Simultaneous Influencing and Mapping for Health Interventions

13 Pages·2015·1.06 MB·English

Simultaneous Influencing and Mapping for Health Interventions
LeandroSorianoMarcolino1,AravindLakshminarayanan2,AmulyaYadav1,MilindTambe1
1UniversityofSouthernCalifornia,LosAngeles,CA,90089,USA
{sorianom,amulyaya,tambe}@usc.edu
2IndianInstituteofTechnology,Madras,TamilNadu,600036,India
[email protected]
Abstract ordertoperformfutureactionstoenhancetheirwell-being,
andbetteractuateintheirpractices(Marsden2005).Asmen-
Influence Maximization is an active topic, but it was
tioned, however, the works in influence maximization are
alwaysassumedfullknowledgeofthesocialnetwork
currentlyignoringthisproblem.
graph. However, the graph may actually be unknown
Eachpersoninasocialnetworkactuallyknowsotherpeo-
beforehand.Forexample,whenselectingasubsetofa
homelesspopulationtoattendinterventionsconcerning ple,includingtheonesshecannotdirectlyinfluence.Hence,
health,wedealwithanetworkthatisnotfullyknown. eachtimeweselectsomeoneforanintervention(tospread
Hence, we introduce the novel problem of simultane- influence),wealsohaveanopportunitytoobtainknowledge
ouslyinfluencingandmapping(i.e.,learning)thegraph. from that person. Therefore, in this work we present for
Westudyaclassofalgorithms,whereweshowthat:(i) thefirsttimetheproblemofsimultaneouslyinfluencingand
traditionalalgorithmsmayhavearbitrarilylowperfor- mappingasocialnetwork.Westudytheperformanceofthe
mance;(ii)wecaneffectivelyinfluenceandmapwhen
classicalgreedyinfluencemaximizationalgorithminthiscon-
the independence of objectives hypothesis holds; (iii)
text,andshowthatitcanbearbitrarilylow.Hence,westudy
whenitdoesnothold,theupperboundfortheinfluence
aclassofalgorithmsforthisproblem,andshowthatwecan
lossconvergesto0.Werunextensiveexperimentsover
effectivelyinfluenceandmapanetworkwhenindependence
fourreal-lifesocialnetworks,wherewestudytwoalter-
nativemodels,andobtainsignificantlybetterresultsin ofobjectivesholds.Fortheinterventionswhereitdoesnot
boththantraditionalapproaches. hold,wegiveanupperboundinourloss,whichconverges
to0.Westudyanapproximationofourmainalgorithm,that
worksaswellbutrequiringfewerassumptions.
1 Introduction
Weperformalargescaleexperimentationusingfourreal
Influencingasocialnetworkisanimportanttechnique,with lifesocialnetworksofhomelesspopulations,whereweshow
greatpotentialtopositivelyimpactsociety,aswecanmodify thatouralgorithmiscompetitivewithpreviousapproachesin
thebehaviorofacommunity.Forexample,wecanincrease termsofinfluence(evenoutperformingtheminhardcases),
the overall health of a population; Yadav et al. (2015), for andissignificantlybetterintermsofmapping.
instance,spreadinformationaboutHIVpreventioninhome-
lesspopulations.However,althoughinfluencemaximization 2 RelatedWork
hasbeenextensivelystudied(Kempe,Kleinberg,andTardos
Theinfluencemaximizationproblemhasrecentlybeenavery
2003; Cohen et al. 2014; Golovin and Krause 2010), their
populartopicofresearch.Normally,themainmotivationis
mainmotivationisviralmarketing,andhencetheyassume
viralmarketinginsocialmedia(likeFacebookorMyPlace).
thatthesocialnetworkgraphisfullyknown,generallytaken
Hence,previousworksassumefullknowledgeofthesocial
fromsomesocialmedianetwork(suchasFacebook).
networkgraph.TheclassicalresultisKempe,Kleinberg,and
However,thegraphsrecordedinsocialmediadonotreally
Tardos(2003),wheretheystudytheapproximationguarantee
representallthepeopleandalltheconnectionsofapopula-
ofagreedyalgorithmforthe“independentcascademodel”.
tion.Mostcritically,whenperforminginterventionsinreal
GolovinandKrause(2010)furtherextendedthatresulttothe
life,wedealwithlargedegreesoflackofknowledge.Nor-
casewherebeforepickingeachnodeweareabletoobserve
mallythesocialagencieshavetoperformseveralinterviews
whichonesarealreadyinfluencedornot.However,theystill
inordertolearnthesocialnetworkgraph(Marsden2005).
assume full knowledge of the graph. Cohen et al. (2014)
Thesehighlyunknownnetworks,however,areexactlythe
focusonhowtoquicklyestimatethepotentialspreadofone
onesweneedtoinfluenceinordertohaveapositiveimpact
node, since running simulations (as needed by the greedy
intherealworld,beyondproductadvertisement.
algorithm)inlargegraphsiscomputationallyexpensive.A
Additionally,learningasocialnetworkgraphisveryvalu-
differentviewwasstudiedbyYadavetal.(2015),wherethey
ableperse.Agenciesalsoneeddataaboutapopulation,in
analyzeamodelwherenodestrytoinfluencetheirneighbors
Copyright(cid:13)c 2016,AssociationfortheAdvancementofArtificial multipletimes,andeachedgehasanexistenceprobability.
Intelligence(www.aaai.org).Allrightsreserved. Wedealherewithadifferentkindofuncertainty,asinour
modelwholeportionsofthegrapharecompletelyunknown. For each node v , we assume we can observe a number
i
Thisworkalsorelatestotheproblemofsubmodularop- γi,whichindicatesthesizeofitsteachinglist.Westudytwo
timization(influenceisasubmodularfunction)byselecting versions: in one γi is the number of nodes in Vi that are
elementswithoutknowledgeofthewholeset.Badanidiyuru notyetinGk (hence,thenumberofnewnodesthatwillbe
etal.(2014)presentanalgorithmforselectingthebestsub- learnedwhenpickingvi).Werefertothisversionas“perfect
set,givingelementsarrivingfroma“stream”.Hence,given knowledge”. In the other, γi := |Vi|, and thus we cannot
enoughtimethefullsetwouldbeseen,andwhichelements knowhowmanynodesinViaregoingtobeneworintersect
arediscovereddoesnotdependonwhichonesareselected. withalreadyknownnodesinVk.Werefertothisversionas
Ourproblemisalsorelatedtotheclassicalmax-k-cover “partialknowledge”.Thepartialknowledgeversionismore
problem (Khuller, Moss, and Naor 1999), where we must realistic,asbypreviousexperiencewemayhaveestimations
pickk subsetsthatmaximizeourcoverageofaset.Inthat ofhowmanypeopleapersonwithacertainprofileusually
case,however,theelementsofeachsubsetisknown.There knows.Westudybothversionsinordertoanalyzehowmuch
isalsoanonlineversionofthemax-k-coverproblem(Alon wemaylosewithpartialknowledge.Notethatwemayalso
etal.2009),wherewepickonesubsetatatime,butanad- havenodeswithemptyteachinglists(γi =0).Theteaching
versaryfixeswhichelementmustbecoverednext.However, listofanodeviisthesetofnodesthatviwillteachusabout
thesetandtheavailablesubsetsarestillknowninadvance. oncepicked,andisnotnecessarilyascompleteasthetrueset
Similarly,Grandonietal.(2008)studythecaseofcoveringa ofallnodesknownbyvi.Somenodescouldsimplyrefuseto
setwhoseelementsarerandomlychosenfromauniverseof provideanyinformation.
