Playing the role of weak clique property in link prediction: A friend recommendation model

1.6 MB·

by  Chuang Ma

Playingtheroleofweak cliqueproperty inlink prediction: A friend recommendation model
Chuang Ma,1 Tao Zhou,2 and Hai-Feng Zhang1,3,4,∗
1School of Mathematical Science, Anhui University, Hefei 230601, China
2WebSciencesCenter,UniversityofElectronicScienceandTechnologyofChina, Chengdu610054, China
3KeyLaboratoryofComputerNetworkandInformationIntegration(SoutheastUniversity),MinistryofEducation,211189,P.R.China
4 CenterofInformationSupport &Assurance Technology, Anhui University, Hefei230601, China
(Dated:January21,2016)
Animportantfactinstudyingthelinkpredictionisthatthestructuralpropertiesofnetworkshavesignificant
impactsontheperformanceofalgorithms. Therefore,howtoimprovetheperformanceoflinkpredictionwith
6
the aid of structural properties of networks is an essential problem. By analyzing many real networks, we
1
findacommonstructureproperty: nodesarepreferentiallylinkedtothenodeswiththeweakcliquestructure
0
(abbreviatedasPWCStosimplifydescriptions). BasedonthisPWCSphenomenon,weproposealocalfriend
2
recommendation (FR) index to facilitate link prediction. Our experiments show that the performance of FR
n index is generally better than some famous local similarity indices, such as Common Neighbor (CN) index,
a
Adamic-Adar(AA)indexandResourceAllocation(RA)index. WethenexplainwhyPWCScangiveriseto
J
thebetterperformanceofFRindexinlinkprediction. Finally,amixedfriendrecommendationindex(labelled
0 MFR)isproposedbyutilizingthePWCSphenomenon,whichfurtherimprovestheaccuracyoflinkprediction.
2

PACSnumbers: 89.75.Hc,89.20.Hh
]
h
p
I. INTRODUCTION sparse and very huge, so Liu et al. presented a local ran-
-
c dom walk method to solve the problem of missing link pre-
o
diction,andwhichcangivecompetitivelygoodpredictionor
s Theresearchoflinkpredictionmainlyfocusesonforecast-
s. ing potentialrelationsbetweennonadjacentnodes, including evenbetterpredictionthanotherrandom-walk-basedmethods
whilehasalowercomputationalcomplexity[14]. Inviewof
c the predictionofthe unknownlinks or the furthernodes[1].
i the local communityfeaturesin many networks, Cannistraci
s Sincethewiderangeofapplicationsoflinkprediction,suchas
et al. proposed an efficient computationalframework called
y recommendingfriendsin online social networks[2], explor-
h localcommunityparadigmtocalculatethelinksimilaritybe-
ing protein-to-protein interactions [3], reconstructing airline
p tweenpairsofnodes[3]. Liuetal. designedaparameter-free
network [4], and boostinge-commercescales, which has at-
[ localblockingpredictorto detectmissing linksin givennet-
tractedmuchattentionrecently[5–7].Theprobabilisticmodel
worksvialocallinkdensitycalculations,whichperformsbet-
1 and machine learning were mainly introduced in link pre-
v diction. The notion of probabilistic link prediction and path terthanthetraditionallocalindiceswiththesametimecom-
6 plexity[15].
analysisusingMarkovchainsmethodwerefirstproposedand
4
evaluatedinRef.[8],andthenMarkovchainsmethodwasfur-
1
therstudied inadaptivewebsites [9]; InRef. [10], Popescul
5
0 etal.studiedtheapplicationofstatisticalrelationallearningto
. linkpredictioninthedomainofscientificliteraturecitations. Sincethestructuralpropertiesofnetworkshavesignificant
1
0 However, the mentionedmethodsfor link predictionwere effectson the performanceof algorithmsin link predictions,
6 mainly based on attributes of nodes. It is known that the there are some literatures have proposed some methods by
1 structure of the network is easier to be obtained than the at- making use of the structural properties of networks. Such
v: tributesofnodes,asaresult,thenetwork-structure-basedlink as the algorithms by playing the roles of hierarchical struc-
i predictionhasattractedincreasingattention. Alongthisline, ture[12], clustering[16], weakties[5]andlocalcommunity
X
Liben-Nowelletal. developedapproachesto link prediction paradigm[3]. However,currentadvancesinincludingstruc-
r basedonmeasuresforanalyzingthe“proximity”ofnodesin turalpropertiesintolinkpredictionarestillnotenough.Inthis
a
a network [11]. Since hierarchical structure commonly ex- paper,byinvestigatingthelocalstructuralpropertiesinmany
istsinthefoodwebs, biochemicalnetworks,socialnetworks real networks, we find a common phenomenon: nodes are
and so forth, a link prediction method based on the knowl- preferentiallylinked to the nodeswith weak clique structure
edge of hierarchical structure was investigated in Ref. [12], (PWCS). Then based on the observedphenomenon,a friend
and they found that such a method can provide an accurate recommendation (FR) index is proposed. In this method,
performance. Zhouetal. proposedalocalsimilarityindex— whenanodejintroducesoneofhisfriendstoanodei,hewill
ResourceAllocation(RA)indextopredictthemissinglinks, notintroducethenodeswhoarealsotheneighborsofnodei.
