Comprehensive routing strategy on multilayer networks

0.45 MB·

by  Lei Gao

Comprehensive routing strategy onmultilayernetworks
Lei Gao,1,2 Panpan Shu,3 Ming Tang∗,1,2,4 Wei Wang†,1,2 and Hui Gao1,2
1WebSciencesCenter,UniversityofElectronicScienceandTechnologyofChina, Chengdu610054, China
2Bigdataresearchcenter,UniversityofElectronicScienceandTechnologyofChina,Chengdu610054, China
3School of Sciences, Xi’an University of Technology, Xi’an 710054, China
4School ofInformationScienceTechnology, EastChinaNormalUniversity, Shanghai 200241, China
(Dated:January20,2017)
Designing an efficient routing strategy is of great importance to alleviate traffic congestion in multilayer
networks. In thiswork, we design an effective routing strategy for multilayer networks by comprehensively
7
consideringtherolesofnodes’localstructuresinmicro-level,aswellasthemacro-leveldifferencesintrans-
1
missionspeedsbetweendifferentlayers.Bothnumericalandanalyticalresultsindicatethatourproposedrouting
0
strategycanreasonablyredistributethetrafficloadofthelowspeedlayertothehighspeedlayer,andthusthe
2
trafficcapacityofmultilayernetworksaresignificantlyenhancedcomparedwiththemonolayerlowspeednet-
n works.Thereisanoptimalcombinationofmacro-andmicro-levelcontrolparametersatwhichcanremarkably
a
alleviatethecongestionandthusmaximizethetrafficcapacityforagivenmultilayernetwork. Moreover, we
J
findthatincreasingthesizeandtheaveragedegreeofthehighspeedlayercanenhancethetrafficcapacityof
9 multilayernetworksmoreeffectively.Wefinallyverifythatreal-worldnetworktopologydoesnotinvalidatethe
1 results.Thetheoreticalpredictionsagreewellwiththenumericalsimulations.

]
h PACSnumbers:89.75.Hc,89.75.Fb,89.40.-a
p
-
c Alleviating the congestion in transportation and com- andreal-worldnetworks. Ourresearchmaystimulatefu-
o munication systems is vital to modern society. For the turestudiesondesigningrealistictransportationandcom-
s
purpose of redistributing the traffic load in a low speed municationmultilayernetworks.
.
s transportation system such as bus net, we can establish
c
i ahighspeedsystem(e.g.,subwaynetwork)inthebusyre-
s
gionsorbetweenthestationswithhightrafficflow,andthe
y I. INTRODUCTION
two systems make up a new multilayer system (i.e., mul-
h
p tilayernetwork). Recentyears,someinvestigationsabout
Manysystemsinmodernsocietycanbedescribedbycom-
[ trafficcongestiononmultilayernetworkswereperformed,
plex networks, such as power grids, transportation networks
which mainly focusedonthe differentrolesoflayersin a
1 and social networks [1–5]. Routing on such networked sys-
macroscopiclevel(e.g., different transmission speeds), or
v
tems to enhance traffic capacity is a significant issue, and
7 the local structures of nodes within the same layer in a
hasbeenwidelystudiedfromtheperspectiveofcomplexnet-
0 microscopiclevel,withouttakingthemintoconsideration
work frameworkoverthe past decades[6–11]. Moststudies
3 comprehensively. To this end, we propose a comprehen-
5 siveroutingstrategyonmultilayernetworkscomposedof aboutroutingarefocusedonmonolayernetworks. Thestud-
0 ies have revealed that traffic congestion is highly related to
a low and a high speed network. We introduce a macro-
. the structures of networks [3, 12, 13]. Generally, there are
1 andamicro-levelparametertoadjusttherolesofnetwork
threewidelyusedtechniquestoenhancethethroughputofthe
0 structuresplayedintheroutingstrategy. Ourstrategyre-
7 distributes the traffic load in low speed layer to the high whole network: (1) modification of network structures [14–
1 17], (2)optimizationoftrafficresourcesallocations[18–20],
speed layer reasonably, and the traffic capacity of multi-
: and (3) designingbetter routingstrategies[9, 21–27]. Com-
v layer networks are thus remarkably enhanced compared
i withthemonolayerlowspeednetworks. Foragivenmul- pare with the first two methods, proposing effective routing
X
strategies seems to be more practical and thus has attracted
tilayer network, an optimal combination of macro- and
r muchinterest. Amongnumerousdifferentkindsofproposed
a micro-level parameters is found. Under these parame-
routing strategies, an efficient routing strategy proposed by
ters, the traffic capacity of the system reaches its maxi-
Yan and his colleagues is widely acknowledged for its sim-
mum value. Moreover, increasing the networks size and
plicity and efficiency [21]. The strategy redistributes traffic
theaveragedegreeofthehighspeedlayercanenhancethe
loadincentralnodestoothernoncentralnodesandimproves
trafficcapacityofmultilayernetworksmoreeffectively.To
the network throughput significantly. Echenique et al. pro-
quantificationallyunderstandtheproposedroutingstrat-
posed a novel traffic awareness protocol (TAP) by consider-
egy, we developed a theoretical approach and a remark-
ing the waiting time of packets, in which a node forwardsa
ableagreementwithnumericsisobservedinbothartificial
packettoitsneighboringnodeaccordingtotheshortesteffec-
tivedistance[28,29]. Somescholarsalsoproposedstrategies
forsystemswithlimitedbandwidth[30,31].
