheoptimizationproblemas
r i i
User (U (t),h (t)),
r r i r i
maximizeU (x (t))−P (t), (2)
r r i r i
P >0
r
On the other hand, network wants to maximize weighted log function of P (t). Therefore, network optimization
r i
problemcanbeformulatedas
NETWORK(A ,p ),
ti ti
maximize (cid:229) P (t)log(x (t)), (3)
r i r i
r∈R,ti
ATx ≤C andx ≥0.
ti ti ti ti
As the network is time varying so, the values of h and x are varying with time. Each user r ∈R, initially
r r
computes the price per unit flow by using the Eq. ((2)) and it is willing to pay, P (t). Utility of each user r is
r i
assumed as a strictly concave function of user’s rate x (t) at that time (t). At each time instant if user wants to
r i i
accessa link e ∈E then, the costwill dependonthe totaldata flow throughthe link at thattime and is givenby
mn
y (t)=V ((cid:229) x (t))where,V (•)isgrowingifitcomesinlargenumberofuser’spathandV (y)isgiven
emn i emn r:emn∈E r i emn e
by
V (y)=c .(y/C )w
emn emn emn
where,c isconstantandassumedwithvalueone,C isthecapacityoflinke ∈E. Now,considerthefollowing
emn emn mn
systemofdifferentialequationforgettingoptimaldatarate(x∗)
dxr(ti) =J (P (t)−x (t) (cid:229) y (t)) (4)
dt r r i r i emn i
i emn∈r
Here,J isproportionalityconstant. Eachuserfirstcomputesit’swillingnesstopayasP (t)then,itadjustsits
r r i
ratebasedontheresponseprovidedbythelinksinthenetworkandtryingtobalanceitswillingtopayandtotalprice.
Eq. ((4))consistsoftwocomponents: asteadyincreaseintherateproportionaltoP (t)andsteadydecreaseinthe
r i
rateproportionaltotheresponsey (t)providedbythenetwork.
emn i
3. Timevaryingcommunicationnetworkmodelandroutingstrategies
Networktopologyandroutingareresponsibleforcongestionandnetworkefficiency. Anoptimalnetworkstruc-
tureisgeneratedusinganefficientTVCN model. Proposedmodelconsidersalltheaspectof growthandalteration
(rewiring& removal)of links in the network. In the established network, users establish dedicated connectionbe-
tween the nodes with whom they want to communicate. These connections are able to manage traffic efficiently,
whichresultsminimizedcongestioninthenetwork.
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3.1. Timevaryingcommunicationnetworkmodel
Timevaryingdatacommunicationnetworksaredesignedbymaintainingspecificsetofrulesandnodesarecon-
nectedviadirectedlinks. Directedlinksmayhavetwocategories: linkswhichcomeoutfromanode(out-flowing)
andotherlinksareincidentaltoanode(in-flowing).Degreedistributionofbothout-flowingandin-flowinglinksfol-
lowaheavytailpowerlawdistributionwithdistinctpowerlawexponents.Intheproposedmodel,thepreferencesare
assignedtoin-flowinglinks. Thenewlyincomingnodewillbeattachedtothenodeendorsedbythehighestnumber
ofnodes[1,5].
3.1.1. NetworkDynamics
AcommunicationnetworkmodelisproposedbyusingtheBarabasi-Albert(BA)model[5]andthemodelusing
thedynamicsofdirectedgraphs[1]. Theproposeddirectednetworkmodelhasfollowingproperties:
1. Networkconsistsofdirectedlinksforconnectingnodes.
2. Networkexpansion,linkrewiringandremovalaredoneatuniquetimescale.
Ateachtimeinstantt ∈T,anewnoden isaddedtothenetwork(expansion)andanumberX(≤n )isselectedfor
i i 0
networkexpansion,rewiringandremoval,wheren isinitialnumberofnodespresentintheseednetwork. Linksare
0
dividedintothreecategories: newlyadded,rewiredandremovedlinks. Distributionofthe linksaredoneusingthe
givensetofrules[26].
Notations:
(a.) b =Fractionoftheevolving(appearfromthenewincomingnodeandappear/disappearintheexistingnetwork)
linksatanytimeinstant. Itinformsabouttheestablishmentofnewconnectionsfromthenewincomingnoden
t
attimet ,0<b <1.
(b.) g =Fractionofthelinks,thosearerewiredintheexistingnetwork,0.5<g ≤1.