andtheirfindingsindicatethatRA indexhasthebestperfor- OurresultsshowthattheperformanceofFRindexissignifi-
manceoflinkprediction[13]. Giventhatmanynetworksare cantlybetterthanCN,AAandRAindicessinceFRindexcan
makegooduseofPWCSinnetworks. Atlast,tofurtherplay
theroleofPWCS,wedefineamixedfriendrecommendation
(MFR)method,whichcanbetterimprovetheaccuracyinlink
∗Electronicaddress:[email protected] prediction.
2
II. PRELIMINARIES CNindex
SCN =|Γ(i)∩Γ(j)|, (3)
Considering an undirected and unweighed network ij
G(V,E),whereV isthesetofnodesandEisthesetoflinks.
AAindex
Themultiplelinksandself-connectionsarenotallowed. For
anetworkwithsizeN,theuniversalsetofallpossiblelinks,
pdaeinrootefdnobdyeUs,,xc,oynt∈ainVin,gwoefaNss(iNg2n−1a)spcaoirres,oSfli,nkacs.coFrodrinegactho SiAjA = X lg(k1(l)), (4)
xy Γ(i)∩Γ(j)
adefinedsimilaritymeasure.Higherscoremeanshighersim-
ilaritybetweenxandy,andviceversa.SinceGisundirected, RAindex
thescoreissupposedtobesymmetry,thatisS = S . All
xy yx
thenonexistentlinksaresortedinadescendingorderaccord-
1
ing to their scores, and the links at the top are most likely SRA = , (5)
ij X k(l)
to exist [13, 14]. To test the predictionaccuracy of each in-
Γ(i)∩Γ(j)
dex, we adopt the approach used in Ref. [13]. The link set
E is randomly divided into two sets E = ET ∪ EP with respectively.
ET ∩EP = ∅. WheresetET isthetrainingsetandissup-
posedtobeknowninformation,andEP isthetestingsetfor
thepurposeoftestingandnoinformationthereinisallowedto III. DATASET
beusedforprediction. Asinpreviousliteratures,thetraining
setET alwayscontains90%oflinksinthiswork,andthere- In this paper, we choose twelve representative networks
maining10%oflinksconstitutethetestingset. Twostandard drawn from disparate fields: including: (1) C. elegans-The
metricsare used to quantifythe accuracyof predictionalgo- neural network of the nematode worm C. elegans [19]; (2)
rithms: areaunderthereceiveroperatingcharacteristiccurve NS-A coauthorship network of scientists working on net-
(AUC)andPrecision[5]. work theory and experiment [20]; (3) FWEW-A 66 compo-
Area under curve (AUC) can be interpreted as the proba- nent budget of the carbon exchanges occurring during the
bility that a randomly chosen missing link (a link in EP) is wet and dry seasons in the graminoid ecosystem of South
givenahigherscorethanarandomlychosennonexistentlink Florid [21]; (4) FWFW-A food web in Florida Bay during
(alinkinU −EP). Whenimplementing,amongnindepen- the rainy season [21]; (5) USAir-The US Air transportation
dentcomparisons,iftherearen′ timesthemissinglinkhasa system [22]; (6) Jazz-A collaboration network of jazz mu-
higherscoreandn′′ timestheyarethesamescore,AUCcan sicians [23]; (7) TAP-yeast protein-protein binding network
bereadasfollow[5]: generated by tandem affinity purification experiments [24];
(8)Power-AnelectricalpowergridofthewesternUS[19];(9)
n′+0.5n′′ Metabolic-AmetabolicnetworkofC.elegans[25];(10)Yeast-
AUC = . (1)
n A protein-proteininteraction network in budding yeast [26];
(11) Router-A symmetrized snapshot of the structure of the
If all the scores generated from independentand identical
Internetat the levelof autonomoussystems [27]; (12)PB-A
distribution,theaccuracyshouldbeabout0.5. Therefore,the
networkof the US politicalblogs[28]. Topologicalfeatures
degreetowhichtheaccuracyexceeds0.5indicateshowmuch
ofthesenetworksaresummarizedinTab.I.
thealgorithmperformsbetterthanpurechance.