possibleelements.Anotherimportantrelatedproblemisthe Additionally,notethatweareassumingtheteachinglist
submodularsecretaryproblem(Bateni,Hajiaghayi,andZadi- andtheneighborlisttobeindependent.Thatis,anodemay
moghaddam2013),whereweagainpickasubsettooptimize teachusaboutnodesthatitisnotabletodirectlyinfluence.
asubmodularfunction.However,inthatcase,wereceiveone For instance, it is common to know people that we do not
elementatatime,andwemustmakeanirrevocabledecision havedirectcontactwith,orwearenot“close”enoughtobe
ofeitherkeepingitornot. abletoinfluence.Similarly,apersonmaynottellusaboutall
Finally,thisworkalsorelatestosequentialdecisionmak- herclosefriends,duetolimitationsofaninterviewprocess,
ingwithmultipleobjectives(Roijersetal.2013).Herewedo or even “shame” to describe some connections. However,
notaimatcomputinganoptimalpolicy(whichiscomputa- somereaderscouldarguethatpeoplewouldbemorelikely
tionallyexpensive),butatstudyingagreedymethod,similar to teach us about their direct connections. Hence, we will
tootherworksininfluencemaximization.Ouralgorithmisa handlethecasewheretheindependencedoesnotholdinour
scalarizationovertwoobjectives,butsuchmethodwasnever empiricalexperimentsinSection4.2.
studiedbeforetoinfluenceandmapsocialnetworks. Simultaneouslytotheproblemofmapping,wealsowant
tomaximizethespreadofinfluenceoverthenetwork.We
considerherethetraditionalindependentcascademodel,with
3 InfluencingandMapping
observation,asinGolovinandKrause(2010).Thatis,anode
Weconsidertheproblemofmaximizingtheinfluenceina maybeeitherinfluencedoruninfluenced.Anuninfluenced
social network. However, we start by knowing only a sub- nodemaychangetoinfluenced,butaninfluencednodewill
graph of the social network. Each time we pick a node to neverchangebacktouninfluenced.Eachtimewepickanode
influence,itmayteachusaboutsubgraphsofthenetwork. foranintervention,itwillchangetoinfluenced.Whenanode
Ourobjectiveistospreadinfluence,atthesametimelearning changes from uninfluenced to influenced, it will “spread”
thenetworkgraph(i.e.,mapping).Wecallthisproblemas the influence to its neighbors with some probability. That
“Simultaneous Influencing and Mapping” (SIAM). In this is, at each edge e there is a probability p . When a node
e
paper,weconsideraversionofSIAMwhereweonlyneedto v changes to influenced, if there is an edge e = (v ,v ),
1 1 2
mapthenodesthatcomposethenetwork.Weassumethatwe thenodev willalsochangetoinfluencedwithprobability
2
alwaysknowalltheedgesbetweenthenodesoftheknown p .Similarly,ifv changestoinfluenced,itwillspreadthe
e 2
subgraph.Forclarity,wewilldefineformallyhereonlythe influencetoitsneighborsbythesameprocess.Influenceonly
version of SIAM that we handle in this work. Therefore, spreads in the moment a node changes from uninfluenced
unlessotherwisenoted,henceforthbySIAMwemeanthe to influenced. That is, a node may only “try” one time to
versionoftheproblemthatisformallydefinedbelow. spreadinfluencetoitsneighbors.AsinGolovinandKrause
Let G := (V,E) be a graph with a set of nodes V and (2010),weconsiderthatwehaveknowledgeaboutwhethera
edgesE.Weperformηinterventions,whereateachonewe nodeisinfluencedornot(butinourcase,wecanonlyknow
pickonenode.Theselectednodeisusedtospreadinfluence aboutnodesinthecurrentknownsubgraphGk).LetIk be
andmapthenetwork.Weassumewedonotknowthegraph the number of influenced nodes after k interventions. Our
G,weonlyknowasubgraphGk =(Vk,Ek)⊂G,wherekis objectiveistomaximizeIk givenηinterventions.Influence
thecurrentinterventionnumber.GkstartsasGk :=G0 ⊂G. mayspreadbeyondGk.Hence,weconsiderIkasthenumber
Foreachnodevi,thereisasubsetofnodesVi ⊂V,which of influenced nodes in the full graph G. We denote as σi
willbecalled“teachinglist”.Eachtimewepickanodev ,the theexpectednumberofnodesthatwillbeinfluencedwhen
i
knownsubgraphchangestoGk :=(Vk−1∪Vi,Ek),where pickingvi(usuallycalculatedbysimulations).
E containsalledgesbetweenthesetofnodesV ∪Viin As mentioned, we want to attend both objectives simul-
k k−1
G.Ourobjectiveistomaximize|V |,givenηinterventions. taneously.Hence,wemustmaximizeboth|V |andI .Itis
k k k
easytoshowthatSIAMisanNP-Completeproblem:
A B
Proposition1. SIAMisNP-Complete.
Connectedgraph
Proof. Letκbeaninstanceoftheinfluencemaximization ofznodes C
problem, with graph G. Consider now a SIAM problem
wherenonodecarriesinformationandG :=G.Ifwecan A’ B’
0
solvethisSIAMproblemwecanalsosolvetheinfluencemax-
imizationproblemκ.Therefore,SIAMisNP-Complete.
Figure1:Agraphwherethetraditionalgreedyalgorithmhas
arbitrarilylowperformance.
AsSIAMisNP-Complete,similarlytopreviousinfluence
maximizationworks(Kempe,Kleinberg,andTardos2003;
GolovinandKrause2010),westudygreedysolutions.Like
anynode.A,B,A’andB’haveemptyteachinglists.C,onthe
theexplorationvsexploitationdilemmasinonlinelearning
otherhand,canteachusaboutaconnectedgraphofznodes.
(Valizadegan,Jin,andWang2011),thefundamentalproblem
Influence-greedy,byrunningsimulationsontheknowngraph,
ofSIAMiswhethertofocusoninfluencingormappingthe
picksnodesAandB,sinceeachcaninfluenceonemorenode.
network.Hence,weproposeasageneralframeworktoselect
Theoptimalsolution,however,istopicknodeC,whichwill
thenodev suchthat:
i
teachusabouttheconnectedgraphofznodes.Then,wecan
pick one node in that graph, and influence z +1 nodes in
v =argmax(c ×σ +c ×γ ) (1)
i 1 i 2 i total.Hence,theinfluence-greedysolutionisonly 4 ofthe
Constantsc andc controlthebalancebetweeninfluenc- z+1
1 2 optimal.Aszgrows,influence-greedywillbearbitrarilyfar
ingormapping.c =1,c =0istheclassicalinfluencemax-
1 2 fromtheoptimalsolution.
imizationalgorithm(“influence-greedy”);c = 0,c = 1,
1 2
on the other hand, only maximizes the knowledge-gain at If we make some assumptions about the distribution of
eachintervention(“knowledge-greedy”).c =c =1isan theteachinglistsacrossthenodes,however,influence-greedy
1 2
algorithmwherebothobjectivesareequallybalanced(“bal- eventuallymapsthefullgraphgivenenoughinterventions.
anced”).Differentweightsmayalsobeused. Letn=|V|,andn =|V |.Weshowtheexpectednumber
k k
Rememberthatwedefinedtwoversionsfortheγ values: ofinterventionstolearnallnnodes(subtractedbyasmall
perfect knowledge, where we know how many new nodes (cid:15),fornumericalstability).Westudythepartialknowledge
a node will teach us about; and partial knowledge, where version. Assume the size of the teaching list of each node
we do not know how many nodes will be new. In order to is drawn from a uniform distribution on the interval [0,u],
better handle the partial knowledge case, we also propose and any node is equally likely to be in a teaching list. We
the“balanced-decreasing”algorithm,wherec constantly considerthatthereisaprobabilityϕthatanodewillhavea
2
decreasesuntilreaching0.Hence,wedefinec as: non-emptyteachinglist.