Usingtheabovenotation,followingsetofrulesmaybedefined:
(i) Totalnumberofnewout-flowinglinksfromthenewappearednode,i(=t)withthenodesexistinginthenetwork
at(t−1)basedonpreferentialattachmentisgivenby,
f (t)=b X
add
(ii) Fewlinksarerewiredintheexistingnetwork.g fractionoftheavailableX linksarechosenforrewiring,
f (t)=g (X−f (t))=g (1−b )X
rewire add
(iii) LastpartoftheremainingsegmentofX areusedfordeletingmostinfrequentlyusedlinks.
f (t)=X−f (t)−f (t)=(1−g )(1−b )X
delete add rewire
Theparameterd istheratioofalteredandaddedlinksinthemodelandisgivenby,
f (t)+f (t)
d = delete rewire
f (t)
add
(1−g )(1−b )X+g (1−b )X
= (5)
b X
(1−b )
=
b
Here,d isindependentofthenumberX andisknownascorrelationparameter.
Whilestudyingthebehaviorofcommunicationnetworks,theconceptofpreferredlinkingtoanodeisdrivenby
the demand of the node to outflow the data into the network. Only few influential nodes have the right to rewire
theirlinks. Nodesrewiretheirinfrequentlyusedlinksandconnectittopreferrednodesfordatacommunication.The
preferenceisgiventotheinfrequentlyusedlinksfortheremovaloflinks. Hence,thelinksattachedtoanodehaving
lowerdegreewillbepreferredforremoval.
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3.1.2. DegreeDistribution
Thescalefreebehaviorisfound,afterstudyingthescalingpropertiesofthetimevaryingnetworkmodel.Amean
field theoretical approach[5] is used to anticipate the growth dynamicsof distinct nodes, which is later applicable
foranalyticalcomputationofconnectivitydistributionandscalingexponents. Networkisgrowingwithtimehence,
someofthepreviouslyexistingnodeswillgetmoretimetoacquirelinksandarebeingtheholdersofhighestdegrees.
Degreek ofthenodeiischangingcontinuouslywithtime,soprobabilityP (k)isinterpretedasrateofchangeofk.
i i i
Ateachtimestamp,fractionsb andg decidethenumberoflinkschosenforexpansion(addition),rearrangementand
removal,andisdecidingparametersofdegreeofthenodei. Analysisisgivenusingfollowingsteps:
1. Afractionb ofthenumberX linksarenewlyaddedlinksattimet.
dk k
i =b X i (6)
dt (cid:229) k
(cid:18) (cid:19)add j j
Effectofnewlyaddedlinksonthedegreeofnodeiiswrittenonthelefthandsideoftheequationandonthe
rightsideb X linksareformedusingpreferentialattachment.
2. Afractiong (1−b )X linksarere-arrangedattimet.
dk 1 1 k 1 k
i =g (1−b )X + 1− i − 1− i (7)
dt n +t n +t (cid:229) k n +t (cid:229) k
(cid:18) (cid:19)rewire (cid:20) 0 (cid:18) 0 (cid:19) j j 0 (cid:18) j j(cid:19)(cid:21)
When few links are rewired then, change in degree of node i depends on three terms: first term shows ran-
domselectionofnodes,secondtermcorrespondstolinkingwithotherexistingnodeshavinghighpreferential
attachmentprobabilityandthirdtermshowsremovaloflinkhavinglowpreferentialattachmentvalue.
3. Afraction(1−g )(1−b )X linksareremovedfromthenetworkattimet.
dk 1 1 k 1
i =−(1−g )(1−b )X + 1− 1− i (8)
dt n +t n +t (cid:229) k n +t
(cid:18) (cid:19)delete (cid:20) 0 (cid:18) 0 (cid:19)(cid:18) j j(cid:19) 0 (cid:21)
Removal of links affect the degree of node i and it is shown in above equation. First term shows random
selectionofanodeandthatlinkwillberemovedwhichhaslowpreferentialattachmentvalue.
Attimet,thesumofdegreesofnodesinthenetworkwillbe:
(cid:229) k =2t[b X+(1−b )(g X−(1−g )X)]=2Xt[b +(1−b )(2g −1)]
j
j
Letc=b +(1−b )(2g −1).