Precision is the ratio of the number of missing links pre-
dicted correctly within those top-L ranked links to L, and IV. UNIVERSALITYOFPWCSPHENOMENON
L = 100 in this paper. If m links are correctly predicted,
thenPrecisioncanbecalculatedas[5]:
TocheckwhetherthePWCSphenomenoncommonlyexists
m in real networks, we divide all links into common links or
Percision= . (2) strong-tie links by judging whether the number of common
L
neighborsbetweenthetwoendpointsislargerthanathreshold
We mainly compare three local similarity indices for link β. Taking Fig. 1 as an example, when we choose β = 3,
prediction, including (1) Common Neighbors(CN) [17]; (2) thelinks{A,B}and{A,C}inFigs.1(a),(b)and(c)canbe
Adamic-Adar(AA)index[18];(3)ResourceAllocation(RA) correspondinglydegeneratedtothesketchesinFig1.(d)-(f),
index [13]. Among which, CN index is the simplest index. wherecommonlinks andstrong-tielinksare markedby fine
AAindexandRAindexhavethesimilarform,andtheyboth linksandboldlinks,respectively.
depress the contribution of the high-degree common neigh- Inthispaper,thethresholdβ ischosensuchthatthenum-
bors, however,Zhouetal. have shownthatthe performance ber of common links and the number of strong-tie links are
ofRAindexisgenerallybetterthanAAindex. approximately equal in each network. Once the value of β
LetΓ(i)betheneighborsetofnodei,|.|bethecardinality is fixed, there are seven possible configurationsfor the con-
oftheset, andk(i)bethedegreeofnodei. ThenCNindex, nectedsubgraphswith3nodes(i.e.,triples[29]),alltheseven
AAindexandRAindexaredefinedas configurationsareplottedinFig.2, wheretheboldlinksand
3
TABLE I: The basic topological features of twelve example
networks.N andM arethetotalnumbersofnodesandlinks,respec-
tively. C andrareclusteringcoefficientandassortativecoefficient,
respectively. H isthedegreeheterogeneity, definedasH = hk2i,
hki2
wherehkidenotestheaveragedegree[29].
Network N M C r H
C.elegans 297 2148 0.308 -0.163 1.801
NS 1589 2742 0.791 0.462 2.011
FWEW 69 880 0.552 -0.298 1.275
FWFW 128 2075 0.335 -0.112 1.237
USAir 332 2126 0.749 -0.208 3.464
Jazz 198 2742 0.633 0.02 1.395
Tap 1373 6833 0.557 0.579 1.644
Power 4941 6594 0.107 0.003 1.45
Metabolic 453 2025 0.655 -0.226 4.485
Yeast 2375 11693 0.388 0.454 3,476 FIG. 1: (Color online) Degenerating the upper sketches into the
Router 5022 6258 0.033 -0.138 5.503 lower cases by judging whether two links {B,A} and {A,C} are
PB 1222 16724 0.36 -0.221 2.971 strong-tielinkorcommonlink. Hereweassumethatifthenumber
ofcommonneighborsbetweenAandB(orAandC)islargerthan
β=3,thenthelinkisstrong-tielink;otherwise,thelinkiscommon
linkintheoppositecase. Finelinesandtheboldlinesin(d)-(f)are
fine lines denote strong-tie links and common links, respec-
thecommonlinksandthestrong-tielinks,respectively.
tively. Let N ,i = 1,··· ,7 be the number of CS ,i =
i i
1,··· ,7(eachCSrepresentsaconfigurationinFig.2)innet-
works. If both of {A,B} and {A,C} are strong-tie links,
thentheprobabilityofnodeB connectingnodeC isdefined
as[30]:
3N +N
P = 4 6 . (6)
1
N +3N +N
1 4 6
Ifonlyoneoflinks{A,B}or{A,C}isstrong-tielink,then
theprobabilityofnodeB connectingnodeC isdefinedas:
2N +2N
P = 6 7 . (7)
2
2N +2N +N
6 7 3 FIG. 2: (Color online) Seven possible configurations of connected
subgraphs with three nodes. Fine lines and the bold lines are the
If neither of them is strong-tie link, then the probability of
commonlinksandthestrong-tielinks,respectively.
