Estimation of Vertex Degrees in a Sampled Network

0.36 MB·

by  Apratim Ganguly

Estimation of Vertex Degrees in a Sampled Network
ApratimGanguly∗ EricKolaczyk
DepartmentofMathematicsandStatistics DepartmentofMathematicsandStatistics
BostonUniversity BostonUniversity
Boston,MA02212 Boston,MA02212
[email protected] [email protected]
7
1
0
2 Abstract

n
a Theneedtoproduceaccurateestimatesofvertexdegreeinalargenetwork,based
J
on observation of a subnetwork, arises in a number of practical settings. We

5 studyaformalizedversionofthisproblem,whereinthegoalis,givenarandomly
2 sampledsubnetworkfromalargeparentnetwork, toestimatetheactualdegree

of the sampled nodes. Depending on the sampling scheme, trivial method of
]
P momentsestimators(MMEs)canbeused. However,theMMEisnotexpected,
A ingeneral, touseallrelevantnetworkinformation. Inthisstudy, weproposea
handfulofnovelestimatorsderivedfromarisk-theoreticperspective,whichmake
.
t more sophisticated use ofthe information in the sampled network. Theoretical
a
assessment ofthe new estimators characterizesunder what conditionstheycan
t
s offerimprovementovertheMME,whilenumericalcomparisonsshowthatwhen
[
suchimprovementobtains,itcanbesubstantial.Illustrationisprovidedonahuman

1 traffickingnetwork.
v
3
0
1 Introduction
2
7
0 Frequentlyitisthecaseinthestudyofreal-worldcomplexnetworksthatweobserveessentially
. a sample from a larger network. There are many reasons why sampling in networks is often
1
0 unavoidable–and,insomecases,evendesirable. Sampling,forexample,haslongbeenanecessary
7 partofstudyingInternettopology[3]. Similarly,itsrolehasbeenlong-recognizedinthecontextof
1 biologicalnetworks,e.g.,protein-proteininteraction[7,11,13],generegulation[17]andmetabolic
: networks [7]. Finally, in recent years, there has been intense interest in the use of sampling for
v
monitoringonlinesocialmedianetworks. See[19],forexample,forarepresentativelistofarticles
i
X inthislatterdomain. Givenasamplefromanetwork,afundamentalstatisticalquestionishowthe
r samplednetworkstatisticsbeusedtomakeinferencesabouttheparametersoftheunderlyingglobal
a network. Parameters of interest in the literature include (but are by no means limited to) degree
distribution, density, diameter, clusteringcoefficient, andnumberofconnectedcomponents. For
seminalworkinthisdirection,see[4,5].
Inthispaper,weproposepotentialsolutionstoanestimationproblemthatappearstohavereceived
significantly less attention in the literature to date – the estimation of the degrees of individual
samplednodes. Degreeisoneofthemostfundamentalofnetworkmetrics,andisabasicnotionof
node-centrality. Derivingagoodestimateofthenodedegree,inturn,canbehelpfulinestimating
otherglobalparameters,asmanysuchparameterscanbeviewedasfunctionsthatincludedegreeas
anargument. Whileanumberofmethodsareavailabletoestimatethefulldegreedistributionunder
networksampling(e.g.,[16,19]),littleworkappearstohavebeendoneonestimatingtheindividual
nodedegrees. Ourworkaddressesthisgap. Formally,ourinterestliesinestimationofthedegreeofa
vertex,providedthatvertexisselectedinasampleoftheunderlyinggraph.
∗CurrentlyatNateraInc.
Therearemanysamplingdesignsforgraphs. See[9,Ch5]forareviewoftheclassicalliterature,
and[1]forarecentsurvey. Canonicalexamplesincludeego-centricsampling[6],snowballsampling,
induced/incidentsubgraphsampling,link-tracingandrandomwalkbasedmethods[10,14]. Under
certainsamplingdesignswhereoneobservesthetruedegreeofthesamplednode(e.g. ego-centric
and one-wave snowball sampling), degree estimation is unnecessary. In this paper, we focus on
induced subgraph sampling, which is structurally representative of a number of other sampling
strategies[19]. Formally, in induced subgraph sampling, a set of nodes is selected according to
independentBernoulli(p)trialsateachnode. Then,thesubgraphinducedbytheselectednodes,i.e.,
thegraphgeneratedbyselectingedgesbetweenselectednodes,isobserved. Thismethodofsampling
sharesstochasticpropertieswithincidentsubgraphsampling(whereintheroleofnodesandedgesis
reversed)andwithcertaintypesofrandomwalksampling[14].