2
(cid:26) c(cid:48) − 1 ×c(cid:48) ×k ifk ≤d Proposition2. Theexpectednumber(kfull)ofinterventions
c2 := 02 d 2 otherwise , (2) forinfluence-greedytolearnn−(cid:15)nodesis log(−n0(cid:15)−n).
log(1−ϕ×u)
2×n
wherec(cid:48) isthedesiredvalueforc attheveryfirstiteration,
2 2 Proof. Sinceinfluence-greedyisnotconsideringγ,itpicks
anddcontrolshowfastc decaysto0.
2 nodes arbitrarily in terms of knowledge-gain. Hence, on
average it selects the expected value of the uniform distri-
3.1 Analysis
bution, u/2. For each node v in a teaching list, the prob-
We begin by studying influence-greedy. It was shown that ability that it is not yet known is n−nk. Therefore, the
whenpickingthenodevwhichargmax(σv)ateachinterven- expected number of nodes known atnone iteration k is:
tion,weachieveasolutionthatisa(1−1/e−(cid:15))approx- E[n ]=ϕ× u × n−E[nk−1] +E[n ].Solvingtherecur-
imation of the optimal solution, as long as our estimation k 2 n k−1
rencegives:E[n ]=n ×(1−ϕ×u)k−n×(1−ϕ×u)k+n.
of σv (by running simulations) is “good enough” (Kempe, k 0 2×n 2×n
Kleinberg,andTardos2003).However,eventhoughtheac- SolvingforE[nk] = n−(cid:15)givesthattheexpectednumber
wtuealcainnflounelnycreunspsriemaudlamtiaoynsgtoobeesytiomnadtethσevkinnothwencgurrarepnhtGGkk,. ofinterventionsis:kfull = lloogg((1−−nϕ02(cid:15)×−×nnu)).
Hence, the previous results are no longer valid. In fact, in
k quicklyincreasesasϕ(oru)decreases.InSection4,
full
thenextobservationweshowthatwecanobtainarbitrarily
westudyexperimentallytheimpactofϕontheperformance
low-performingsolutionsbyusinginfluence-greedy.
ofinfluence-greedy.
Observation1. Theperformanceof influence-greedycan Now,let’slookatbalanced.Clearly,itwilllearnthefull
bearbitrarilylowinaSIAMproblem. graphwith alower numberofexpected interventionsthan
influence-greedy. However, although intuitively balanced
Weshowwithanexample.ConsiderthegraphinFigure1,
mayseemreasonable,itsperformancemayalsoquicklyde-
andassumewewillrun2interventions(i.e.,pick2nodes).
gradeifweassumepartialknowledge(i.e.,γ =|Vi|).
Thereisaprobability1tospreadinfluenceinanyedge.Our i
initialknowledgeisV ={A,A(cid:48),B,B(cid:48),C}.AandBcanin- Proposition3. Theperformanceofthebalancedalgorithm
0
fluenceA’andB’,respectively.However,Ccannotinfluence degradesasn →n,ifγ =|Vi|.
k i
Proof. Eachnodeintheteachinglistofanodev hasprob- terventions where VGood (cid:54)= ∅, balanced selects a node in
i k
ability n−nk ofbeingayetunknownnode.Hence,theex- VGood,whileinfluence-greedymakesamistakeintermsof
n k
pectednumberofunknownnodesthatwillbelearnedbypick- |Vσ∩Vγ|
mappingwithprobability1− k k .
Ainsgnano→denw,iEth[nteeawch]i→ngl0is.tHsieznecγe,iwis:hEen[nnew→]=nγ,bi×alann−ncnekd. Weconsidernowtheprobabil|iVtkyσ|ofVGood (cid:54)= ∅acrossk
k k k
picksanodevthatmaximizesσv+γv,thusmissingtoselect interventions. Clearly, if |Vkσ| > nk/2, and |Vkγ| > nk/2,
nodesv (ifavailable)withσ > σ ,σ +γ < σ +γ , wehaveVGood (cid:54)= ∅.Ifnot,notethatask → ∞,n → n.
o o v o o v v k k
withnoactualgainsinmapping. Therefore,Vγ → V (sinceallγ → 0,allnodeswillhave
k k
thesameteachinglistsize),thusVGood →Vσ (cid:54)=∅.Hence,
Thisproblemdoesnothappenintheperfectknowledge k k
theprobabilityofVGood (cid:54)=∅goesto1ask →∞.
version.Sincetheγvaluesonlyincludenewnodes,γ →0as k
Let’s study now the case when VGood = ∅. Let v be
nk →n,forallγ.Hence,intheperfectknowledgeversion, the node in Vσ picked by influence-kgreedy, and v be1the
balancedconvergestothesamebehaviorasinfluence-greedy k 2
askincreases.Inordertoapproximatethisbehaviorforthe node in Vk \ Vkσ with the largest γ2. Since VkGood = ∅,
partialknowledgecase,weproposethebalanced-decreasing we must have that σ1 > σ2, and γ2 > γ1. However, as
algorithm, where the constantly decreasing c2 “simulates” long as γ2 −γ1 < σ1 −σ2, balanced still selects v1 (or
thedecreasingγ values. anevenbetternode).Intheworstcase,theexpectedvalue
forγ istheexpectedmaximumoftheuniformdistribution:
Wenowshowthatbalancedcanmatchtheperformance 2
E[γ ] = u −u /(n +1) ≤ u . γ , on the other hand,
of influence-greedy in terms of influence, but at the same 2 k k k k 1
hastheexpectedvalueoftheuniformdistributionE[γ ]=
timemappingbetterthenetwork.Aswejustdiscussedthat 1
u /2. Hence, as long as σ −σ > u /2, in expectation
perfectknowledgecanbeapproximatedbyusingbalanced- k 1 2 k
balanced stillpicksv (oranevenbetternode).Moreover,
decreasing,wefocushereontheperfectknowledgecase.We 1
whenbalanceddoesnotpickv ,ourlossintermsofinfluence
showthatwhentheindependenceofobjectiveshypothesis 1
atinterventionk isatmostu /2.Sinceγ → 0asn → n,
holds(definedbelow),balancedplaysthesameasinfluence- k k
u /2→0ask →∞.
greedyorbetter,whileinfluence-greedymaystillfailinterms k
ofmapping.Ifthehypothesisdoesnothold,ourinfluence
Proposition4showsthatwemayexperiencelossinonein-
lossatoneinterventionwillbeboundedbyu/2→0.