nodeB connectingnodeC is:
3N +N
P = 5 7 . (8)
3
3N +N +N
5 7 2 RouterandPBnetworks,whereP > P > P . Asaresult,
1 2 3
First,wedefineasubgraphwithnnodesbeaweakclique we can state that PWCS phenomenon is more significant in
thenumberoflinksamongthennodesisratherdense,which thesesixnetworks.
isanextendeddefinitionofn-cliquewhereallpairsofnodes
are connected. Next, by calculating the probability of node
B connectingC, we canjudgewhetherthe phenomenonthat V. FRIENDRECOMMENDATIONMODEL
nodesarepreferentiallylinkedtothenodeswithweakclique
structure(i.e.,PWCSphenomenon)commonlyexistsinanet- Given that PWCS phenomenon commonly exists in real
work. We say thatthe PWCS phenomenonexistsin the net- networks,whethercanwedesignaneffectivelinkprediction
work if P > P and P > P . Moreover, we say that the methodbasedonthisphenomenon. Consideringthecasesin
1 2 1 3
PWCS phenomenonis significant if P > P > P , other- Fig. 3, where node 3 ask its neighbor node 2 to introduce a
1 2 3
wise,thePWCSphenomenonisweakasP >P ≥P . friendtoit. Sincethenumberofcommonneighborsbetween
1 3 2
Table IIreportsthevaluesofP , P andP inthetwelve node2andnode3inFig.3(c)islargerthanthatofinFig.3(b)
1 2 3
real networks (labelled as RN) and the values on the corre- andisfurtherlargerthanthatofin Fig. 3(a), inotherwords,
spondingnullnetworks(labelledNN)arealsocomparatively the strength of link {2,3} in Fig. 3(c) is the strongest. Ac-
shown. One can find that P > P and P > P in eleven cording to PWCS phenomenon, the probability (labelled by
1 2 1 3
networks except for Metabolic network (P < P , marked f )ofnode1(callnominee,greencolor)beingintroduced
1 3 123
by red color). However,in the correspondingnullnetworks, tonode3(callacceptor,redcolor)bynode2(callintroducer,
P ≈P ≈P . Also,forC.celegans,FWEW,FWFW,Power, bluecolor)inFig.3(c)shouldbelargerthanthatofinFig.3
1 2 3
4
TABLEII: Thevalues of P1, P2 andP3 in12real networks (RN)
andthecorresponding nullnetworks(NN)arereported. Resultsin
NNaremarkedinItalic.ResultsinnetworkswithsignificantPWCS,
i.e.,P1 >P2 >P3areshowninbluecolor,andresultsinMetabolic
aremarkedbyredcolorduetoitsspecificity.
Network Network P1 P2 P3 FIG.3:(Coloronline)TheroleofPWCSontheprobabilityoff123.
RN 0.2351 0.1654 0.1519 Node2(bluecolor,callintroducer)wanttointroducenode1(green
C.elegans
NN 0.1011 0.0970 0.0953 color,callnominee)tonode3(redcolor,callacceptor).Thenumber
RN 0.9292 0.2392 0.5970 commonneighborbetweennode2andnode3in(a),(b)and(c)is0,
NS NN 0 0.0037 0.0045 1and2,respectively. AccordingtoEq.(9),onehas(a)f123 =1/3;
RN 0.5998 0.4832 0.2504 (b) f123 = 1/2; (c) f123 = 1. Namely, the probability of node
FWEW NN 0.7421 0.7627 0.7647 1beingintroduced tonode 3in(c)islargerthan(b)andisfurther
largerthanin(a).
RN 0.4191 0.3532 0.1230
FWFW
NN 0.5220 0.5259 0.5207
RN 0.7008 0.1519 0.2355
USAir
NN 0.0752 0.0765 0.0797
RN 0.6902 0.3968 0.4503 neighbors.