Theproblemofestimatingdegreesofsamplednodeshasbeengivenaformalstatisticaltreatment
in[18],forthespecificcaseoftraceroutesamplingasaspecialcaseoftheso-calledspeciesproblem
[2]. Tothebestofourknowledge,asimilarlyformaltreatmenthasnotbeenappliedmoregenerally
for other, more canonical sampling strategies. However, a similar problem would be estimating
personalnetworksizeforagroupofpeopleinasurvey. Somepriorworksinthisdirection[8,12]
considerestimatorsobtainedbyscalinguptheobserveddegreeinthesamplednetwork,inthespirit
ofwhatwetermamethodofmomentsestimatorbelow. Butnospecificgraphsamplingdesigns
arediscussedinthesestudies. Wefocusonformulatingtheproblemusingtheinducedsubgraph
samplingdesignandexploitnetworkinformationbeyondsampleddegreetoproposeestimatorsthat
arebetterthannaivescale-upestimators. Keytoourformulationisarisktheoreticframeworkused
toderiveourestimatorsofthenodedegrees,throughminimizingfrequentistorBayesrisks. This
contributionisaccompaniedbyacomparativeanalysisofourproposedestimatorsandnaivescale-up
estimators,boththeoreticalandempirical,inseveralnetworkregimes.
Wenotethatwhensamplingiscoupledwithfalsepositiveandfalsenegativeedges,e.g.,incertain
biologicalnetworks,ourmethodsarenotimmediatelyapplicable. Samplingdesignsthatresultinthe
selectionofafractionofedgesfromtheunderlyingglobalnetwork(inducedandincidentsubgraph
sampling,randomwalksetc.) areourprimaryobjectsofstudy. Weuseinducedsubgraphsampling
as a rudimentary but representative model for this class and aim to simultaneously estimate the
truedegreesofalltheobservednodeswithaprecisionbetterthanthatobtainedbytrivialscale-up
estimatorswithnonetworkinformationused.
2 DegreeEstimationMethods
Let us denote by G0 = V0,E0 a true underlying network, where V0 = 1, ,N . This
{ ··· }
networkisassumedstaticand,withoutlossofgenerality,undirected. Thetruedegreevectorisd0 =
(cid:0) (cid:1)
(d0, ,d0 )T. ThesamplednetworkisdenotedbyG = (V ,E )where,againwithoutlossof
1 ··· N ∗ ∗ ∗
generality,weassumethatV = 1, ,n . Writethesampleddegreevectorasd =(d , ,d ).
∗ { ··· } ∗ ∗1 ··· ∗n
Throughoutthepaper,weassumethatwehaveaninducedsubgraphsample,with(known)sampling
proportionp.
Itiseasytoseefromthesamplingschemethatd B(d0,p). Therefore,themethodofmoments
∗i ∼ i
T
estimator(MME)ford0 isdˆMME = d∗i. Thus,dˆ = dˆMME, ,dˆMME isanaturalscale-
i i p MME 1 ··· n
upestimatorofthedegreesequenceofthesamplednode(cid:16)s. Inthissection,w(cid:17)eproposeaclassof
estimators that minimize the unweighted (cid:96) -risk of the sampled degree vector and discuss their
2
theoreticalproperties. Weaimtodemonstrate,underseveralconditions,thattheriskminimizersare
superiortotheregularscale-upestimators,theformertakingintoaccounttheinherentrelationships
insidethenetwork.
Wenotethatalthoughamaximumlikelihoodapproachtoestimationisperhapsintuitivelyappealing,
aclosedformderivationoftheMLEinthissettingisprobitive. Anotheroptionistolookatmarginal
likelihoods. ButtheMLEbasedonunivariatemarginallikelihoodsareessentiallyequivalenttothe
MMEforthissamplingscheme. Wewillfrequentlyusethethefirstandsecondmomentsofthe
sampleddegreevectorinourestimationmethods. Thefollowinglemmawillbeuseful.
2
Lemma2.1. Underinducedsubgraphsampling,themeanandcovariancematrixoftheobserved
degreevectorare
E(d∗)=pd0 (1)
Var(d∗)=p(1 p)D0 (2)
−
wherethediagonalsofD0 ared0, ,d0 andthe(i,j)-thoff-diagonalisdenotedbyd0 , which
1 ··· n ij
denotesthenumberofcommonneighborsofnodeiandnodej inthenetworkG0.
2.1 FrequentistRiskMinimization
Adoptingthestandarddefinitionof(unweighted)frequentist(cid:96) riskofanestimatorθˆofaparameter
2
θ0,i.e., (θˆ,θ0)=E θˆ θ0 2,thefrequentistrisksarecalculatedforageneralclassofestimators.
R || − ||
We also define (θˆ,θ0) := E θˆ θ0 21(G∗ ) , a restricted risk function assuming the
RA || − || ∈A
sampledgraphG∗ isrestrictedto(cid:16)someclass . Ourpr(cid:17)oposedcandidatesaretheelementsinthe
A
classoflinearfunctionsoftheobserveddegreevectorthatminimizetheriskortherestrictedrisk
w.r.t. someclass. Itisexpectedthattheoptimalestimatorwillbeafunctionoftheparameterand
henceanother(naive)estimatorwillneedtobepluggedin. Ourfinalestimatewillthenbeaplug-in
riskminimizer.