LetVσbeasubsetofV whereeachv ∈Vσmaximizesσ tervention,whencomparinginfluence-greedywithbalanced.
inthecukrrentinterventionkk.Similarly,letVkγ ⊂V ,where However,thelossisboundedbyuk/2,whichgoesto0asthe
eachv ∈Vγ maximizesγ inthecurrentintekrventikonk.As numberofinterventionsgrows.Moreover,whenwedonot
k updatetheγvalues,wecanusethebalanced-decreasingalgo-
before, we consider that the teaching list size of a node is
rithmtosimulatethesameeffect.Additionally,inProposition
givenbyauniformdistribution,butsinceγ tendstodecrease
4weconsideredthesamegraphstatesateachintervention
ateachintervention,wedenotetheintervalas[0,u ].
Clearly,anynodeinthesetVGood := Vσ ∩Vkγ should acrossbothalgorithms.Inpractice,however,sincebalanced
k k k is able to map the network faster, any loss experienced in
beselected,astheymaximizebothobjectives.Hence,when
VGood (cid:54)=∅itispossibletosimultaneouslymaximizebothob- thebeginningwhenkislowcanbecompensatedbyplaying
k betterlaterwithfullknowledgeofthegraph,whileinfluence-
jectives,andthuswesaythattheindependenceofobjectives
greedy may still select nodes with lower σ due to lack of
hypothesisholds.Sincewearestudyinggreedy-algorithms,
knowledge.AsnotedinObservation1,lackofknowledgeof
bothbalancedandinfluence-greedylackoptimalityguaran-
thefullgraphcanmakeinfluence-greedyplaywitharbitrarily
tees.Hence,wefocushereona“local”analysis,andshow
low performance. In Section 4.2 we perform an empirical
that given a set of k possible interventions (with the same
analysisassumingapowerlawmodelfortheteachinglists,
graphstateacrossbothalgorithmsateachintervention),bal-
andwenoteherethatourmainresultsstillhold.
ancedisabletopicknodesthatspreadasmuchinfluenceas
influence-greedy.Moreover,whenbalancedpicksadifferent
4 Results
node,ourlossisboundedbyu /2.Asu →0withk →∞,
k k
ourlossalsoconvergesto0. Werunexperimentsusingfourreallifesocialnetworksofthe
Proposition4. Balancedselectsnodesthatspreadasmuch homelesspopulationofLosAngeles,providedbyusfrom
influenceasinfluence-greedy,if|Vσ| > n /2and|Vγ| > EricRice,fromtheSchoolofSocialWorkoftheUniversity
k k k
n /2, or as k → ∞. Influence-greedy, on the other hand, ofSouthernCalifornia.Allthenetworksarefriendship-based
k
selectsworsenodesthanbalancedintermsofmappingwith socialnetworksofhomelessyouthwhovisitasocialagency.
|Vσ∩Vγ| Thefirsttwonetworks(A,B)werecreatedthroughsurveys
probability1− k k .Moreover,whenbalancedselects
|Vσ| and interviews. The third and fourth networks (Facebook,
k
a node with worse σ than influence-greedy, the expected
MySpace) are online social networks of these youth cre-
influencelossisboundedbyu /2,which→0ask →∞.
k atedfromtheirFacebookandMySpaceprofiles,respectively.
Computationfortheworkdescribedinthispaperwassup-
Proof. Asbalancedplaysargmax(σ+γ),ifthereisanode
portedbytheUniversityofSouthernCalifornia’sCenterfor
v ∈ VGood, balanced picks v. Influence-greedy, however,
k High-PerformanceComputing(hpc.usc.edu).
selects an arbitrary node in Vσ. Hence, it picks a node
k We run 100 executions per network. At the beginning
v ∈ VGood withprobability |Vkσ∩Vkγ|.Therefore,forallin- of each execution, 4 nodes are randomly chosen to com-
k |Vσ|
k
poseourinitialsubgraph(G ).Asmentioned,weconsider
0
tccisohtiuivydeare,setrrderwiinunanetelgtlkhatanotelhlwolersweapnsy4laeuesc0tdlwe0tkgsonceearooxkgcwnersrsoacttspsurhisahtmeiiaon(uelnGtlldsst,fkga)o.)nweu.seIerWnobsupeeotslthoclwnyeitoaeht(laeitepcnnhrepeaetdetthwntesihdsoiem,nirxkowriseld(ae.searuFsvaltoavteosrienflracdcaobloegaulnnreer--- Influence000001......567890 ● ● ● ●● ●IKBBn●naaflolluaa●wennnlcce●ceededdg●−−egd−r●egecrere●deeyad●syin●g● Knowledge00001.....24680 ● ● ●●● ●IKBBn●naaflolluaa●wennnlcc●eceededd●g−−egd●−rege●crerede●eyads●yin●g
athttp://teamcore.usc.edu/people/sorianom/a16-ap.pdf)we 0.4 0.0
0 10 20 30 40 0 10 20 30 40
showtheindividualresultsforeachnetwork.Inallgraphs, Intervention Number Intervention Number
theerrorbarsshowtheconfidenceinterval,withρ = 0.01. (a)Influence (b)Knowledge
Whenwesaythataresultissignificantlybetterthananother,
wwwoietrhkmρiesa≤:n104w.20i,t11h,8us8tn,al3tei3sss,ti1cn0ao5lte;sdfiogorntiAhfie,crBawn,icsFeea.caTecbhcoeoorskdizianengodftMoeayacShtp-atnecesett-, owledge01..80 ● ● ● ● ● ● ● ● ● ● ● ●
n
respectively.Weevaluateupto40interventions. + K0.6 ● ● Influence−greedy
(“IWnfleumenecaes”u)reanthdeppeerrcceennttaaggee ooffiknnflouwenncenoindetshe(“nKetnwoowrkl- uence 0.4 ● KBBnaaollaawnnlcceeedddg−ed−egcrreeeadsying
edge”)forinfluence-greedy,knowledge-greedy,balancedand Infl0.2
balanced-decreasing(withc(cid:48)2 =1.0,andd=5).Inorderto 0 I1n0tervent2io0n Num3b0er 40
estimatetheexpectedinfluencespread(σ )ofeachnode,we (c)Influence+Knowledge
v
run1000simulationsbeforeeachintervention.Estimatingthe
expectedinfluencethroughsimulationsisacommonmethod Figure2:Resultsof4realworldnetworksacrossmanyinter-
intheliterature.Inourcase,thesimulationsareruninthecur- ventions,forp=0.5andϕ=0.5(uniformdistribution).
rentknownsubgraphG ,althoughtheactualinfluencemay
k
gobeyondG .InfluencednodesinG\G willbeconsidered
k k
whenwemeasureInfluence,butwillnotbeconsideredinour inFigure2(a),theInfluenceobtainedbyinfluence-greedy,
estimationofσ .Concerningtheteachinglistsize(γ ),we balanced,andbalanced-decreasingareverysimilar.Infact,
v v
considerittoholdthenumberofnewnodesthatwouldbe out of all 40 interventions, their result is not significantly
learnedifvisselected,forbalancedandknowledge-greedy differentinanyofthem(andtheyaresignificantlybetterthan
(i.e.,perfectknowledge).Forbalanced-decreasing,wecon- knowledge-greedyinaround75%oftheinterventions).This
siderγ toholdthefullteachinglistsize,includingnodes showsthatbalancedisabletosuccessfullyspreadinfluence
v
thatarealreadyknown(i.e.,partialknowledge).Therefore, inthenetwork,whileatthesametimemappingthegraph.We
wecanevaluateifbalanced-decreasingapproximateswell canalsonoticethataperfectknowledgeaboutthenumberof
balanced,whenperfectknowledgeisnotavailable. newnodesintheteachinglistsisnotnecessary,asbalanced-
Wesimulatetheteachinglists,sincetherearenorealworld decreasingobtainedcloseresultstobalanced.