Jazz
NN 0.2734 0.2790 0.2804 Withtheabovepreparations,thesimilarityindexSFRfora
ij
RN 0.7862 0.2969 0.3673 pairofnodesiandj isdefinedas
Tap
NN 0.0141 0.0136 0.0146
RN 0.2781 0.0854 0.0686 f +f
Power SFR = ij ji, (11)
NN 0 0 0 ij 2
RN 0.1630 0.0760 0.1643
Metabolic
NN 0.0409 0.0374 0.0389 whichguaranteesSFR =SFR.
ij ji
RN 0.5945 0.1498 0.1530 The sketches in Fig. 4 is given to show how to calculate
Yeast
NN 0.0089 0.0090 0.0086 the similarity between node 1 and node 2 based on the FR
RN 0.1992 0.0254 0.0022
Router index. Also,thered,blueandgreennodesdenotetheaccep-
NN 0 0 0
tors, introducersand nominees, respectively. Node 2 can be
RN 0.3998 0.1247 0.0855
PB introducedtonode1bynode3(seeFig.4(a))ornode4(see
NN 0.0457 0.0454 0.0441
Fig.4(b)). Whennode3isanintroducer(seeFig.4(a)),who
will introducenodes2, 5 and 7 (greencolor) to node 1 with
(b),andthenfurtherlargerthanthatofinFig.3(a). Toreflect equalprobability,butexcludesnode4,i.e.,f231 =1/3. Sim-
thementionedfact,wedefinef betheprobabilityofibeing ilarly, whennode4 is anintroducer(see Fig. 1(b)),whojust
ilj
introducedtojbytheircommonneighborl,whichisgivenas: introducesnodes 2 and 6 (green color) to node 1 with equal
probability, but excludes node 3, i.e., f = 1/2. There-
241
1 fore, the probabilityf = 1/3+1/2 = 5/6. Likely, from
f = . (9) 21
ilj
k(l)−1−|Γ(l)∩Γ(j)| Figs.5(c)and(d),thevalueoff =1/2+1/2=1. There-
12
fore,theFRsimilarityindexisSFR =SFR =11/12.
Based on the definition in Eq. (9), the values of f in 12 21
123 CombingEqs.(9),(10)and(11),theadvantagesofFRin-
Fig.3(a),(b)and(c)are1/3,1/2and1, respectively. Thatis
dexcanbesummarizedas:1)similartomanylocalsimilarity
tosay,theprobabilityf canreflectthePWCSphenomenon
ilj indices,thesimilaritybetweenapairofnodesincreaseswith
inrealnetworks.
thenumberofcommonneighbors;2)likeAAindexandRA
More importantly, Eq. (9) addresses two important facts:
index,FRindexdepressesthecontributionofthehigh-degree
first, since node l will not introducenode j to j, as a result,
commonneighbors;3)mostimportantly,FRindexcanmake
1 is subtracted in denominator of Eq. (9); second, in social
useofthePWCSphenomenoninmanyrealnetworks;4)FR
communication,whenafriendintroduceoneofhisfriendsto
indexhashigherresolutionthanotherlocalsimilarityindices.
me,heshouldintroducehisfriendsbutexcludingthecommon
Forinstance,thesimilaritiesSCN,SAAandSRAarethesame
friends. Therefore,thecommonneighborssetbetweenj and 13 13 13
inFigs.3(a),(b)and(c). Yet,thevalueofSFR inFig.3(c)is
l (i.e., Γ(l)∩Γ(j)) should be subtracted in denominator of 13
largerthanFig.3(b),andisfurtherlargerthanFig.3(a).
Eq. (9). For instance, in Fig. 3(c), node2 will notintroduce
node3tonode3,andnodes4and5shouldnotbeintroduced
tonode3.
VI. RESULTS
Let f be the weight of node i being introduced to node
ij
j (weuseweightratherthanprobabilitysincef maylarger
ij
than1),whichiswrittenas: In this section, the comparisonof FR index with CN, AA
andRAindicesintwelvenetworksissummarizedinTab. III.
1
As shown in Tab. III, FR index in general outperforms the
f = f = . (10)
ij X ilj X k(l)−1−SCN other three indices in link prediction, regardless of AUC or
l∈Γ(i)∩Γ(j) l∈Γ(i)∩Γ(j) jl
Precision. Thehighestaccuracyineachlineisemphasizedin
Here the value of f increases with the numberof common bold.
ij
5
Moreover,thecorrelationofrankingvaluesbetweenFRin-
dex and RA index is given in Fig. 5, where the percentage
values in x or y axis is the top percentage of ranking val-
uesbasedonPrecision. Asaresult,asmallpercentagevalue
meansahigherrankingvalue.Fig.5describesthatahighRA
rankingvalue of links givesrise to a highFR rankingvalue.