Now,combiningtheeqns. (6)to(8)wegetthechangeindegreeofnodeiwithrespecttotimet.
dk b k g (1−b )k
i i i
= +
dt 2ct 2ct
2 1 k k
−(1−b )(1−g )X − − i + i
n +t (n +t)2 2ct(n +t) 2ct(n +t)2
(cid:20) 0 0 0 0 (cid:21)
b g (1−b ) k 1
= + i −((1−b )(1−g )X)
2c 2c t t
(cid:18) (cid:19)
(forlarget)
Letusassumeq = b +g(1−b ) andq =−(1−b )(1−g )X
1 2c 2c 2
Hence,theaboveequationcanbere-writtenas
dk k 1
i =q i +q (9)
1 2
dt t t
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Thesolutionofthe Eq. ((9)) isderivedbytakinginitialconditionthatnodeiappearsattimet with X connections
i
k(t)=X andisgivenas:
i i
q q t q1
2 2
k(t)= − + k(t)+
i q i i q t
1 (cid:18) 1(cid:19)(cid:18) i(cid:19)
q q t q1
2 2
= − + X+
q q t
1 (cid:18) 1(cid:19)(cid:18) i(cid:19)
Ateachtimestamp,anodeisaddedintothenetworkhence,Probabilitydensityoft is:
i
1
P(t)=
i i
n +t
0
Theprobabilitythatanodehastotalk(t)connectionsandwhichissmallerthank,P(k(t)<k),canbewrittenas:
i i
q q t q1
2 2
P(k(t)<k)=P(− + X+ <k)
i q q t
1 (cid:18) 1(cid:19)(cid:18) i(cid:19)
X+qq2 q11
=P t > 1 t
i q 
k+q2 !
1
 
 X+qq2 q11
=1−P t ≤ 1 t
i q 
k+q2 !
1
 
 
TheprobabilitydensityofkisP(k)andcanbewrittenas:
¶ P(k(t)<k)
i
P(k)=
¶ k
(10)
t q q1 q −(1+q1 )
2 1 2 1
= X+ k+
q (n +t) q q
1 0 (cid:18) 1(cid:19) (cid:18) 1(cid:19)
Now,assumea =−(1+ 1 ).
q
1
Thevalueofthescalingexponentoftherealworldnetworkliesbetween2and3,hence,fordynamiccommunication
network,exponent(a )mustliewithin2<a ≤3.Constraintona willbefulfilledif0.5<q <1. AsP(k)isalways
1
q
positiveso,X+ 2 shouldbepositive.
q
1
3.2. RoutingStrategies
Eachnodeinthecommunicationnetworkgeneratespacketswithacertainrateandcandeliverpacketsaccording
tothecapacityofthatparticularnode. Capacityofthenetwork(C)iscalculatedbysummingupthecapacityofall
theindividualnodesinthecommunicationnetwork. Aggregatevaluesofthepacketgenerationrateofallthenodes
aretermedastheloadofthenetwork.
N−1
(cid:229)
L= l
v
v=0
here,l = loadatnodev
v
When, load (L) exceeds the capacity (C) of the network then, it will become congested with the packets. If all
the packets are sent throughthe shortest paths to their destinationsthen, some nodes may appear frequentlyin the
formationofshortestpathanditwillbecomecongested.Inthelargecommunicationnetworks,multipleuserswantto
establishaconnectionbetweenasource(S)anddestination(D). Theremayexistmultipleshortestpathss (S→D)
z
fromStoD,wherez=1,2,...,c andc istotalnumberofshortestpathsbetweenSandD. Intheshortestpath,anode
may lie with high betweennesscentrality. Sending data throughthe shortestpath will lead to the congestionin the
8
network(mostoftheuserpairswilltrytofollowthesamepath). Therefore,itisimportanttoinvestigatea shortest
paths (S→D)betweentheuserpairssuchthatoverallbetweennesscentralityofthenodesappearsinthepath
q [s (S→D)]= (cid:229) g
g z v
v:v∈s z(S→D)
here,g = betweennessofnodev
v
shouldbeminimum. Hence,wewanttofindoutapathwhosenodesshouldnotbethepartoflargenumberofother
shortestpaths(lowbetweennesscentrality).Hence,itisdefinedby,
min{∀z:q (s (S→D))}
g z
Apartfromroutingstrategyusingbetweennesscentralityofthenodes,alocalstrategyisusedbyconsideringinfluence
of the neighbornodes. If a heavily loaded nodeis connectedwith some other nodesthen, it will make all the suc-
ceedingnodescongested. Heavilycongestedprecedingnodeshavenegativeimpactonsuccessornodes. Theimpact
ofonenodetoothersismeasuredbyEigenvectorcentralityofthenodev,
1 (cid:229)
x = a x
v k vj j
j∈N
Here,k islargestEigenvalueofthematrixAandifnodevand jareconnectedthen,a =1. IfanodeismostEigen
vj
vectorcentral,thenitwillbehighlycongestedandhence,leastreputed.So,amongalltheshortestpath,thatpathwill
bereputedifaggregatevalueofthereputationofthepathq [s (S→D)]
r
q [s (S→D)]= (cid:229) 1
r z
x
v:v∈s z(S→D) v
x = Eigenvectorcentralityofnodev
v
ismaximizedandandthefunctionisdefinedby,
max{∀z:q (s (S→D))}.