∗Correspondenceto:[email protected] Withtheavailabilityofbigdata,scholarsfoundthatmodern
†Correspondenceto:[email protected] infrastructures are actually significantly interact with and/or
depend on each other, which can be described as multilayer
2
(multiplex) networks [32–35]. For example, to redistribute II. MODEL
thetrafficloadinalowspeedtransportationnetwork,wecan
buildanewhighspeednetworkinthebusyregionsorbetween
A. Networkmodel
thehighflowstations, andthetwomonolayernetworkscon-
stitute a multilayernetwork. Researchershavedemonstrated
Themultilayernetworkconsiderediscomposedbytwolay-
thatthedynamicsof[35]andon[36–39]multilayernetworks
ers with N and N nodes respectively. Layer A represent
aremarkedlydifferentfrommonolayernetworks. Untilvery A B
the low speed network, and layer B is the high speed net-
recently, some researchers studied the traffic dynamics on
work. In general condition, the expense of building a high
multilayer complex networks, i.e., how to alleviate the traf-
speed network is far more than that of low speed network,
ficcongestioninordertoenhancethemultilayernetworkca-
thusthesizeofhighspeednetworkissmaller.Forexample,in
pacity [40–45]. Interestingly, Sole´-Ribalta et al. [44] devel-
therailway-airlinemultilayernetwork,wherethespeed(cost)
opedastandardizedmodeloftransportationinmultilayernet-
of the airline network is faster (more) than that of the rail-
works, and showed that the structure of multiplex networks
waynetwork. Thus,thesizeoftheairlinenetworkissmaller
caninducecongestiononaccountoftheunbalanceofshortest
than the railway network, and all airline stations are located
pathsbetweenlayers.MorrisandBarthe´lemy[40]analyzeda
atpointswhichcanbeconsideredasnodesintherailwaynet-
multilayernetworkconsistsoftwolayers,andshowedthatit
work,butnoviceversa[47]. Forsimplicity,we assume that
ispossibletoobtainanoptimalcommunicationmultiplexby
thenodesinthehighspeedlayerBarearandomsubsetofthe
balancingtheeffectsbetweendecreasingtheaveragedistance
low speed layer A [40]. We use the uncorrelated configura-
andcongestiononaverysmallsubsetofedges.
tionmodel(UCM)[48]togeneratethelowspeedlayerA,and
Thestructuresofmultilayernetworksbringnewchallenges usetheErdo¨-Re´nyi(ER) networks[49] torepresentthe high
when we proposeeffectiveroutingstrategies, and the task is speed layer B. The multilayer network is generated as fol-
markedlydifferentfrom monolayernetworks. On one hand, lows: (1)BuildlayerAusingtheUCM methodwith power-
multilayer networks can relieve the traffic congestion of the lawdegreedistributionsP(k) k−γ, whereγ is thedegree
∼
low speedlayerbyusingthe highspeed layer,howevercon- exponent.WesetthesizeoflayerAasNA,theminimumde-
gestion may be induced in the high speed layer [46]. Al- greeiskmin = 2,andthemaximumdegreeiskmax √NA.
∼
though establishing high speed transportation networks can (2)RandomlyselectNB (NB 6 NA) nodesin layerA, and
improvethetrafficcapacityoflowspeednetwork,howtorea- match these nodes one-to-one. This means that each pair of
sonablytheredistributetrafficloadsisanessentialissue. On thetwomatchednodesvAo andvBo areactuallythesamenode
theotherhand,whendesigningeffectiveroutingstrategieswe butin two differenttransportmanners. Both of them can be
should (1) take the intra-layer structures into considerations denotedasacouplednodevo. Orsay,acouplednodevo has
from microscopic perspective, and (2) consider the efficien- tworeplicanodesinlayersAandB,whicharedenotedbyvAo
cies of different layers from a macroscopic view. Previous and vBo respectively. (3) Constructa ER networkas the sec-
investigationsabouttrafficcongestiononmultilayernetworks ondlayerB byusingtheselectedNB nodesinstep (2),i.e.,
mainlyfocusedon themacroscopicdifferencesbetweenlay- eachpairoftheserandomlyselectednodesareconnectedwith
ers[40],orthelocalstructureofnodeswithinthesamelayer a probabilityp. Accordingto the abovethree steps, a multi-
inamicroscopiclevel[43],withouttakingbothoftheminto layer network can be built. Note that every node in layer B
considerationcomprehensively. In this work, we proposean hasitscounterpartnodeinlayerA,buttheinverseisnottrue.
comprehensiveroutingstrategyonmultilayernetworksbyin- Wedenotethedegreedistributionofthemultilayernetworkas
corporatingthemacroscopicdifferenceofspeedbetweenlay- P(−→K)=P(k ,k ),wherek andk denotethedegreesin
A B A B
ersandmicroscopicdistinctionsamongdifferentnodesinthe layerAandB respectively.Foranodev inlayerAwithout
A
samelayer. Wefindthatourroutingstrategycanredistribute counterpartinlayerB,wehavek =0.Anillustrationofthe
B
thetrafficloadinlowspeedlayertohighspeedlayerreason- multilayernetworkisshowninFig.1(a).