However,ahighFRrankingvalueoflinksmayinducealow
RA ranking value of links. Take Tap and Yeast networks as
examples, based on FR index, some links have higher rank-
ingvalues,howevertheircorrespondingrankingvaluesbased
onRAindexmayverysmall(seetheregionsmarkedbypink
dashboundaryinFigs.5(g)and(j)).
By analyzing a typical case in the Yeast network (see
Fig. 6), where two nodes A and B are the neighbors of in-
troducerC(infact,therehasalinkconnectingAandBinthe
FIG. 4: (Color online) Calculation of the similarity S1F2R between Yeast network). Since links {A,C} and {B,C} are strong-
node1andnode2.Nodes1and2canbeintroducedbytheircommon tie links. WhenusingFR index,thesimilaritySFR is rather
AB
neighbors3and4.(a)node3introduceshisfriendstonode1.Only large, whichcanpredictthe existenceof link{A,B}. How-
neighbornodes2,5,7canbeintroducedtonode1butexcludesnode ever,forRAindex,sincethelargedegreevalueofintroducer
4, sincenode 4has been afriend of node 1. Thus, theprobability C, the similarity SRA is very small, such an existing link
of node 3 introducing node 2 to node 1 is: f231 = 1/3; (b) node {A,B}cannotbeacAcBuratelypredictedbyRAindex.
2 is introduced to node 1 by node 4, here only nodes 2 and 6 can
be introduced to node 1. As a result, the probability f241 = 1/2;
(c)node1isintroducedtonode2bynode3,hereonlynodes1and
5canbeintroduced tonode 1. Asaresult, theprobabilityf132 = VII. ROLEOFPWCS
1/2; (d)node1isintroducedtonode2bynode4,hereonlynodes
1 and 6 can be introduced to node 1. As a result, the probability WehavevalidatedthattheFRindexbasedonPWCSphe-
f141 =1/2.Wehavef21 =1/3+1/2bycombing(a)and(b),and nomenoncanimprovetheperformanceoflinkprediction,and
f12 =1/2+1/2bycombing(c)and(d).SotheFRsimilarityindex the reasonswere also analyzed. Here we wantto knowhow
isS1F2R =S2F1R =11/12. the strength of PWCS affects the performance of RA index
and FR index. For this purpose, we propose a generalized
friendrecommendation(GFR)index,whichisgivenas:
TABLE III: Comparison of SFR with SCN, SAA and SRA in 12
networks, includingAUCandPrecision. Thehighestvalueineach 1 1 1
rowismarkedinbold. SiGjFR = 2 X (cid:0)k(l)−αSCN + k(l)−αSCN(cid:1)(,12)
Network Metric CN AA RA FR l∈Γ(i)∩Γ(j) jl il
AUC 0.8501 0.8663 0.8701 0.8756
C.elegans
Precision 0.1306 0.1374 0.1315 0.1504 where parameter 0 ≤ α ≤ 1 is used to uncover the role of
AUC 0.9913 0.9916 0.9917 0.9916 PWCS. As α = 0, Eq. (12) returns to RA index, that is,
NS
Precision 0.8707 0.9731 0.9712 0.9832 SGFR = SRA. When α = 1, the difference between FR
ij ij
AUC 0.6868 0.6939 0.7017 0.7595 methodand GFR method is the absence of 1 in the denomi-
FWEW
Precision 0.1415 0.1551 0.1664 0.2763 natorsofEq.(12),therefore,wecansimplyviewGFRindex
AUC 0.6074 0.6097 0.6142 0.6623
FWFW is the same as FR index when α = 1. As a result, with the
Precision 0.0837 0.0853 0.082 0.1798 increasing of α from zero to one, SSFR index can compre-
AUC 0.9558 0.9676 0.9736 0.9752
USAir hensivelyinvestigatetheroleofPWCS ontheRA indexand
Precision 0.606 0.6218 0.6337 0.6586
FRindex.