2.1.1 UnivariateRiskMinimization
Hereweestimatethenodedegreesindividually,assumingthattheestimatefortheithnodeisofthe
formdˆ =c d ,wherec isascalarandd istheobserveddegreeinthesample.Sinced B(d0,p),
i i ∗i i ∗i ∗i ∼ i
whered0isthetruedegreeoftheithnode,
i
(dˆ,d0)=Bias2(c d )+Var(c d )=(c pd0 d0)2+p(1 p)c2d0 .
R i i i ∗i i ∗i i i − i − i i
Differentiatingw.r.t. ciandequatingto0,wegettheoptimalc∗i = pd0+d0i1 p. PluggingintheMME
i −
2
ofd0,wegettheplug-inunivariateriskminimizerdˆ = d∗i .
i,u,P p(d∗i+1−p)
Taylorexpandingtheaboveformula(duringTaylorexpansionsoffunctionsofd ,wewillassume
∗i
thatd isconcentratedarounditsmean,sothattheTaylorexpandedapproximationisclose)and
∗i
takingexpectation,weseethat
2 1
E(cid:16)dˆi,u,P(cid:17)=E(cid:20)p(d∗id+∗i1−p)(cid:21)= p1E(cid:20)d∗i (cid:16)1+ 1d−∗ip(cid:17)− (cid:21)≈ p1E(cid:104)d∗i (cid:16)1− 1d−∗ip(cid:17)(cid:105)=d0i − 1−pp .
The above calculation suggests that an adjustment needs to be made to dˆ by bias-correction,
i,u,P
sothatitsriskbecomescomparabletothatofdˆMME. Infact,wewillshowinProposition3.1that
i
our bias-corrected plug-in estimator has a lower risk than MME when the true degree is bigger
thanalowerbound,whichcanbeexpressedasaclosedformfunctionofthesamplingproportion.
Ultimately,ourproposedunivariateriskminimizerisgivenby
d 2 1 p
dˆ = ∗i + − (3)
i,u
p(d +1 p) p
∗i −
2.1.2 MultivariateRiskMinimization
Weextendtheideapresentedintheprevioussectiontothemultivariatecase,inordertominimizethe
overall(cid:96) sumoverallsamplednodes. Therationaleforthisextensionistoexploitthecovariance
2
structurewederivedinLemma2.1inestimatingthedegreevector. Accordingly,weconsiderall
estimatesoftheformdˆ =Ad ,whereAisann nmatrix. UsingLemma2.1,wegetthe(cid:96) risk
∗ 2
×
R(dˆ,d0)=(pA I)d0d0T(pA I)T +p(1 p)AD0AT
− − −
=A p2d0d0TAT +p(1 p)D0 AT p d0d0TAT +Ad0d0T +constant.
− −
Themultivariat(cid:16)eriskminimizerisdefinedas(cid:17) (cid:16) (cid:17)
2
A∗ =argminA ni=1E dˆi−d0i =argminAtr R(dˆ,d0) .
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:80)
3
Differentiatingtheobjectivefunctionw.r.t. Aandequatingitto0,weget
1
A =pd0d0T p2d0d0T +p(1 p)D0 − .
∗
−
PluggingintheMMEofd0andD0,we(cid:16)gettheplug-inmultivariate(cid:17)riskminimizer
1 1
dˆm = d∗d∗T d∗d∗T +D∗ − d∗ , (4)
p
(cid:16) (cid:17)
whered denotesthenumberofcommonneighborsofnodeiandnodej inthesample,andD is
∗ij ∗
givenbyamatrixwhosediagonalsared andwhoseoff-diagonalsared ,i,j 1, ,n , i=j.
∗i ∗ij ∈{ ··· } (cid:54)
2.2 BayesRiskMinimization
In this section, we propose a Bayesian solution to our estimation problem, by putting a prior on
thedegreedistribution. Theprincipalmotivationbehindthisapproachisthedesiretoincorporate
additionalinformationonglobalnetworkstructure,wherethenaturalcandidateinthiscontextis
thedegreedistribution. Incasesuchasubjectivepriorisnotavailable, anestimateofthedegree
distributionmaybeused. Weproposeandanalyzeestimatorsbasedonbothknown(subjective)and
estimateddegreedistributionsbelow.