ably, and the traffic capacity of multilayer networks are re-
markablyenhancedcomparedwiththe monolayerlowspeed
networks. Foragivenmultilayernetwork,thereisanoptimal
B. Routingmodel
combinationofmacro-levelparameterandmicro-levelparam-
eterthatmaximizethetrafficcapacity.Increasingthesizeand
averagedegree of the high speed layer can enhancethe traf- Inourmodel,weassumeallnodesinlayerAaretreatedas
ficcapacityofmultilayernetworksmoreeffectively.Numeri- bothhostsand routersforgeneratinganddeliveringpackets,
calresultsonartificialmultilayernetworksaswellasthereal while the nodesin layer B can onlydeliver packets. We as-
Work-Facebookmultilayernetworkagreewellwithouranal- sumethatacouplednodevo candeliverpacketsbetweenlay-
ysis. erswithinfinitybandwidthandnotimeconsumptionthrough
tworeplicanodesvo andvo. Forsimplicity,eachnodeinthe
A B
Theoutlineofthepaperisasfollows. InSec. II,wegivea layerA(B)hasthesamemaximumpacketdeliveryabilityC
A
detaileddescriptionofourroutingstrategyonmultilayernet- (C ). That’stosay,ateachtimestepeachnodev candeliv-
B A
works.InSec. III,wesuggesttheoreticalanalysis. InSec. IV, eryC packetstoitsneighborsinlayerAifithasnocounter-
A
wepresentoursimulationresults. SectionVsummarizesour partin layerB. Otherwise, its counterpartnodev canalso
B
resultsandconclusions. transmitC packetsinlayerB. WesetC = C = C = 1
B A B
3
headofit’squeueanddeliverthepackettothenextstop
A A in layer A (B). So a couplednodevo at mostprocess
s i j t s i j t 2 packets per time step through two replica nodes vAo
Origin Destination and vBo in layers A and B. For a non-coupled node
v (i.e.,nodev hasnocounterpartinlayerB),itcan
A A
processonlyC = 1packet. Whenapacketarrivesat
A
B B itsdestination,itisremovedfromthesystem;otherwise
itisqueued.
i j t i j t
(a) T=0 (b) T=1
C. Comprehensivemultilayerroutingstrategy
A A
s i j t s i j t By integrating different roles of nodes in micro-level, as
wellasdifferenttransmissionspeedoflayersinmacro-level,
weproposeacomprehensivemultilayerrouting(CMR)strat-
egy, which can remarkably enhance the traffic capacity of
B B multilayer networks. We denote that a path between nodes
sandtas
i j t i j t
p(s t):=s v0,v1, ,vl , ,vd−1,vd t, (1)
(c) T=2 (d) T=3 → ≡ F F ··· F ··· F F ≡
where vl is the l-st stop and node vl belongsto layer F
F F ∈
FIG. 1: An illustration of multilayer network where the nodes of A,B ,anddisthenumberofstopsinthispath. Similarto
{ }
layerBformasubsetofthenodesofnetworkA. EdgesoflayerA Ref.[21], wedenotean‘efficientpath’foranypathbetween
areshowninorange,edgesofnetworkBareshowninred,andnodes nodessandtas
incommontobothlayersareconsideredtobecouplednodes(shown
bydashedlines).Highlightedingreen,apathbetweenthe‘origin’s n−1
andthe‘destination’tisrepresented,andthearrowsshowapacket L(p(s→t),αF,βF)= XαF[k(vFl )]βF. (2)
deliveryprocessonthepathwithoutcongestion. Couplednodescan l=0
deliver packets between layers with no time consumption. (a) At
T = 0, a packet is generated with the randomly chosen‘origin’ s where k(vl ) is the degree of node vl in layer F. The effi-
F F
and‘destination’tinlayerA,andapath(highlightedingreen)from cientpathbetweens andt is correspondingto the route that
nodestotischosenaccordingtothepre-givenroutingstrategy. (b) makesthe sum L(p(s t),α ,β ) minimum. If there are
F F
T = 1,‘origin’sdeliversthepackettothenextstopithroughthe severalefficientpathsb→etweentwonodes,wechooseoneran-
edge(s,i)inlayerA. (c)T = 2,nodeideliversthepacket tothe domly. The efficient path is related to the macro-parameter
next stop j through the edge (i,j) in layer B. (d) T = 3, node
α andmicro-parameterβ . Themacro-parameterα 0
j deliversthepacket toits‘destination’ tthroughtheedge(j,t)in F F F ≥
controls packet transmission speed in layer F, and reflects
layerB,andthepacketisremovedfromthesystem.
the macro-level transmission speed difference between lay-
ers. Thesmallervalueofα ,thefastertransmissioninlayer
F
F. Theparameterα (α )correspondstotheslower(faster)
inthispaperforsimplicity. A B
networkandtheratioα /α controlstherelativetimespends
Duetothefinitedeliveryabilityofnodes,aqueueofbuffers B A
each jump in layer B compared with layer A. The micro-
isneededforeachnodeto accommodatepacketswaitingfor
parameter β determines the tendency of packets’ favour to
beingdelivered.Thetransportprocessesisasfollows: F
small-degreeorlarge-degreenodesinlayerF,andreflectsthe
1. Packet generation. At each time step, R number of micro-leveldifferencebetweennodesinthesamelayer.Large
packetsaregeneratedwithrandomlychosenoriginsand degree(smalldegree)nodesinlayerF arepreferentiallytobe
destinations in layer A. For each packet, a path from thenextstopwhenβF < 0(βF > 0). WhenβF = 0,nodes
thesourcetothedestinationischosenaccordingtothe withdifferentdegreeshavethesameprobabilitytobethenext
comprehensivemultilayerroutingstrategy(tobeintro- stop. Fig.1illustratestheroutingonmultilayernetworks.
ducedinthenextsubsection). Ifthereareseveralpaths
between these two nodes, we choose one randomly.