dataavailableyet(weonlyhavedataabouttheconnections Figure2(b)showstheresultsintermsofKnowledge.All
inthefourreallifesocialnetworks).Westudytwomodels: algorithmsclearlyoutperforminfluence-greedywithstatisti-
(i)uniform,whichfollowstheassumptionsofourtheoretical calsignificance.Moreover,theresultforknowledge-greedy,
analysis;(ii)powerlaw,whichconsidersthatnodesaremore balancedandbalanced-decreasingarenotsignificantlydif-
likely to teach us about others which are close to them in ferent in any of the interventions. This shows that we are
thesocialnetworkgraph.Wepresentthesecondmodelto abletosuccessfullymapthenetwork(aswellasknowledge-
showthatourconclusionsholdirrespectiveoftheuniform greedy), but at the same time spreading influence success-
assumption.Foreachnode,wedecidewhetheritwillhavea fully over the network (as well as influence-greedy), even
non-emptyteachinglistaccordingtoaprobabilityϕ.Werun inthepartialknowledgecase.Hence,theindependenceof
experimentsusingdifferentcombinationsofϕ,probability objectives hypothesis seems to hold at most interventions
ofinfluencep,andc1andc2values. in the networks, since we could maximize both objectives
simultaneously,aspredictedinProposition4.Givenenough
4.1 UniformModel interventions,however,influence-greedyisalsoabletomap
Undertheuniformmodel,ifanodehasateachinglist,we thenetwork,aswediscussedinProposition2.
fix its size according to a uniform distribution from 0 to It is also interesting to note that even though influence-
0.5×|V|.Eachnodeinthegraphisalsoequallylikelytobe greedyhasmuchlessinformationaboutthenetwork(with
intheteachinglistofanodev .Weconsiderheretheteaching significantly lower mapping performance in around 16 in-
i
listandtheneighborlisttobeindependent,aspeoplemay terventions), it is still able to perform as well as the other
knowothersthattheycannotinfluence,andtheymayalso algorithms in terms of Influence. Observation 1, however,
not tell us all their connections, as described before. The showedthatitsInfluenceperformancecanbearbitrarilylow.
case where the teaching list and the neighbor list are not Aswediscusslater,forsomeparametrizationsweactually
independentisconsideredinSection4.2. found that influence-greedy has significantly lower results
Werunseveralparametrizations.Figure2showstheresult thantheotheralgorithmsintermsofInfluenceaswell.
at each intervention for ϕ = 0.5 and p = 0.5. As we see Inordertocomparetheresultsacrossdifferentparametriza-
tions, we calculate the area under the curve (AUC) of the e40
v
graphs.Thecloserthecurvesareto1.0thebetter,hencean ur30
C
AUCof39(thatis,alwaysat1.0acrossall40interventions) der 20 Influence−greedy
wouldbean“ideal”result.InFigure3(a)weshowtheresults Un Knowledge−greedy
forafixedinfluenceprobabilityvalue(p=0.5),butdifferent ea 10 Balanced
Ar Balanced−decreasing
teachingprobability(ϕ)values.Firstwediscusstheresults 0
ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
in terms of Influence (left-hand side of the graph). As we Influence Knowledge
cansee,exceptforknowledge-greedy,allalgorithmsobtain (a)Changingteachingprobability
very similar results. However, for ϕ = 0.1, the Influence
forbalancedandbalanced-decreasingisslightlybetterthan e40
v
influence-greedy,intheborderlineofstatisticalsignificance ur30
C
(ρ = 0.101and0.115,respectively).Moreover,wecansee der 20 Influence−greedy
thatϕdoesimpacttheinfluencethatweobtainoverthenet- Un Knowledge−greedy
work,althoughtheimpactisnotbig.Forinfluence-greedy, ea 10 Balanced
Ar Balanced−decreasing
from ϕ = 0.5 to ϕ = 1.0, the difference is only statisti- 0
p=0.1 p=0.5 p=0.1 p=0.5
cally significant with ρ = 0.092. However, from ϕ = 0.1 Influence Knowledge
to ϕ = 0.5 the difference is statistically significant with (b)Changinginfluenceprobability
ρ = 3.26×10−27.Similarly,forallotheralgorithmsthere
isasignificantdifferencefromϕ = 0.1toϕ = 0.5,while Figure3:ResultsofInfluenceandKnowledgefordifferent
fromϕ = 0.5toϕ = 1.0thedifferenceisonlysignificant teachingandinfluenceprobabilities(uniformdistribution).
withρ<0.1(exceptforknowledge-greedy,itsdifferenceis
notsignificantbetweenϕ=0.5andϕ=1.0). 8
Let’s look at the results in terms of Knowledge, on the Influence−greedy Balanced
6 Knowledge−greedy Balanced−decreasing
right-hand side of Figure 3 (a). We can see that ϕ has a et
gr4
muchbiggerimpactinourmapping,asexpected.Knowledge- e
R
greedy, balanced and balanced-decreasing are all signifi- 2
cantlybetterthaninfluence-greedy.However,wecannotice 0 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
thatthedifferencebetweeninfluence-greedyandtheother p = 0.1 p = 0.5
algorithmsdecreasesasϕincreases.Similarly,whencompar-
Figure4:Regretfordifferentteachingandinfluenceproba-
ingknowledge-greedy,balancedandbalanced-decreasing,
bilities(uniformdistribution).Lowerresultsarebetter.
wecannoticethatthedifferencebetweenthealgorithmsalso
decreases as ϕ increases. For both ϕ = 0.1 and ϕ = 0.5,
however, the algorithms are not significantly different. In-
terestingly, when ϕ = 1, because of the lower variance, inthefigure,thelowertheresultthebetter.Ontheleft-hand
knowledge-greedy and balanced become significantly bet- sideweshowtheregretforp = 0.1anddifferentϕvalues,
ter than balanced-decreasing, even though the differences whileontheright-handsideweshowforp=0.5.Allalgo-
betweenthealgorithmsdecreases. rithms(exceptknowledge-greedy)haveasimilarregret,asit
In Figure 3 (b), we keep ϕ = 0.5, and change p. In the wouldbeexpectedbasedonthepreviousresults.However,
left-handsideweseetheresultsforInfluence.Asexpected, wecannoticeherethattheregretforbalancedandbalanced-
thereisclearlyasignificantdifferencewhenpchangesfrom decreasingwhenϕ=0.1andp=0.5isactuallylowerthan
0.1 to 0.5. However, we can notice that the difference be- influence-greedy. The difference is statistically significant,
tween the algorithms does not change significantly when withρ=0.069andρ=0.092,respectively.Hence,wecan
pchanges.Inbothcases,thedifferencesbetweeninfluence- actually have a significantly better influence on the social
greedy,balancedandbalanced-decreasingarenotsignificant. networkgraphthanthetraditionalgreedyalgorithm,when
Additionally, in both cases all algorithms are significantly theteachingprobabilityislow.However,whenϕ=0.1and
betterthanknowledge-greedy.IntermsofKnowledge(right- p=0.1,eventhoughtheregretforbalancedandbalanced-
handsideofthefigure)weseethattheinfluenceprobability decreasingisstilllowerthaninfluence-greedy,itisnotsig-
hasnoimpactinanyalgorithm,asitwouldbeexpected.For nificantanymore(asthereislargervariancewhenp=0.1).