First,letusassumethatweknowthedegreedistributionπ()oftheunderlyingnetwork. Underthe
·
assumptionthatthetruedegreeofnodeifollowsπ(),andunderinducedsubgraphsamplingofG,
·
theconditionaldistributionofd d isB(d ,p). ThenitcanbeeasilyshownthattheBayesestimator
∗i| i i
undersquareerrorlossis
dˆB = di≥d∗i di dd∗ii (1−p)diπ(di) . (5)
i (cid:80) di≥d∗i (cid:0)dd∗ii (cid:1)(1−p)diπ(di)
(cid:80) (cid:0) (cid:1)
Ifthetruedegreedistributionisnotknown,thenitneedstobeestimated,forexampleusingtechniques
described in orsimilar to [19]. Let πˆ() be a “reasonable"estimator for π(). Then an empirical
· ·
Bayesestimatorisgivenby
dˆEB = di≥d∗i di dd∗ii (1−p)diπˆ(di) . (6)
i (cid:80) di≥d∗i (cid:0)dd∗ii (cid:1)(1−p)diπˆ(di)
Generallyspeaking,ifξ(d ;d )denote(cid:80)sthedis(cid:0)trib(cid:1)utionofd givend ,thenthisempiricalBayes
∗i i ∗i i
estimatecanbeexpressedas
d ξ(d ;d )πˆ(d )
dˆEB = di≥d∗i i ∗i i i .
i ξ(d ;d )πˆ(d )
(cid:80) di≥d∗i ∗i i i
(cid:80)
Theseestimatorstaketheformofaweightedmean,asexpectedforBayesestimatesunderquadratic
loss. Theweightsarefunctionalsofbothsamplingdesignandthedegreedistribution. Forthelatter
estimator,onlytheestimateddegreedistributioncomesintoplay,andthustheproposedempirical
Bayesestimatorincorporatesthesamplingandsamplednetworkinformation.
3 RiskAnalysis
Inthissection, wepresentresultsontherelativeperformanceofourproposedestimatorsfroma
risk-theoreticperspective,andwediscussseveralconditionsunderwhichoneoutperformstheother.
Alltheseestimateswillbebenchmarkedagainsttheregularscale-upestimatedˆ . Proofsmaybe
MME
foundinthesupplementarymaterials.
3.1 RiskofFrequentistEstimates
In the first part of our risk analysis, we look at the (cid:96) frequentist risk of our proposed univariate
2
andmultivariateestimators. Ourmainresultsinthissectionwillcomparetheriskincurredbyour
proposed estimators to the scale up estimator and discuss conditions under which our proposed
estimatorsperformbetter.
4
Proposition3.1. Assumingd0i > 1−pp,wehaveR dˆi,u,d0i <R dˆMi ME,d0i .
(cid:16) (cid:17) (cid:16) (cid:17)
Inotherwords,theunivariateriskminimizerdˆ willoutperformtheMMEwhenthetruedegreed0
i,u i
issufficientlylarge.
Proposition3.2. Letusdenotetheclassofallsampledgraphsofsizen(whered∗i ≥1foralli,i.e.,
thereisnoisolatednode)asG . Alsoassumethatthereexistsan0<α 1suchthat
n∗ 0 ≤
T
G1∗,n = G ∈Gn∗ : Normalizedeigenvectorsv1,v2,··· ,vnof d∗d∗ +D∗ satisfy
(cid:110) 1Tv √nα i(cid:16) (cid:17)
i 0
≥ ∀
n3α2 (1 p)λ (D(cid:9))
G2∗,n =G ∈Gn∗ : |E(G)| 2|nE(0G1)| +n ≥1− −||d0m||2in 
−
(cid:16) (cid:17)
are nonempty. Then we have RG1∗ 2,n dˆm,d0 ≤ RG1∗ 2,n dˆMME,d0 over sampled graphs
∩ ∩
belongingtoG1∗ 2,n =G1∗,n G2∗,n. (cid:16) (cid:17) (cid:16) (cid:17)
∩
Scrutinyoftheconditionsin(cid:84)Proposition3.2,alongwithdefinitionofthesetG ,revealsageneral
1∗ 2,n
characterizationofthegraphswheretheproposedmultivariateestimatorperfo∩rmsbetter. Itistobe
noticedthatdˆ shrinksdˆMME bysomefactor. Thetermontherightsideoftheinequalityinthe
m
definitionofG providesalowerboundontheshrinkagefactorandthetermontheleftdecreasesas
2∗,n
thecardinalityofE( )increases,i.e.,thegraphbecomeslesssparse. Hence,theproposedestimator
G
canbeexpectedtoworkbetterthanthestandardscale-upestimatorundertheassumptionofsparsity
ofthesampledgraph. Thiswillalsobedemonstratedinthesimulationsection.
Theeigenvectorconditionimposesageometricconstraintonthesampledegree-degreematrixD .
∗
Whatitessentiallymeansisthattheanglebetweentheeigenvectorsof d d T +D and1should
∗ ∗ ∗
besmallerthanarccos(α0). Or,inotherwords,byselectinganα0suffi(cid:16)cientlysmallb(cid:17)utpositive,our
classofsampledgraphsarerestrictedwheretheassociatedmatrix d d T +D haseigenvectors
∗ ∗ ∗
atleastarcsin(α0)angleawayfromanyorthogonaldirectionto(cid:16)1. Thus,oure(cid:17)stimatorperforms
betterforsparsegraphsatisfyingamildgeometriccondition.