Each newly created packet is placed at the end of the III. THEORETICALANALYSIS
queueofitssourcenodev if thenextstop isinlayer
A
A,orqueuedatthecounterpartnodevo ifthenextstop Fromtheperspectiveofstatisticalphysics,weusetheorder
B
isinlayerB. parameterH(R)tocharacterizethecongestiononmultilayer
networks[21],
2. Packet processing. The first-in-first-out(FIFO) rule is
adopted to hand each queue. At each time step, node C ∆W
v (v )canprocessC =1(C =1)packetfromthe H(R)= lim h i, (3)
A B A B t→∞R ∆t
4
where∆W =W(t+∆t) W(t), isaveragevalueover IV. RESULTS
− h···i
∆t, andW(t) is the totalnumberof accumulatedpacketsin
thesystemattimet. FromthevaryingofH withR,wewill We introduce four parameters to investigate the effective-
knowthecriticalpointRc(tobecomputedlater)abovewhich ness of CMR strategy. First, we introducea generalizedpa-
the congestion occurs. For a small value of R, the number rametercouplingbasedonRef.[40],whichisusedtodescribe
of generated and delivered packets are balanced, i.e., every howwelltwolayersareusedtotransmitthepackets.Herethe
packetcanbetransportedtotheirdestinations,thusH(R) = couplingisdefinedas
0.ForalargevalueofR(i.e.,R>R ),thecongestionoccurs
c σst(α ,β )
and the numberof accumulatedpackets increaseswith time, P B F F
s6=t
so H(R) > 0. The critical traffic capacity R is the most λ= , (7)
c σst(α ,β )
significantparameter of a transportationnetwork, which can P F F
s6=t
beusedtoevaluatetheperformanceofaroutingstrategy,i.e.,
thelarger,thebetter. where σst(αF,βF) is the numberof efficient paths between
TocomputethevalueofRc,wefirstdefinetheefficientbe- nodessandtforgivenvaluesofαF andβF,andσBst(αF,βF)
tweennesscentralities(EBC)ofnodesinmultilayernetworks isthenumberofefficientpathsthatcontainsatleastoneedge
as in layer B. Specifically, we have σBst(αF,βF) = 0 when
everyefficientpathbetweennodessandtonlyusestheedges
σst(α ,β ,v) inlayerA. Forthecaseofλ 0,mostpacketsaretransported
g(αF,βF,v)=X σst(αF ,βF ) , (4) onlybylayerA,withoutusin≈gtheedgesinlayerB. Withthe
s6=t F F increaseofλ,morepacketsaretransportedbyusingtheedges
inB.
where σst(α ,β ) is the number of efficient paths be-
F F Secondly,wedefineδas
tween nodes s and t for given values of α and β , and
F F
σst(αF,βF,v) is the number of efficient paths that pass PesBt(αF,βF)
node v. The larger value of g(αF,βF,v), the more effi- δ = s6=t , (8)
cient paths that pass node v. As a result, node v needs to PesAt(αF,βF)
processmore packetsand hasa largerprobabilityto be con- s6=t
gested. We denote nodes with high values of EBC as high- whereest(α ,β )[est(α ,β )]isthenumberofedgesbe-
A F F B F F
load (HL) nodes, and similarly denote nodes with low val- longingtolayerA(B)intheefficientpathsbetweennodess
ues of EBC as low-load (LL) nodes. A coupled node vo andtforgivenvaluesofα andβ . Whenδ 0,mostedges
F F
≈
can deliver the packets to its neighbors in both layers A thatareusedtodeliverpacketsbelongtolayerA. Themore
and B, and it has two values of EBCs g(αF,βF,vAo) and edgesinlayerBareusedtotransportpackets,thelargervalue
g(αF,βF,vBo),wherevAo (vBo)representsnodevo inlayerA of δ. The definitions of λ and δ look similar, the difference
(B). If node vo or vo overload, traffic congestion will oc- isthatcouplingλrepresenttheproportionofalltheefficient
A B
cur at node vo. Thus, node vo’s EBC in the system is the pathsinsystemthatcontainedgesinlayerB,whileδ isthat,
maximum value of g(αF,βF,vAo) and g(αF,βF,vBo), i.e., inallefficientpaths,theratioofedgesinBandA.
g(α ,β ,vo) = max g(α ,β ,vo),g(α ,β ,vo) . For Thirdly, we define the average length of efficient paths to
F F { F F A F F B }
a non-coupled node v (i.e., node v has no counterpart in capturetheeffectivenessoftheCMRstrategyas
A A
layerB),itcanonlydeliverthepacketstoneighborsinlayer
1
A,anditsEBCcanbeexpressedasg(αF,βF,vA). d = Xdst(αF,βF), (9)
h i N (N 1)
At every time step, the system will generate R packets in A A− s6=t
layer A. We can get the average number of packets that a
where dst(α ,β ) is the length or jumps of efficient paths
nodevneedstoprocessas F F
between node s and t for given values of α and β . For
F F
example,thelengthofselectedpathbetweennodessandtin
g(α ,β ,v)
Θ =R F F . (5) Fig.1is3. Thesmallerof d , thelessaveragejumpsofthe
h Fi N (N 1) h i
A A packetsarrivethedestination.
−
Toimprovenetworktrafficcapacity,theaveragepacketde-
WhenR R ,thereisnoaccumulatedpacketsatanynodein
≤ c liverytime T mustbeminimized.Thedefinitionof T is
thesystem,i.e., ΘF C. WhenR>Rc,trafficcongestion h i h i
willoccuratsomheHiL≤nodes,i.e., ΘF >C. Sincethenode 1 n
with the largest EBC value has thhe larigest probability being T = lim XTp, (10)
h i t→∞n
congested, and combining the condition C = 1, the critical p=1
packetgeneratingnumberRc shouldfulfill wherenisthenumberofarrivedpacketsatagiventimeand
T isthepacketdeliverytimeofpacketp. Thedeliverytime
p
R = NA(NA−1), (6) ofeachpacketconsistsofthetravellingtimefromtheorigin
c gmax(αF,βF) to the destination and the waiting time in the queue of the
congestednodes. WhenRislessthanR , T onlydepends
c
h i
wheregmax(αF,βF)isthelargestvalueofEBCinthesystem on the travelling time which is relatively small, while when
forthegivenα andβ . R>R , T increaseswithRrapidly.