allalgorithms,thedifferencebetweenp=0.1andp=0.5 Additionally, we note that the difference in regret for bal-
isnotstatisticallysignificant. ancedandbalanced-decreasingisalwaysnotsignificant.It
Wealsocomparetheregret obtainedbythedifferental- is also interesting to note that for some parametrizations,
gorithms at different influence probabilities and teaching theregretofinfluence-greedyisactuallycloseto0,which
probabilityvalues.First,weruntheinfluence-greedyalgo- meansthatinsomecaseslackofknowledgeofthefullgraph
rithm,butconsideringthatweknowthefullgraph(thatis, doesnotsignificantlyharmtheinfluenceperformance.When
G := G). Although that solution is not optimal, it is the p=0.1,andϕ=0.5orϕ=1.0,theregretisnotsignificant
k
bestknownapproximationoftheoptimal,hencewecallit (ρ = 0.410, and ρ = 0.78, respectively). For p = 0.5 and
“perfect”.WecalculatetheAUCforperfect,anddefinethe ϕ=1.0,theregretisintheborderlineofnotbeingsignificant
regretofanalgorithmxas:AUC −AUC .Wean- (ρ=0.102).Inallothercases,theregretissignificant.
Perfect x
alyzetheregretintermsofInfluenceinFigure4.Notethat, We discussed 4 algorithms, but our framework can ac-
tually generate a variety of behaviors by using differ-
ent c and c values. We tested 6 more combinations: 1.0 1.0
{dtHh(iede0n.pnc5reo,1e,2tvfi)oi,onb(dus1ies,nr0agv2l.eg5ao)sg,riogi(t1onh,dimfi1pc.s5aair)nna,tm(td1hei,eft2rffie)oz,rauetrn(i2ocs,neo0sco.ii5fan)lc,1cn(oe2amtn,w1dpo)ac}rr2k,issdobognuertaswpnwihtohset. Influence0000....6789 ● ● ● ●● ●IKBn●naflolua●wennlce●cededg●−eg−r●geree●deyd●y● ● Knowledge000...468 ● ● ●● ● IK●nnflo●uwe●nlecd●eg−e●g−rge●reed●eyd●y ●
seemtobeacrucialproblemforthebalancedalgorithm. 0.5 Balanced−decreasing 0.2 ● Balanced
Balanced−decreasing
0.4 0.0
0 10 20 30 40 0 10 20 30 40
4.2 PowerLawModel Intervention Number Intervention Number
(a)Influence (b)Knowledge
Inordertoshowthatourconclusionsstillholdunderdifferent
models, we also run experiments considering a power law
distributionfortheteachinglists.Thepowerlawisavery e1.0
suitablemodelforarangeofrealworldphenomena.Infact, wledg0.8 ● ● ● ● ● ● ● ● ●
Andriani and McKelvey (2007), in a very comprehensive no ● ●
lairteermatoudreelseudrvineyt,hleisltiste8r0atduirfefebryenptokwinedrslaowfpdhiesntroibmuetinoanws,haincdh ce + K0.6 ● ●● IKnnflouwenlecdeg−eg−rgereedeydy
n0.4
halfofthemaresocialphenomena.Forexample,ithasbeen ue ● Balanced
showntobeagoodmodelforsocialnetworks,co-authorships Infl0.2 Balanced−decreasing
0 10 20 30 40
networks, the structure of the world wide web and actor Intervention Number
movieparticipationnetworks.Apowerlawmodelalsoseems (c)Influence+Knowledge
suitableinourcase,aswecanexpectthatapersonwillbe
Figure5:Resultsof4realworldnetworksacrossmanyinter-
verylikelytoteachusaboutthepeoplewhoshehasadirect
ventions,forp=0.5andϕ=0.5(powerlawdistribution).
connection with, and less and less likely to report people
thatarefurtherawayinthegraph.Hence,whengenerating
theteachinglistofanodev inourexperiments,eachnode
i
v (v (cid:54)= v ) will be in its teaching list according to the wasnotsignificantlyworsethantheotheralgorithmsafter
o o i
followingprobability:p :=(a−1.0)×h−a,where1.0< iteration20(ρ≤0.1).Hence,asexpected,influence-greedy
o o
a ≤ 2.0,andh istheshortestpathdistancebetweennode becomesrelativelyworsethantheotheralgorithmswhenwe
o
v andv .a−1.0representstheprobabilityofaneighbor assumeamodelwheremappingisharder.
i o
node v (i.e., h = 1) being selected. If node v and v We calculate the AUC, in order to compare different
o o i o
arecompletelydisconnected,weseth := |V|.Underthis parametrizations. Figure 6 (a) shows the result for a fixed
o
modeltheprobabilityofapersonteachingusaboutanother influence probability value (p = 0.5), and different teach-
isalwaysstrictlygreaterthan0,eventhoughitmaybevery ingprobability(ϕ)values.Asbefore,exceptforknowledge-
smalliftherespectivenodesareverydistantinthegraph. greedy, all algorithms in general obtain similar results in
Wefixa=1.8(80%probabilityofeachofanodes’neigh- termsofInfluence.Wenotice,however,thatforϕ=0.1,the
borsbeinginitsteachinglist).Weshowresultsfora=1.2 resultforbalancedandbalanced-decreasingisactuallysig-
in the appendix, for the interested reader (and our conclu- nificantlybetterthaninfluence-greedy(ρ=0.064and0.038,
sionsstillholdinthealternativeparametrization).Similarly respectively).Again,wealsonoticethatϕsignificantlyim-
asbefore,Figure5showstheresultateachinterventionfor pactstheinfluencethatweobtainoverthenetwork,although
ϕ=0.5andp=0.5.Aswecansee,ourmainconclusions theimpactissmall.Forallalgorithms,theimpactissignif-
still hold in the power law model. The Influence obtained icant from ϕ = 0.1 to ϕ = 0.5. For influence-greedy, the
byinfluence-greedy,balanced,andbalanced-decreasingare differenceisstatisticallysignificantwithρ=1.39×10−31,
very similar. Out of 40 interventions, their results are not while for knowledge-greedy, balanced and balanced de-
significantlydifferentin39(ρ≤0.05),andtheyaresignif- creasing, there is statistically significant difference with
icantlybetterthanknowledge-greedyinaround60%ofthe ρ = 4.529×10−6, 3.258×10−14 and 8.429×10−20,re-
interventions(ρ≤0.05). spectively.However,theimpactofϕincreasingfrom0.5to
Thistime,however,balanced-decreasingobtainedworse 1isnotthatsignificantforknowledge-greedy,balancedand
resultsthanbalancedintermsofKnowledge.Althoughup balanced-decreasing (ρ = 0.41,0.02,0.03, respectively),
to iteration 4, knowledge-greedy, balanced and balanced- whileforinfluence-greedythechangehassignificantimpact
decreasingarenotsignificantlydifferent;bothknowledge- (ρ=10−6).IntermsofKnowledge,wecanseethatallalgo-
greedyandbalancedaresignificantlybetterthanbalanced- rithmsaresignificantlybetterthaninfluence-greedyforallϕ
decreasingafterthatiteration.ItishardertoobtainKnowl- values(withρ≤3.464015×10−19).However,thistimewe
edge under the power law model than under the uniform noticethatknowledge-greedyandbalancedaresignificantly
model (all algorithms converge slower to 1.0 than before). betterthanbalanced-decreasingforallϕ.Asmentioned,a
Hence,balanced-decreasingwouldrequireaslowerdecay differentdecayspeeddisnecessaryinthiscase.