3.2 RiskofBayesEstimate
TheperformanceoftheBayesestimatorsisevaluatedhereunderseveralconditionsandnetwork
paradigms. Notethattheseestimatorsarecomparedtotheregularscale-upestimatorwithrespect
to their frequentist risk functions. We start with our estimator in its most general form and state
conditionsonthepriordegreedistributionthatwillensurelowerrisk. Fromthat,weassessitsrisk
whenthepriordegreedistributionisreplacedwithanappropriateestimate. Wealsoexplicitlyderive
theBayesestimatorfortheErdös-Rényiclassofrandomgraphsandstateconditionsunderwhichthe
Bayesestimatoryieldslowerriskthanthescale-upestimator.
Proposition3.3. Letd0i bethetruedegreeofsamplenodei,andd∗i,theobserveddegree. Denoteby
G theclassofsampledgraphswherethefollowingtwoconditionshold:
B∗
p(1 p) N 1
E π2(di)≤ (N 1− d0)2d0i whend0i ≤ 2− ; and (7)
d(cid:88)i≥d∗i − − i
 p(d ,d)π(d )
di≥d∗i ∗i i i p , (8)
p(d ,d ) ≥
(cid:80) di≥d∗i ∗i i
wherep(d∗i,di)=(cid:80) dd∗ii (1−p)di.ThenRGB∗ dˆBi ,d0i ≤RGB∗ dˆMi ME,d0i underinducedsubgraph
sampling. (cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:17)
Theconditions(7)and(8)essentiallyconstrainthetailbehaviorofthepriordegreedisbution. The
first condition ensures that the tail decays at a rate such that it is not too “thick” and the second
5
conditionensuresthatitisnottoo“thin”. Asd0becomesbigger,theRHSincondition(7)becomes
i
smallerandthatisreminiscentofthesparsitypropertyoftheunderlyinggraph,meaningthatnota
lotofnodescanhaveveryhighdegree,anobservationconsistentwithsparsegraphs. Ontheother
hand,theLHSinthecondition(8)canbeinterpretedasthemeanofthetailprobabilitiesweightedby
theposteriordistribution. ThishastobeboundedawayfromzeroinorderfortheBayesestimateto
havelowerriskthantheMME.
Inrealproblems,wherethetruedegreedistributionisunknown,oneeitherhastochooseπsubjectively
orusethedatatocomeupwithareasonableestimate. Estimatingπforageneralcaseisbeyondthe
scopeofthispaperandwillnotbediscussedhere. Forouranalysis,wewilljustassumethatwehave
anestimateofthedegreedistributionatourdisposal(e.g.,[19]),denotedbyπˆ. Usingπˆ willgiveus
ourproposedempiricalBayesestimatedˆEB,thebehaviorofwhichcanbedescribedasfollows.
i
Proposition3.4. Letπˆ()beanestimateofπ()suchthat πˆ π <(cid:15).Thenunderassumption
· · (cid:107) − (cid:107)
(8),withπreplacedbyπˆ,wehave ∞
di (1 p)diπˆ(d ) di (1 p)diπ(d ) < (cid:15)(1−p)d∗i (9)
(cid:12)(cid:12)(cid:12)d(cid:88)i≥d∗i (cid:18)d∗i(cid:19) − i −d(cid:88)i≥d∗i (cid:18)d∗i(cid:19) − i (cid:12)(cid:12)(cid:12) pd∗i+1
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)d(cid:88)i≥d∗i di(cid:18)dd∗ii(cid:19)(1−p)diπˆ(di)−d(cid:88)i≥d∗i di(cid:18)dd∗ii(cid:19)(1−p)diπ(cid:12)(cid:12)(di)(cid:12)(cid:12)(cid:12)< (cid:15)(1pd−∗i+p2)d∗i (d∗i +1−p) (10)
(cid:12) (cid:12)
Th(cid:12)us,itfollowsthat (cid:12)
(cid:12) (cid:12)
dˆEi B −dˆBi < (cid:15)(1−p)d∗i + (cid:15)(1−p)d∗i(d∗i +1−p) (11)
(cid:12)(cid:12)(cid:12) dˆBi (cid:12)(cid:12)(cid:12) pd∗i+1 di≥d∗i di dd∗ii (1−p)diπ(di) pd∗i+2 di≥d∗i dd∗ii (1−p)diπ(di)
(cid:80) (cid:0) (cid:1) (cid:80) (cid:0) (cid:1)
Itiseasilyseenthatwiththeassumption(8),theupperboundin(11)canbesimplifiedto
dˆEi B −dˆBi < (cid:15)(1−p)d∗i + (cid:15)(1−p)d∗i(d∗i +1−p) .