F F c
h i
5
A. Artificialmultilayernetworks of Rc. To understandthe non-monotonousphenomenon,we
need to check what happenswhen varyingβ . When β is
B B
small(large),thevaluesofλandδarelarge(small)asshown
In this subsection, we perform extensive numerical sim-
in Figs. 3(a) and (b). This indicates that packets are more
ulations on artificial multilayer networks. We set the size
likely to be transmitted in layer B (A). For a smallvalue of
of layer A as N = 1000, degree exponents γ = 3.0,
A
β , manycouplednodesareusedtotransmitpackets. Simi-
the minimum degree kmin = 2, and the maximum degree B
lartotheeffectivestrategyonmonolayernetworks,preferen-
kmax √NA. ThesizeoflayerB isNB =500andaverage
∼ tially transmittingthe packetsthroughsmall degreenodesin
degree k = 6. All the results are obtained by averaging
B
h i layerB couldimprovethetrafficcapacityofthesystem[21],
over20 differentnetworkrealizations, with 100independent
andR thusfirstincreaseswithβ . Foralargevalueofβ ,
runsoneachrealization. c B B
mostpacketsaretransmittedonlayerA,whichdecreasesthe
usage of coupled nodes in transmitting the packets, and Rc
0.10 (a) 80 (c) 30 thusdecreases. InFig.2(d),wefurtherverifyinwhichlayer
B=0.1 R c20 thecongestionoccursfordifferentvaluesofβB. Tothisend,
0H.05 BB==00..47 R c60 10 0 1 A 2 wwehesnetwtheechdeeclikvetrhyeaubpiplietyrloimfnitocdaepsaicnitlyayoefrlAaye(Br B)is(Ain)fi,ni.iet.e,
BB==11..03 40 Boo( B)=0.7 asinmaluylsaitsion CanAd g→(α∞,βand,vgo()α=F,βgF(α,vo,)β=,vgo()α]F. ,WβFe,fivnBod)th[CatBco→nge∞s-
Rc( B)=79 F F F F A
0.00 20 tionoccursinlayerB(A)forsmall(large)valuesofβ ,since
20 40 60 80 100 120 0 1 2 3 4 5 B

60 layerB (A)hasasmallercriticalnetworkthroughput. From
(b) (d) CB simulation
120 CB analysis what we discussed above, we can see that compared to the
T 40 BB==00..14 Rc80 CCAA sainmaluylsaitsion inseotlwatoerdklioswremspaerekdabnleytwimoprkroAve,dthaetscoampaecpiatyraRmcetoefrsm,suilntcileaytheer
B=0.7 trafficloadofthelowspeedlayerAisredistributedtothehigh
20 BB==11..03 40 speed layer B reasonably. The system capacity is affected
by both layersA and B, and dependsnon-monotonicallyon
20 40 60 80 100 120 0 1 2 3 4 5 micro-parameter β . The theoretical predictions agree well
R B B
withthenumericalsimulationsinbothFigs.2(c)and(d).
FIG.2: TheefficiencyofCMRstrategyonartificialmultilayernet-
We further study the effects of network size N and av-
works when αB = 0.5. (a) The order parameter H and (b) the B
eragedegree k of layer B (i.e., increasingthe numberof
average delivery time T of packets versus R. (c) The trafficca- B
h i
h i couplednodesand edgesin the high speed network)on sys-
pacityRcofmultilayernetworksand(d)theupperlimitcapacityof
layer B (A)asafunctionof theβB. Theinsetof (c)isthetraffic temcapacityinFig.4. AsshowninFigs.4(a)and(c),wefind
capacity Rc of monolayer networks (layer A) as a function of βA thatthemaximumtrafficcapacityRco(αB)whenαB =0.5in-
when αA = 1. The theoretical analysis results are obtained from creaseswith kB andNB,sincethenumberofefficientpaths
h i
Eq.(6).WesetotherparametersasNA =1000,γ =3.0,kmin=2, (coupled nodes) increase. That’s to say, increasing the size
kmax √NA,NB =500, kB =6,αA =1,andβA =1. andaveragedegreeof the high speed layerenhancethe traf-
∼ h i
fic capacity of multilayer networkseffectively. We note that
Wefirstfocusontheeffectsofmicro-parameterβB onthe the optimal micro-parameter βBo(αB) decreases with hkBi
effectivenessofCMRstrategyinFig.2. Sinceβ = 1isop- [seeFig.4(b)],butdoesnotchangewithNB [see Fig.4(d)].
A
Again,ourtheoreticalpredictionsagreewellwiththenumer-
timalvalue withoutlayerB for the case of α = 1 [see the
A
icalsimulations.