AUC 0.9563 0.963 0.9717 0.9714
Jazz The effect of α on the Precision in all twelve networks is
Precision 0.8247 0.8401 0.8192 0.8406
plotted in Fig. 7. As illustrated in Fig. 7, several interesting
AUC 0.9538 0.9545 0.9548 0.955
Tap
Precision 0.7594 0.78 0.7818 0.8659 phenomenaandmeaningfulconclusionscanbesummarized:
AUC 0.6249 0.6251 0.6245 0.6248 First,exceptforMetabolicnetwork,thePrecisionforthecase
Power
Precision 0.1215 0.0952 0.0801 0.1275 ofα > 0isfarlargerthanthecaseofα = 0(i.e.,RAindex)
AUC 0.9248 0.9565 0.9612 0.9623 inallother11networks.SinceP >P andP >P inthese
Metabolic 1 2 1 3
Precision 0.2026 0.2579 0.3219 0.3302 11networks,whichindicatesthatPWCSphenomenoninnet-
AUC 0.9158 0.9161 0.9167 0.9172 workscanensurethehigheraccuracyofFRindex(i.e.,α=1)
Yeast
Precision 0.6821 0.6958 0.4988 0.8041 inlinkprediction;Second,Metabolicnetworkhasnon-PWCS
AUC 0.6519 0.6523 0.652 0.6519
Router phenomenonsinceP1 > P2 andP2 < P3,andFig.7(i)sug-
Precision 0.1144 0.1104 0.0881 0.0592
gests that Precision decreases with the value of α. In other
AUC 0.9239 0.9275 0.9286 0.9309
PB words, FR index is invalid in network with non-PWCS phe-
Precision 0.4205 0.3782 0.2509 0.3454
nomenon,whichagainemphasizesthe importanceofPWCS
6
FIG.5:(Coloronline)ThecorrelationofrankingvaluesbetweenFRindexandRAindexbasedonPrecision.Thepercentagevaluesinx-axis
and y-axis are the top percentage ranking values of FR index and RA index, respectively. The regions marked by pink dash boundary in
subfigures(g)and(j)correspondtothecasesinwhichsomelinkshavehigherFRrankingvaluesbuthavelowRArankingvalues.
in link prediction; At last, by systematically comparing the issignificant. Unfortunately,themaximalvalueαinEq.(12)
subfiguresinFig.7,onecanseethat,whenthenetworkswith is one. the denominator may be negative if α > 1. So we
weakPWCSP >P ≥P (i.e.,theinsetsarelightredback- design a new index to further explore the role of significant
1 3 2
ground,seeFig.7(b),(e),(f),(g)and(j)),Precisionincreases PWCS.
withαatfirstandthendecreaseswhenαisfurtherincreased SinceEq.(12)canberewrittenas
(except Fig. 7(e)). However, when P > P > P (i.e.,
1 2 3
networks with significant PWCS, the insets are white back- 1 2k(l)−SCN −SCN
SGFR = il jl (13)
ground,seeFigs.7(a),(c),(d),(h),(k)and(l)),Precisional- ij 2 X (k(l)−SCN)(k(l)−αSCN)
waysincreaseswiththevalueofαevenwhenα=1.0. l∈Γ(i)∩Γ(j) jl il
In view of this observation, we can conjecture the role of whenα=1. TofurtherplaytheroleofPWCS,anothersimi-
PWCScanbefurtherexploredwhenthePWCSphenomenon larityindex,calledstrongfriendrecommendation(labelledas
7
ofP ,P andP indifferentnetworks.Tothisend,wedesign
1 2 3
amixedfriendrecommendation(labelledMFR)index:
SSFR, P >P >P ;
ij 1 2 3

SiMjFR =  SiFjR, P1 >P2,P1 >P3,P2 <P3;
SRA, otherwise. (15)
 ij
Table 5 lists the results of MFR index and FR index in 7
networks (since MFR index is the same to FR index when
P > P , P > P and P < P , in this case, it is unnec-
1 2 1 3 2 3
essary to compare the two indices). The results in Tab. V
indicate that, compared with FR index, MFR index can fur-
therimprovetheaccuracyoflinkprediction.