(cid:12)(cid:12)(cid:12) dˆBi (cid:12)(cid:12)(cid:12) d∗ipd∗i+2 di≥d∗i dd∗ii (1−p)di pd∗i+3 di≥d∗i dd∗ii (1−p)di
Assumingalargenetwork,thesum(cid:80)inthed(cid:0)en(cid:1)ominatorcanbeapp(cid:80)roximate(cid:0)db(cid:1)y (1−p)d∗i . Thenthe
pd∗i+1
upperboundis
(cid:15) (cid:15)(d +1 p) (cid:15) 1 d +1 p
+ ∗i − = + ∗i − .
d p p2 p d p
∗i (cid:18) ∗i (cid:19)
Fromtheabovediscussion, itisevidentthatif(cid:15) = o(p2/n), dˆEB dˆB foralliandhencetheir
i ≈ i
riskfunctionswillalsobeclose. Thus,usingProposition3.3,itisexpectedthatRGB∗ dˆEiB,d0i (cid:46)
RGB∗ dˆMi ME,d0i (cid:16) (cid:17)
(cid:16) (cid:17)
3.2.1 Illustration: Erdös-RényiGraphs
ItiswellknownthattheasymptoticdegreesinErdös-RényigraphmodelsfollowaPoissondistribu-
tion,understandardconditions. Inthissection,westudytheeffectsofusingaPoissonpriordegree
distributionforlargeErdös-Rényigraphs. ThegoalistodemonstratetheefficacyoftheBayesian
approachcomparedtoscale-upestimatesasinthelastsection. However,studyingspecificmodels
likeErdös-RényiwillgiveusmoreinsightabouttheperformanceoftheproposedBayesestimate. In
thisscenario,thepriorπ()isgivenby
·
λdi
π(d )=e λ ,
i −
d !
i
where λ is the prior mean. For a large Erdös-Rényi graph with number of nodes N and edge
probabilityp ,λ Np . Wedenote,byP(k,µ),theshiftedPoissondistributiononk,k+1, ,
e e
≈ ··· ∞
whosep.m.f. isgivenby
µx k
f(x)=e µ − 1 (x).
− (x k)! {k,k+1,···}
−
6
ItiseasytocheckthatwithaPoisson(λ)priorond ,theposteriordistributionisP (d ,λ(1 p)).
i ∗i −
HencetheBayesestimatewithrespecttothequadraticlossfunctionis
dˆB =d +λ(1 p) .
i ∗i −
Proposition3.5. Assuming
1+p 1 λp 1+p 1 λp
λ+ +1 d0 λ+ + +1 ,
p (cid:32)2 −(cid:115)1+p (cid:33)≤ i ≤ p (cid:32)2 (cid:115)1+p (cid:33)
thequadraticriskoftheBayesestimatorusingaPoisson(λ)priorissmallerthanthatoftheMME.
Theaboveresultshowsthatifthesamplednodeissuchthatitstruedegreebelongstoaneighborhood
aroundthemeanoftheunderlyingdegreedistribution,thentheBayesestimatorisuniformlybetter
thantheMME.Incasetheunderlyingmeanisunknown,itcaneasilybeestimatedfromthesample.
(e.g.,forknownN,λˆ = Npˆ = N E(G )/ n .)Ifλˆ isaconsistentestimatorofλinthesense
e e | ∗ | 2
thatλˆ P λwhenN ,n andn/N (cid:0) (cid:1)p,thentheempiricalBayesestimator
→ →∞ →∞ →
dˆEB =d +λˆ(1 p)
i ∗i −
willconvergeinprobabilitytotheBayesestimatorinthesensethat dˆEB dˆB P 0. Hence,the
i − i →
resultofProp. (3.5)isexpectedtohold. Thiswillalsobedemonstrate(cid:12)dinthesim(cid:12)ulations.
(cid:12) (cid:12)
(cid:12) (cid:12)
4 Simulations
Foroursimulationstudy,welookattwodifferentregimesofnetwork–Erdös-Rényirandomgraphs
andheavytaileddegreedistributions.
4.1 Erdös-Rényinetwork
Wecomparefourmethodsofestimation-theregularMME,univariateriskminimizer,multivariate
riskminimizerandtheBayesestimate. AspriorsinBayesestimation,weusebothexponentially
decaying(Poisson)andpolynomiallydecayingdegreedistributionaspriors. Table1recordsthe
Euclidean distance between the true and estimated degree vectors across some combinations of
graphsizeN,edgestrengthp andsamplingproportionp. Theerrorsareaveragedover50different
e
samplesfromeachgivengraphG. Fromtheoutput,itisclearthattheBayesestimatorswithtrue
λ and estimated λ outperform other estimators by a very wide margin in terms of (cid:96) risk. Also,
2
ourtheoreticalpredictioninthediscussionfollowingProposition3.2wasthatthemultivariaterisk
minimizer(MRM)worksbetterthantheMMEforsparsegraphs. Thisisexperimentallyverified
in this simulation, since we see that the relative risk of MRM compared to MME decreases as
thesparsityoftheunderlyinggraphincreases,i.e.,asp decreases. Themethodwithlowesttotal
e
quadraticlossisshowninredforeachcondition.