insetofFig.2(c)],we setα = 1andβ = 1. Throughex-
A A
tensivenumericalsimulations,wefindthatothervaluesofα All results above are obtained when macro-parameter
A
andβ donotqualitativelyaffecttheeffectivenessofthepro- α =0.5,wenextstudytheeffectsofmacro-parameterα in
A B B
posedCMRstrategy.Wesetα /α <1(i.e.,α 1)here, Fig. 5. We find that Ro(α ) dependsnon-monotonicallyon
whichindicatesthatajourneyoBnthAehighspeedlBay≤erBisfa- α [Ro(α ) first increcaseBwith α and then decrease], and
B c B B
voredforajourneyinlayerA.FromFigs.2(a)and(b),wefind the corresponding optimal micro-parameter βo(α ) mono-
B B
thatfordifferentvaluesofmicro-parameterβ ,boththeorder tonically decreases with α and reaches a suitable packets’
B B
parameterH andaveragepacketdeliverytime T monoton- preference to layer B. For small (large) values of α , λ
B
h i
icallyincreaseswithR. AbovethethresholdR , H and T and δ are large (small) as shown in the insets of Figs. 5(a)
c
h i
are finite, andincrease with R. Importantly,we find thatR and (b) respectively. Importantly, we find that the system
c
exhibits a non-monotonously varying with β as shown in reachesamaximumtrafficcapacityR⋆ attheoptimalmacro-
B c
Fig. 2(c), and the system existsan optimalvalueβo(α ) = and micro-level parameters combination (α⋆,β⋆), and the
B B B B
0.7 at which the traffic capacity R reaches the maximum number of delivered packets by each layer reach to a bal-
c
valueRo(α ) = 79whenα = 0.5. Theaveragelengthof ance. From Figs. 5(a) and (b), we obtain R⋆ = 79 at
c B B c
efficientpaths d reachestheminimumvalueatthesamepa- (α⋆,β⋆) = (0.5,0.7). Without the high speed network B,
h i B B
rameters[see Fig. 3(c)]. Specifically, R first increaseswith themaximumtrafficcapacityoflowspeednetworkis33[see
c
β , and peaks at βo(α ) = 0.7, and then decreases. The theinsetofFig.2(b)]. Themaximumtrafficcapacityofsys-
B B B
theoretical predictions agree well with the numerical values tem is improved about 2.5 times once the network B is in-
6

120
(a) 200 (c)
1.0 (a)
100 150
)B
o R(c 80 100
analysis 50 analysis
simulation simulation
0.5 60
0
0 10 20 30 200 400 600 800 1000

1.0 (b) analysis 1.2 (d) analysis
simulation simulation
0.8
0.0 )B 0.8
0 1 2 3 4 5 o (B 0.6
0.4
0.4
3
(b) 0.2 0.0
0 10 20 30 200 400 600 800 1000
<kB> NB
2
FIG.4:CMRstrategyonartificialmultilayernetworkswithdifferent
kB andNB. ThemaximumcapacityRco(αB)(top),andthecor-
hrespiondingoptimalmicro-parameterβBo(αB)(bottom)asafunction
1 of kB [(a)and(b)]andNB [(c)and(d)]. Wesetotherparameters
h i
as NA = 1000, γ = 3.0, kmin = 2, kmax √NA, αA = 1,
∼
βA =1,αB =0.5,NB =500[(a)and(b)], kB =6[(c)and(d)].
h i
0
0 1 2 3 4 5 TABLEI:StructuralpropertiesofWorknetworkandFacebooknet-
8 work, including number of nodes N, number of edges E, mean
degree k , maximum degree kmax, degree heterogeneity Hk =
k2 / kh2i,diameterD,averageshortestdistanceL,correlationco-
7 ehfficiiehntir,clusteringcoefficientc,andmodularityQ.
Network N E k kmax Hk D L r c Q
h i
d 6 Work 60 194 6.5 27 1.7 4 2.4 −0.218 0.64 0.46
Facebook 32 124 7.8 15 2.3 4 2.0 0.003 0.54 0.34
5
(c) fectiveness of our proposed CMR strategy on a real-world
4 multilayernetwork, which is a social network of Employees
0 1 2 3 4 5 ofComputerScienceDepartment(ECSD)atAarhusUniver-
sity[50]. Themultilayersocialnetworkconsistsoffivekinds
B ofonlineandofflinerelationships(Facebook,Leisure,Work,
Co-authorship, Lunch), and we choose the Work and Face-
FIG.3:Theparameters(a)couplingλ,(b)ratioofedgesthatusedin
book relationshipsas layer A and layer B, respectively. We
layersBandAδ,and(c)averagejumps d betweennodesVSβB
h i denotethisreal-worldnetworkasWork-Facebookmultilayer
onartificialmultilayernetworks. WesetotherparametersasNA =
1000, γ = 3.0,kmin = 2,kmax √NA,NB = 500, kB = 6, network. LayerAcomposesof60nodesand194edges,and
αA =1,βA=1,andαB =0.5. ∼ h i layerB has32nodesand124edges. Somestructuralproper-
tiesofthetwonetworksarepresentedinTableI.
We study the effectiveness of the CMR strategy on the
duced. Although establishing high speed transportation can Work-Facebook multilayer network in Fig. 6. Since the ex-
improvethetrafficcapacityoflowspeednetwork,ourresults tremely complicated structures of networks, there has two
indicatethatareasonableredistributionoftrafficloadisanes- peaks of R versus β , at which the traffic capacity is very
c B
sentialissue. The theoreticalpredictionsagreewell with the large,andtheRo(α )correspondstothesecondpeak. Com-
c B
numericalsimulations. paredtotheisolatedWorknetwork,thecapacityR ofWork-
c
Facebook multilayer network are improved when Facebook
network joins in the system [see the inset of Fig. 6(a)], and
B. Real-worldnetworks the system capacity is affected by both Work and Facebook
networks[seeFig.6(b)]. InFig.7,wefindthatRo(α )ver-
c B
A wide range of systems in the real world have multi- sus α exhibits a nonmonotonicpattern [see Fig. 7(a)], and
B
ple subsystems and layers of connectivity, which can be de- the corresponding optimal micro-parameter βo(α ) mono-
B B
scribed as multilayer networks [32–35]. We verify the ef- tonicallydecreaseswithα [seeFig.7(b)]. Similartothear-
B
7

80 14
8
(a) (a)
6c
12 R
* *
) 75 B =0.5, B =0.7 4
B c
( 0.99 R 10 2
o 0 1 2
c
R A
70 0.98 8
analytsis
0.0 0.5 1.0 1.5 2.0
B 6 simulation
65
0.0 0.5 1.0 1.5 2.0 0 1 2 3 4
1.2
3
(b) 50 (b)

2 40
)0.8
B c
1 R 30 CB simulation
o ( B 0 1 2 CB analysis
B 20 CA simulation
0.4
analysis CA analysis
10
simulation
0
0.0 0.5 1.0 1.5 2.0
0 1 2 3 4
B B
FIG.5:CMRstrategyonartificialmultilayernetworkswithdifferent FIG.6: CMR strategyonWork-Facebook multilayer networkwith
αB. (a)Rco(αB)and(b)βBo(αB)asafunctionofαB. Theinsets αB = 0.5. (a)ThetrafficcapacityRc, and(b)theupper limitca-
of (a) and (b) respectively exhibit λ and δ versus αB when βB = pacity of layer A (Work) and layer B (Facebook) as a function of
βBo(αB).WesetotherparametersasNA =1000,γ =3.0,kmin = theβB. Theinsetof(a)isthetrafficcapacityRc ofisolatedWork
2,kmax √NA,NB =500, kB =6,αA =1,βA=1. networkasafunctionofβAwhenαA =1.WesetαA=1,βA =1,
∼ h i
andαB =0.5.