speed(i.e.,higherd)inthiscase,inordertoperformbetter. In Figure 6 (b), we show different values of p for ϕ =
Wecanalsonoticethatallalgorithmsaresignificantlybet- 0.5.Asbefore,intermsofInfluencethedifferencebetween
terthaninfluence-greedyinalliterations,intermsofKnowl- influence-greedy,balancedandbalanced-decreasingisnot
edge.Notethatundertheuniformmodel,influence-greedy significant, and all algorithms are significantly better than
e40 Weshowtheoreticallyandexperimentallythatanalgorithm
v
ur30 whichlocallymaximizesbothinfluenceandknowledgeper-
C
Under 20 IKnnflouwenlecdeg−eg−rgereedeydy foofrminsflausenwceel,laasndanaisnflwueelnlcaes-oanklynogwreleeddygael-goonrliythgmreiendtyermaps-
ea 10 Balanced proachintermsofknowledge.Wepresentanapproximation
Ar Balanced−decreasing
0 ofouralgorithmthatgraduallydecreasestheweightgivento
ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
Influence Knowledge knowledge-gain,whichrequiresfewerassumptions.Results
(a)Changingteachingprobability showthattheapproximationworkswell,andallalgorithms
canevensignificantlyinfluencemorenodesthanthetradi-
e40 tionalgreedyinfluencemaximizationalgorithmwhennodes
v
ur30 havealowknowledgeprobability.
C
nder 20 Influence−greedy Acknowledgments: ThisresearchwassupportedbyMURI
Area U10 KBBnaaollaawnnlcceeedddg−ed−egcrreeeadsying wgroaunltdWlik9e1t1oNtFha-1n1k-E1-r0ic3R32ic,eafnodrtbhyesIoUcSiaSlTnFe.twTohreksaduathtao.rs
0
p=0.1 p=0.5 p=0.1 p=0.5
Influence Knowledge References
(b)Changinginfluenceprobability
Alon,N.;Awerbuch,B.;Azar,Y.;Buchbinder,N.;andNaor,
Figure6:InfluenceandKnowledgefordifferentteachingand J.S. 2009. Theonlinesetcoverproblem. SIAMJournalon
influenceprobabilities(powerlawdistribution). Computing39(2).
Andriani, P., and McKelvey, B. 2007. Beyond gaussian
8 averages:Redirectinginternationalbusinessandmanagement
Influence−greedy Balanced
6 Knowledge−greedy Balanced−decreasing researchtowardextremeeventsandpowerlaws. Journalof
et InternationalBusinessStudies38(7).
gr4
Re Badanidiyuru, A.; Mirzasoleiman, B.; Karbasi, A.; and
2 Krause, A. 2014. Streaming submodular maximization:
0 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1 Massivedatasummarizationonthefly. InKDD.
p = 0.1 p = 0.5 Bateni,M.;Hajiaghayi,M.;andZadimoghaddam,M. 2013.
Submodularsecretaryproblemandextensions. TALG9(4).
Figure7:Regretfordifferentteachingandinfluenceproba-
bilities(powerlawdistribution).Lowerresultsarebetter. Cohen,E.;Delling,D.;Pajor,T.;andWerneck,R.F. 2014.
Sketch-basedinfluencemaximizationandcomputation:Scal-
ingupwithguarantees.Technicalreport,MicrosoftResearch.
knowledge-greedy. In terms of Knowledge, the influence Golovin,D.,andKrause,A. 2010. Adaptivesubmodularity:
probabilitydoesnotaffectknowledge-greedy,balancednor Anewapproachtoactivelearningandstochasticoptimiza-
balanced-decreasing significantly, as expected. This time, tion. InCOLT.
however,influence-greedyobtainsasignificantlybetterresult Grandoni, F.; Gupta, A.; Leonardi, S.; Miettinen, P.;
forp=0.1thanforp=0.5.Thismayhappenbecausefor Sankowski,P.;andSingh,M. 2008. Setcoveringwithour
p=0.1influence-greedyhasahighertendencyofselecting eyesclosed. InFOCS.
nodeswithahighnumberofneighbors,whichalsotendsto
Kempe,D.;Kleinberg,J.;andTardos,E. 2003. Maximizing
betheoneswithlargeteachinglists.Notethatthisdoesnot
thespreadofinfluencethroughasocialnetwork. InKDD.
happenwhentheteachingandneighborlistsareindependent.
Khuller,S.;Moss,A.;andNaor,J.S. 1999. Thebudgeted
Figure7showstheregret(lowerresultsarebetter).Wecan
maximumcoverageproblem. InformationProcessingLetters
noticesimilarresultsasbefore:allalgorithmshavesimilar
70(1).
regret(exceptforknowledge-greedy),andtheregretforbal-
Marsden,P.V. 2005. Recentdevelopmentsinnetworkmea-
ancedandbalanced-decreasingwhenϕ=0.1andp=0.5
surement. InCarrington,P.J.;Scott,J.;andWasserman,S.,
isagainsignificantlylowerthaninfluence-greedy(ρ=0.019
eds.,Modelsandmethodsinsocialnetworkanalysis.Cam-
and0.009,respectively).Thistime,however,wecannotice
bridgeUniversityPress.
thatforsomeparametrizationsbalanced-decreasingisactu-
allythealgorithmwiththelowestregret.Forp = 0.5and Roijers,D.;Vamplew,P.;Whiteson,S.;andDazeley,R.2013.
ϕ=0.5,balanced-decreasingisbetterthanbalancedwith A survey of multi–objective sequential decision–making.
ρ = 4.7×10−4.Hence,eventhoughbalanced-decreasing JournalofArtificialIntelligenceResearch48.
performedrelativelyworsethanundertheuniformmodelin Valizadegan,H.;Jin,R.;andWang,S. 2011. Learningto
termsofKnowledge,itisactuallythebestalgorithminterms tradeoffbetweenexplorationandexploitationinmulticlass
ofInfluenceforsomeparametrizations. banditprediction. InKDD.
Yadav,A.;Marcolino,L.S.;Rice,E.;Petering,R.;Winetrobe,
5 Conclusion H.;Rhoades,H.;Tambe,M.;andCarmichael,H. 2015. Pre-
Weintroducedthenovelproblemofsimultaneouslyinfluenc- ventingHIVspreadinhomelesspopulationsusingPSINET.
ingandlearningthegraphof(i.e.,mapping)asocialnetwork. InIAAI.