4.2 ScaleFreeNetwork
Wecomparedfourmethodsofestimationinsimulatedscalefreenetworkswhichfollowapower
lawdegreedistribution. AspriorsinBayesestimation,wecomparedthetruepolynomialpriorand
quadraticprior. Wecomputedthel distancesacrosssomecombinationsofsparsity(denotedbys,
2
givenbytheratiooftotaledgestoallpossibleedges),samplingproportionpandheavinessofthetail
ofthedegreedistribution,controledbym. TheresultsareshowninTable2. TheBayesestimatorsor
themultivariateriskminimizersworkbetterthantheotherestimators. Oneimportantthingtoobserve
hereisthatforthemostsparsegraph,theBayesestimatorwithtruepriorworksthebestandass
increases,multivariateriskminimizersworkbetterthantherest,butthereishardlyanyimprovement
overMME.Again,themethodwithlowesttotalquadraticlossisshowninredforeachcondition.
5 HumanTraffickingNetwork
InFebruary2015, theDefenseAdvancedResearchProjectsAgency(DARPA),anagencyofthe
U.S.DepartmentofDefense,announcedtheMemexprograminresponsetotheuseoftheInternetin
7
pe,p↓,N→ N=1000 s,p↓,N→ N=1000
MME URM MRM Bayes MME URM MRM Bayes
Pois.(λ) Pois.(λˆ) Poly. TruePrior Quad.Prior
pppeee===000...123,,,ppp===000...111 244919262...202922 244919051...018458 244818938...720685 11923016...038326 11924589...940592 244919262...406824 ss=ss===00.215.2%%%%,,,,pppp====0000...111.1,,,,mmmm====2222.5 249435228...614.10380 238235584...732.26968 248433907...792.28377 238133292....22271936 238233212....22071976
pe=0.4,p=0.1 588.18 587.84 586.40 152.94 168.99 588.02 s=1%,p=0.1,m=2.5 92.91 82.89 91.50 81.93 78.72
pe=0.1,p=0.2 284.08 283.67 282.76 119.87 122.73 284.24 s=5%,p=0.1,m=2.5 210.04 214.70 208.22 231.68 219.55
pe=0.2,p=0.2 389.15 389.07 386.87 164.30 166.84 389.55 s=0.2%,p=0.1,m=3 41.52 28.75 39.36 21.71 22.61
pe=0.3,p=0.2 485.09 485.07 481.82 187.43 190.55 485.63 s=1%,p=0.1,m=3 89.40 79.98 88.07 83.39 75.46
pe=0.4,p=0.2 527.37 527.28 527.68 205.47 210.42 527.07 s=5%,p=0.1,m=3 209.97 213.30 208.25 242.90 217.87
Table 1: Erdös-Rényi Simulation Results: λ is Table2: ScaleFreeSimulationResults
thetruemeanusingknownp . λˆistheestimated
e
meanusinganestimatepˆ ofp .
e e
humantrafficking,especiallychatforums,advertisementsandjobservicessections. DARPA-funded
researchdeterminedthetraffickingindustryspent$250Mtopostmorethan60Madvertisementsover
atwo-yeartimeframe[15]. Indexingandcross-referencingtheadswiththesamecontactnumber,
similaraddressorzipcodeshelpidentifyandtracktheillegaltraffickingactivities. Thisleadstoa
massivebackgroundnetworkstructurewhereeachnoderepresentsanadvertisementandanedge
betweentwonodesarecreatediftheysharecertainfeatures. Itisnotunreasonabletoexpectthat,in
surveillanceofnetworkslikethis,samplingmaywellarise,eitherbychoiceorbycircumstance. We
mimicthissituationbypretendingthatthisunderlyingnetworkgeneratedbytheMemexprogram
is unknown to us and sampling it using induced subgraph sampling. The nodes associated with
traffickingactivitiesareflaggedinthedata. Thereare31,248nodes,ofwhich12,387areflaggedand
thereare10,200,838edges. Ourgoalwastoestimatethetruedegreesofflaggednodesthatwesawin
oursample. Wecomparedthe(cid:96) distanceofregularscale-upestimators,andourproposedunivariate,
2
multivariateandBayesestimators. FortheBayesestimator,anumberofpolynomialpriorswere
takenintoconsiderationwithvaryingdegreeofdecay,denotedbyα. TheresultsareshowninTable
3. Almosteverythingworksbetterthanthenaivescale-upestimatorintermsoftotal(cid:96) loss,although
2
therelativeimprovementismoremodestthaninsimulation.