tificialnetworks,wefindthatthesystemreachesamaximum
traffic capacity R⋆ = 15 at the optimal micro- and macro-
c
level parameters (α⋆,β⋆) = (2.1,0.7). The fluctuation of
B B by considering different transmission speeds of layers from
curvesinFigs.6and7iscausedbytheextremelycomplicated
the macroscopicview (by adjusting a macro-parameterα ),
structuresof both Work and Facebooknetworks. We should F
and different roles of nodes from the perspective of micro-
notethatthetheoreticalpredictionsmarkedlywellagreewith
scopic structure (controlled by a adjustable micro-parameter
thenumericalsimulations.
β ). We thenperformedextensivenumericalsimulationson
F
both artificial and real-world networks. We found that our
routingstrategy can redistributethe traffic loadin low speed
V. DISCUSSIONS layer to high speed layer reasonably, and the traffic capac-
ityofmultilayernetworkareremarkablyenhancedcompared
Forthepurposeofalleviatingthecongestionofalowspeed withthemonolayerlow speednetwork. Inaddition,thesys-
transportation network, an intuitive way is to built a new tem capacity is affected by both layers A and B, and de-
high speed network in busy regions or among the high flow pendsnon-monotonicallyonmicro-parameterβ andmacro-
B
nodes. The low and high speed networks constitute a mul- parameter α . For a given multilayer network, the system
B
tilayer network. How to reasonable redistribution of traffic reachesamaximumtrafficcapacityR⋆ attheoptimalmicro-
c
load to maximize the multilayer network is an essential is- and macro-level parameters (α⋆,β⋆). Moreover, we found
B B
sue and full of challenges. In this work, we first proposed that increasing the size and the average degree of the high
a comprehensivemultilayernetworkrouting(CMR) strategy speed layer B enhances the transport capacity of multilayer
8
networksmore effectively. The theoreticalpredictionsagree
wellwiththenumericalsimulationsinbothartificialandreal-
15 (a) worldnetworks.
14
)
B
13
(
o
c
R
12 Awisewaytoalleviatetrafficcongestionformultilayernet-
analysis
worksisdesigningeffectivemultilayernetworkroutingstrat-
simulation egy. Our results exhibit a way to reasonable redistribute the
11 traffic load. In this work, we proposed an effective strategy
whichconsidersthelocalstructuresofdifferentnodes,aswell
0 1 2 3
as the transmission speeds of differentlayers. We study our
proposedstrategy on multilayernetworksincludingtwo lay-
2.5 ers,anditcanremarkablyimprovethesystems’trafficcapac-
analysis
ity.Ourresearchmaystimulatefuturestudiesondesigningre-
simulation
alistictransportationandcommunicationmultilayernetworks,
2.0
suchas,consideringdifferentdeliveryabilitiesofnodes,lim-
)
B ited traffic resources, transmission cost of layers, and multi-
1.5 layernetworkswithmorethantwolayers.
(
o Acknowledgments
B
1.0
0.5 (b)
0 1 2 3
B
FIG.7: CMRstrategyonWork-Facebook multilayer networkwith Thiswork was supportedby the NationalNaturalScience
differentαB. (a)ThemaximumcapacityRco(αB)foragivenαB, Foundationof China (GrantNos. 11575041and 61673086),
and (b) the corresponding optimal micro-parameter βBo(αB) as a andtheFundamentalResearchFundsfortheCentralUniver-
functionofmacro-parameterαB.WesetαA=1,βA =1. sities(GrantNo. ZYGX2015J153).
[1] S.H.Strogatz,Nature410,268(2001). [12] R.Guimera,A.Arenas,A.D´ıaz-Guilera,andF.Giralt,Physical
[2] R.AlbertandA.-L.Baraba´si, Reviewsofmodernphysics74, ReviewE66,026704(2002).
47(2002). [13] R. Guimera`, A. D´ıaz-Guilera, F. Vega-Redondo, A. Cabrales,
[3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. andA.Arenas,PhysicalReviewLetters89,248701(2002).
Hwang,Physicsreports424,175(2006). [14] B.Danila,Y.Yu,J.A.Marsh,andK.E.Bassler,PhysicalRe-
[4] M. Newman, Networks: an introduction (Oxford university viewE74,046106(2006).
press,2010). [15] B.Danila,Y.Yu,J.A.Marsh,andK.E.Bassler,Chaos:AnIn-
[5] A.-L.Baraba´si,Networkscience(CambridgeUniversityPress, terdisciplinaryJournalofNonlinearScience17,026102(2007).