VIII. CONCLUSIONS
In summary, by analyzing the structural propertiesin real
FIG.6:(Coloronline)AtypicalcaseintheYeastnetworkisconsid-
networks, we have found that there exists a common phe-
eredtoemphasize thedifference between FRindex andRA index,
nomenon: nodes are preferentially linked to the nodes with
wherenodesA,BandCarethenode1175,421and205intheYeast
weak clique structure. Then we have proposeda friend rec-
network.Twolinks{A,C}and{B,C}shareacommonendpointC,
andbothofthemarestrong-tielinks. Therefore,thesimilaritySAFBR ommendationmodeltobetterpredictthemissinglinksbased
isratherlarge. However,whenusingRAindex,therankingnumber onthesignificantphenomenon.Throughthedetailedanalysis
ofSARBAisverylowowingtothelargedegreevalueofnodeC,caus- and experimental results, we have shown that FR index has
ingthefailureofRAindexinpredictingsuchanexistinglink. Red severaltypicalcharacteristics: First,FRindexisbasedonthe
nodes,greennodesandbluenodesaretheneighborsofA,BandC informationofcommonneighbors,whichisalocalsimilarity
(includingthemselves), respectively. Purplenodesarethecommon index. Thus, the algorithm is simple and has low complex-
neighborsofAandC;whitenodesarethecommonneighborsofA, ity; Second, the common neighbors with small degrees has
BandC.
greatercontributionsthanthecommonneighborswithlarger
degrees;Third,FRindexcantakefulladvantageofthePWCS
phenomenon,andsoforth.
SFR)index,isgiveninfollowing Furthermore,wealsoproposedanSFRindextofurtherim-
provetheaccuracyoflinkpredictionwhennetworkshavesig-
1 2k(l) nificantPWCSphenomenon. Atlast,byjudgingwhetherthe
SSFG = . (14)
ij 2 X (k(l)−SCN)(k(l)−αSCN) networkshavesignificantPWCS,weakPWCSornon-PWCS
l∈Γ(i)∩Γ(j) jl il phenomenon, we have also proposed a mixed friend recom-
mendationindexwhichcanincreasetheaccuracyoflinkpre-
Combing Eq. (13) with Eq. (14), we can find that two sub-
dictionindifferentnetworks.Inthiswork,wemainlyapplied
trahendsSCN and SCN in the numeratorofEq. (13) are re-
il jl FRindextounweighedandundirectednetworks,andhowto
moved.SoEq.(14)canbetterplaytheroleofPWCS.
generalizeourFRindextoweighted[31,32]ordirectednet-
WeconjecturethattheperformanceofSFRindexisbetter
works[33]isourfurtherpurpose.
thanGFRindexwhenP >P >P (i.e.,significantPWCS),
1 2 3
andworsethanthatofGFRindexwhenP <P andP <P
1 2 1 3
(i.e.,non-PWCS).However,itisdifficulttodistinguishwhich
one has better performancewhen P > P ≥ P (i.e., weak Acknowledgments
1 3 2
PWCS). As presented in Tab. IV, Precision in 12 networks
validatesourconjecture. This work is supported by the National Natural Science
Synthesizingtheaboveresults,wecanfindthattheranking Foundation of China (Grant Nos. 61473001), and par-
of P , P and P hasdeterminanteffecton the performance tiallysupportedbyopenfundofKeyLaboratoryofComputer
1 2 3
of the proposedindex. Inspiredby thisclue, we may design NetworkandInformationIntegration(SoutheastUniversity),
auniversalindicatortodolinkpredictionbasedonthevalues MinistryofEducation(No.K93-9-2015-03B).
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TABLEIV:ThecomparisonofPrecisionbetweenSFRindexandGFRindex(α=1)in12networks.Theresultssuggestthattheaccuracyof
linkpredictioncanbefurtherimprovedbySFRindexwhenthenetworkshavesignificantPWCS.Thehighestvalueineachcaseismarkedas
bold.
P1 >P2 >P3
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SFR(α=1) 0.1577 0.2912 0.2057 0.1658 0.112 0.4353
P1 >P3 ≥P2 P1 <P3,P2 <P3
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GFR(α=1) 0.9804 0.6807 0.8532 0.8568 0.8178 0.3064
SFR(α=1) 0.9744 0.6866 0.8739 0.8485 0.8587 0.2912
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P2 andP3. ThebackgroundofinsetiswhitecolorwhenP1 > P2 > P3;thebackground ofinsetislightredcolorwhenP1 > P3 ≥ P2.
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TABLEV:ComparisonofPrecisionbetweenFRindexandWFRindexin7networks.Thehighestvalueineachcaseisgiveninbold.
Metric Index C.elegans FWEW FWFW Power Router PB Metabolic
FR 0.8756 0.7595 0.6623 0.6248 0.6519 0.9309 0.9623
AUC
WFR 0.8771 0.7771 0.6878 0.6247 0.6516 0.9314 0.9612
FR 0.1504 0.2763 0.1798 0.1275 0.0592 0.3454 0.3302
Precision
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