p MME URM MRM Bayes
α= 0.1 α= 0.5 α= 1
− − −
p=0.005 3451.364 3436.64 3447.24 3687.26 3541.94 3450.97
p=0.01 3427.55 3397.71 3427.88 3451.86 3412.12 3428.59
p=0.02 4462.937 4448.33 4461.64 4492.83 4450.71 4462.31
Table3: SamplingfromHumanTraffickingNetwork
6 Discussion&FutureResearch
In this paper, we addressed the problem of estimation of true degrees of sampled nodes from an
unknowngraph. Weproposedaclassofestimatorsfromarisk-theoryperspectivewherethegoal
wastominimizetheoverall(cid:96) riskofthedegreeestimatesforthesamplednodes. Weconsidered
2
estimatorsthatminimizebothfrequentistandBayesriskfunctionsandcomparedthefrequentist(cid:96)
2
risksofourproposedestimatortothenaivescale-upestimator. Thebasicobjectiveofproposing
theseestimatorswastoexploittheadditionalnetworkinformationinherentinthesampledgraph,
beyondtheobserveddegrees. Ourtheoreticalanalyses,simulationstudiesandrealdatashowclear
evidenceofsuperiorperformanceofourestimatorscomparedtoMME,especiallywhenthegraphis
sparseandthesamplingratioislow,mimickingthereal-worldexamples.
Thereareanumberofwaysourcurrentworkcouldbeextended. Firstly,atheoreticalanalysisofthe
BayesestimatorsunderpriorsforrandomgraphmodelsbeyondErdös-Rényiisdesirable,although
likely more involved. Secondly, although induced subgraph sampling serves as a representative
structuralmodelforacertainclassofadaptivesamplingdesigns,thespecificdetailsofthesufficiency
conditionsdiscussedinthispapercanbeexpectedtovaryslightlywiththeothersamplingdesigns
(e.g., incident subgraph or random walk designs) . Finally, the success of the Bayesian method
appearstorelyheavilyuponappropriatechoiceofpriordistribution,asobservedinourtheoretical
analysis and computational experiments. It would be of interest to explore the performance of
theempiricalBayesestimateinconjunctionwiththenonparametricmethodofdegreedistribution
8
estimation proposed in [19]. More generally, the method in [19] can in principle be extended to
estimate individual vertex degrees. But the computational challenge of implementation and the
correspondingriskanalysiscanbeexpectedtobenontrivial.
9
Estimation of Vertex Degrees in a Sampled Network:
Supplementary-A: Proofs
1 Proofs
1.1 ProofofLemma2.1
Proof. LetS bethesetofsamplednodes. Seethatd = I(k S). Hence,d B d0,p .
∗i k∈Nei ∈ ∗i ∼ i
P (cid:0) (cid:1)
E d d =E I(k S) I(l S)
∗i ∗j  ∈ ! ∈ 
(cid:0) (cid:1) kX∈Nei l∈XNej
  
=E I(k S) + I(k S)I(l S)
 ∈   ∈ ∈ 
k∈NXei∩Nej (k,l)∈(NeiX∪NeXj)\(Nei∩Nej)
=d0p+ d0d0 d0 p2   
ij i j − ij
(cid:0) (cid:1)
Notethatd0 isthecardinalityofthefirstsetofnodes(byitsdefinition)and(d0d0 d0 )isthatof
ij i j − ij
thesecond. Theprobabilitythatanodeisselectedininducedsubgraphsamplingispandsinceeach
nodeisselectedindependently,thejointprobabilitythattwonodesareselectedisp2. Hence,
Cov d ,d =d0 p(1 p)
∗i ∗j ij −
(cid:0) (cid:1)
1.2 ProofofProposition3.1
TakingTaylorexpansionupto2ndorder,weget
1
E dˆi,u = 1−p p + p1E d∗i 1+ 1d−p −
(cid:16) (cid:17) " (cid:18) ∗i (cid:19) #
1 p 1 (1 p)2 (1 p)2
≈ −p + pE d∗i −(1−p)+ −d ≈d0i + p−2d0
(cid:20) ∗i (cid:21) i
WeonlyconsiderTaylorexpansionupto2ndorderbecausetheexpectationofhigherordertermscan
beneglectedassumingd0issufficientlylarge. Hence,weget
i
(1 p)2
Bias dˆ ,d0 = −
i,u i p2d0
i
(cid:16) (cid:17)
Similarly,weapproximatethevariancebyTaylorexpansionandget
1 (1 p)2
Var dˆi,u ≈ p2Var d∗i + −d
(cid:16) (cid:17) (cid:18) ∗i (cid:19)
1 1 1
= Var(d )+(1 p)4Var +2(1 p)2Cov d ,
p2 ∗i − d − ∗i d
(cid:20) (cid:18) ∗i(cid:19) (cid:18) ∗i(cid:19)(cid:21)
10