2016). [16] Z.Liu,M.-B.Hu,R.Jiang,W.-X.Wang,andQ.-S.Wu,Physi-
[6] A.Arenas,A.D´ıaz-Guilera,andR.Guimera,PhysicalReview calReviewE76,037101(2007).
Letters86,3196(2001). [17] G.-Q. Zhang, D. Wang, and G.-J. Li, Physical Review E 76,
[7] L.Zhao,Y.-C.Lai,K.Park,andN.Ye,PhysicalReviewE71, 017101(2007).
026125(2005). [18] Y. Xia and D. Hill, EPL (Europhysics Letters) 89, 58004
[8] Z.Wu,L.A.Braunstein,S.Havlin,andH.E.Stanley,Physical (2010).
reviewletters96,148702(2006). [19] L.Xiang,H.Mao-Bin,L.Jian-Cheng,D.Jian-Xun,andS.Qin,
[9] D.DeMartino,L.DallAsta,G.Bianconi,andM.Marsili,Phys- ChinesePhysicsB22,018904(2013).
icalReviewE79,015101(2009). [20] G.-Q. Zhang, S. Zhou, D. Wang, G. Yan, and G.-Q. Zhang,
[10] M.Barthe´lemy,PhysicsReports499,1(2011). PhysicaA:StatisticalMechanicsanditsApplications390,387
[11] J.Wu,K.T.Chi,F.C.Lau,andI.W.Ho,IEEETransactionson (2011).
CircuitsandSystemsI:RegularPapers60,3303(2013). [21] G. Yan, T. Zhou, B. Hu, Z.-Q.Fu, and B.-H.Wang, Physical
9
ReviewE73,046108(2006). S.Havlin,Nature464,1025(2010).
[22] M.Tang,Z.Liu,X.Liang,andP.Hui,PhysicalReviewE80, [37] W. Wang, M. Tang, H. Yang, Y. Do, Y.-C. Lai, and G. Lee,
026114(2009). Scientificreports4,5097(2014).
[23] B.Tadic´andM.Mitrovic´,TheEuropeanPhysicalJournalB71, [38] J.Go´mez-Gardenes, I.Reinares,A.Arenas, andL.M.Flor´ıa,
631(2009). Scientificreports2,620(2012).
[24] X.Ling,M.-B.Hu,R.Jiang,andQ.-S.Wu,PhysicalReviewE [39] J. Aguirre, R. Sevilla-Escoboza, R. Gutie´rrez, D. Papo, and
81,016113(2010). J.Buldu´,Physicalreviewletters112,248701(2014).
[25] M.TangandT.Zhou,PhysicalreviewE84,026116(2011). [40] R.G.Morrisand M.Barthelemy, Physicalreview letters109,
[26] A. Rachadi, M. Jedra, and N. Zahid, Chaos: An Interdisci- 128703(2012).
plinaryJournalofNonlinearScience23,013114(2013). [41] J.Zhou,G.Yan,andC.-H.Lai,EPL(EurophysicsLetters)102,
[27] Y.Gan,M.Tang,andH.Yang,TheEuropeanPhysicalJournal 28002(2013).
B86,1(2013). [42] O.Yagan,D.Qian,J.Zhang,andD.Cochran,IEEEJournalon
[28] P.Echenique,J.Go´mez-Garden˜es,andY.Moreno,PhysicalRe- SelectedAreasinCommunications31,1038(2013).
viewE70,056105(2004). [43] F. Tan, J. Wu, Y. Xia, and K. T. Chi, Physical Review E 89,
[29] P.Echenique,J.Go´mez-Gardenes,andY.Moreno,EPL(Euro- 062813(2014).
physicsLetters)71,325(2005). [44] A.Sole´-Ribalta,S.Go´mez,andA.Arenas,Physicalreviewlet-
[30] Z.-X. Wu, W.-X. Wang, and K.-H. Yeung, New Journal of ters116,108701(2016).
Physics10,023025(2008). [45] M. Li, M.-B. Hu, and B.-H. Wang, arXiv preprint
[31] W.-X. Wang, Z.-X. Wu, R. Jiang, G. Chen, and Y.-C. Lai, arXiv:1607.05382(2016).
Chaos: AnInterdisciplinary Journal of Nonlinear Science19, [46] C.D.Brummitt,R.M.DSouza,andE.Leicht,Proceedingsof
033106(2009). theNationalAcademyofSciences109,E680(2012).
[32] J. Gao, S. V. Buldyrev, H. E. Stanley, and S. Havlin, Nature [47] C.-G.Gu, S.-R.Zou, X.-L.Xu, Y.-Q. Qu, Y.-M.Jiang, H.-K.
physics8,40(2012). Liu,T.Zhou,etal.,PhysicalReviewE84,026101(2011).
[33] M.Kivela¨,A.Arenas,M.Barthelemy,J.P.Gleeson,Y.Moreno, [48] M.Catanzaro,M.Bogun˜a´,andR.Pastor-Satorras,PhysicalRe-
andM.A.Porter,Journalofcomplexnetworks2,203(2014). viewE71,027103(2005).
[34] K.-M.Lee,B.Min,andK.-I.Goh,TheEuropeanPhysicalJour- [49] P.Erd6sandA.Re´nyi,Publ.Math.Inst.Hungar.Acad.Sci5,
nalB88,1(2015). 17(1960).
[35] S. Boccaletti, G. Bianconi, R. Criado, C. I. Del Genio, [50] M. Magnani, B. Micenkova, and L. Rossi, arXiv preprint
J.Go´mez-Garden˜es,M.Romance,I.Sendin˜a-Nadal,Z.Wang, arXiv:1303.4986(2013).
andM.Zanin,PhysicsReports544,1(2014).
[36] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and