r z
OptimalpathbetweenSandD
q s (S→D)]= (cid:229) g x
gr z v v
v:v∈s z(S→D)
iscalculatedbyconsideringbothbetweennesscentralityandreputation. Apathisoptimalifit’selementswillhave
leastbetweennesscentralaswellasreputedandisgivenby,
min{∀z:q (s (S→D))}.
gr z
AsnapshotoftheproposedmethodisdiscussedinFig. 1. Twousers,U andU wanttosenddatafromnode9to
1 2
node4andfromnode10tonode5,respectively. Theirrespectiveroutesare9−1−4,9−5−4and10−1−5,10−
2−5.Boththeusersavoidcongestedrouteshence,U avoidstheroute9−1−4andU prefers10−2−5.FromFig.
1 2
1,itisevidentthatnode1ishighlycongestednodeinthenetwork,sobothusersavoidtosenddatathroughnode1
andselectotherpossibleoptionsavailableforsendingdatatotheirdestinednode. Packetgenerationrate(PGR)ofa
nodedependsonthepacketforwardingrateandbetweennesscentralityofthenode. Packetforwardingratedepends
onthecapacityofanode,thataffectsbandwidthofalink.
Capacity of a linkC is dependent on the degrees k and k of node m and n. It can be approximated by a
emn m n
power-lawdependence[27]
a
C =b(k k )
emn m n
a isthedegreeinfluencedexponentwhichdependsonthetypeofnetworksandbisapositivequantity.Threepossible
relationsbetweencapacityC andexponenta : (i)ifa >0then,dataistransmittedthroughhighdegreenodes(ii)
emn
whena <0then,datawillbesentthroughlowdegreenodesand(iii)a =0showsdegreeindependenttransmission.
9
Users: U1 = U9,4 and U2 = U10,5
2 U routes: (9-1-4) and (9-5-4)
1
10 7 U routes: (10-1-5) and (10-2-5)
2
Congested
9
1 3 λ1 = µ1(Ng1−1)
C16 = k1k6
µ1 = 1+ βNg1
r1 = e11
Cong. ∝ g
4 5 Cong. ∝ 1
r
Cong. = congestion
g = betweenness
8 k = degree, C= capacity,
6
λ = packet generation rate
Highest Betweenness µ= packet forwarding rate
Central Node
Figure1:Blockdiagram
Thevalueofa intheproposedmodelistakenpositivevaluewithconstanta =1andbisalsoequalto1. Reputation
andbetweennesscentralityaredecidingparameterforevaluatingcongestion. Reputationofanodeisinverseofthe
Eigenvectorcentralityofanode. While,congestionisproportionallyrelatedtobetweennesscentralityandinversely
relatedwithreputation.
Algorithmicstepsaregivenforexpansion,rewireandremovaloflinksintheTVCN(Algorithm1).
Algorithm1NetworkEvolution
1: Input: Anumberofnodes(n0)forcreatingseednetwork,n(≤n0),b ,g andT.
2: Output:Timevaryingdatacommunicationnetworks.
3: whilet≤T do
4: Expandthenetworkwithonenodeateachtimeinstantt.
5: forx: 1to fadd(t)do
6: Attachthenewlyaddednodewithnodehavinghighestinflowinglinkattachmentprobabilityintheexisting
network.
7: endfor
8: for y: 1to frewire(t)do
9: Removetheinfrequentlyusedlinkofaninfluencednodeandattachtothenodehavinghigherinflowinglink
probability.
10: endfor
11: forz:1to fdelete(t)do
12: Removetheinfrequentlyusedlinkofarandomlyselectednode.
13: endfor
14: endwhile
Steps involved in Algorithm 1, are used to design TVCN. In the time varying network, various users want to
communicatedata from a source S to destination D. User gives informationabout the source and destination, and
accordinglyS→Dpairsaregenerated. Incrementsinthenumberofusersleadtothenetworkintocongestedstate.
Therefore,ouraimistofindefficientroutingpathssuchthatmaximumnumberofusersaregettingbenefitedwitha
10