A Appendix
40
e
A.1 Resultsforeachnetwork urv30
C
SinceinSection4wepresentedourresultsacross4differ- der 20 Influence−greedy
n
entsocialnetworkgraphs,inthissectionwepresentthose U Knowledge−greedy
individuallyforeachnetwork,fortheinterestedreader.As- ea 10 Balanced
Ar Balanced−decreasing
sumingauniformmodelfortheteachinglists,wecanseethe 0 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
resultsfornetworkAinFigures8,9,10;fornetworkBin Influence Knowledge
Figures11,12,13;fortheFacebooknetworkinFigures14, (a)Changingteachingprobability
15,16;andfinallyfortheMySpacenetworkinFigures17,18,
40
19.Assumingapowerlawmodel,wecanseetheresultsfor e
v
networkAinFigures20,21,22;fornetworkBinFigures23, Cur30
24,25;fortheFacebooknetworkinFigures26,27,28;and der 20 Influence−greedy
finallyfortheMySpacenetworkinFigures29,30,31.Aswe Un Knowledge−greedy
cansee,theresultsforeachnetworkshowsimilartendencies ea 10 Balanced
Ar Balanced−decreasing
astheresultsacrossallnetworkspresentedinthemainpaper. 0
p=0.1 p=0.5 p=0.1 p=0.5
Influence Knowledge
(b)Changinginfluenceprobability
nfluence00001.....67890 ● ● ● ●● ●IKn●nflou●wenle●cdeg●−eg−r●geree●deyd●y● ● nowledge0001....4680 ● ●●● ●IKn●nflou●wenl●ecde●g−eg●−rge●reed●eyd●y ● Ffuonigrifudorirefmf9er:deRinsettrstiuebalutcstihooinnf.gInaflnudenincfleuaenndceKpnroowblaebdiglietiiens,naestswuomrkinAg
I0.5 BBaallaanncceedd−decreasing K0.2 ● BBaallaanncceedd−decreasing 8
0.4 0.0 Influence−greedy Balanced
0 10 20 30 40 0 10 20 30 40 6 Knowledge−greedy Balanced−decreasing
Intervention Number Intervention Number et
(a)Influence (b)Knowledge egr4
R
2
wledge01..80 ● ● ● ● ● ● ● ● ● ● ● 0 ϕ=0.1 ϕ=p 0=. 50.1 ϕ=1 ϕ=0.1 ϕp == 00..55 ϕ=1
o ●
n
+ K0.6 ● ● Influence−greedy Figure 10: Regret in network A for different teaching and
e Knowledge−greedy influenceprobabilities,assuminguniformdistribution.
enc0.4 ● Balanced
u Balanced−decreasing
Infl0.2
0 10 20 30 40
Intervention Number
Ff0o.i5rg,uinaresflsu8ue:mnRcineegspuurltonsbif(faocob)rrmiIlnnifltedyutiwesptnorc=irebku+0At.iK5oa,nncot.erwoalsceshdgimneganpyroinbtaebrivleitnytiϕon=s, Influence0000001.......4567890 ● ● ● ●● ●IKBBn●naaflolluaa●wennnlcce●ceededdg●−−egd−r●egecrere●deeyad●syin●g● Knowledge00001.....24680 ● ● ●●● ●IKBBn●naaflolluaa●wennnlcc●eceededd●g−−egd●−rege●crerede●eyads●yin●g
0.3 0.0
0 10 20 30 40 0 10 20 30 40
A.2 Additionalresultsforpowerlawdistribution Intervention Number Intervention Number
(a)Influence (b)Knowledge
In Section 4.2 we presented results for the power law dis-
tributionfora = 1.8.Inthissection,wepresentresultsfor
abei=ng1i.n2a(ttheaacthisin,gealicsht).nWeigehsbhoorwhiansFaig2u0r%e3p2rothbeabreilsiutyltoaft wledge01..80 ● ● ● ● ● ● ● ● ● ● ●
eshaochwitnhteerAveUnCtiorensfuoltrsϕfo=rd0i.f5fearenndtppa=ram0.e5t.rIiznaFtiiognusreof3p3,awnde e + Kno0.6 ● ●● IKnnflouwenlecdeg−eg−rgereedeydy
ϕ.Finally,wepresenttheregretforthiscaseinFigure34. nc0.4 Balanced
e
Aswecansee,ourmainconclusionsstillholdforadifferent Influ0.2 ● Balanced−decreasing
parametrizationofa. 0 10 20 30 40
Intervention Number
(c)Influence+Knowledge
Figure11:ResultsfornetworkBacrossmanyinterventions,
forinfluenceprobabilityp=0.5,teachingprobabilityϕ=
0.5,assuminguniformdistribution.
40 40
e e
v v
ur30 ur30
C C
der 20 Influence−greedy der 20 Influence−greedy
n n
U Knowledge−greedy U Knowledge−greedy
ea 10 Balanced ea 10 Balanced
Ar Balanced−decreasing Ar Balanced−decreasing
0 0
ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
Influence Knowledge Influence Knowledge
(a)Changingteachingprobability (a)Changingteachingprobability
40 40
e e
v v
ur30 ur30
C C
der 20 Influence−greedy der 20 Influence−greedy
n n
U Knowledge−greedy U Knowledge−greedy
ea 10 Balanced ea 10 Balanced
Ar Balanced−decreasing Ar Balanced−decreasing
0 0
p=0.1 p=0.5 p=0.1 p=0.5 p=0.1 p=0.5 p=0.1 p=0.5
Influence Knowledge Influence Knowledge
(b)Changinginfluenceprobability (b)Changinginfluenceprobability
Figure12:ResultsofInfluenceandKnowledgeinnetworkB Figure15:ResultsofInfluenceandKnowledgeinFacebook
fordifferentteachingandinfluenceprobabilities,assuming network for different teaching and influence probabilities,
uniformdistribution. assuminguniformdistribution.
8
14
Influence−greedy Balanced
6 Knowledge−greedy Balanced−decreasing 12 Influence−greedy Balanced
et et10 Knowledge−greedy Balanced−decreasing
gr4 gr8
Re Re6
2 4
2
0 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1 0 ϕ=0.1 ϕ=0.5 ϕ=1 ϕ=0.1 ϕ=0.5 ϕ=1
p = 0.1 p = 0.5 p = 0.1 p = 0.5
Figure 13: Regret in network B for different teaching and Figure16:RegretinFacebooknetworkfordifferentteaching
influenceprobabilities,assuminguniformdistribution. andinfluenceprobabilities,assuminguniformdistribution.
nfluence00001.....67890 ● ● ●● ●IKn●nflou●wenle●cdeg●−eg−r●geree●deyd●y● ● nowledge0001....4680 ● ●●● ●IKn●nflou●wenl●ecde●g−eg●−rge●reed●eyd●y ● nfluence00001.....67890 ● ● ● ●● ●IKn●nflou●wenle●cdeg●−eg−r●geree●deyd●y● ● nowledge0001....4680 ● ●●● ●IKn●nflou●wenl●ecde●g−eg●−rge●reed●eyd●y ●
I0.5 ● BBaallaanncceedd−decreasing K0.2 ● BBaallaanncceedd−decreasing I0.5 BBaallaanncceedd−decreasing K0.2 ● BBaallaanncceedd−decreasing
0.4 0.0 0.4 0.0
0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40
Intervention Number Intervention Number Intervention Number Intervention Number
(a)Influence (b)Knowledge (a)Influence (b)Knowledge
edge1.0 ● ● ● ● ● ● ● ● ● ● ● edge1.0 ● ● ● ● ● ● ● ● ● ● ●
wl0.8 ● wl0.8 ●
o o
+ Kn0.6 ● ● Influence−greedy + Kn0.6 ● ● Influence−greedy
ence 0.4 ● KBnaolawnlceeddge−greedy ence 0.4 ● KBnaolawnlceeddge−greedy
u Balanced−decreasing u Balanced−decreasing
Infl0.2 Infl0.2
0 10 20 30 40 0 10 20 30 40
Intervention Number Intervention Number
(c)Influence+Knowledge (c)Influence+Knowledge
Figure14:ResultsforFacebooknetworkacrossmanyinter- Figure17:ResultsforMySpacenetworkacrossmanyinter-
ventions,forinfluenceprobabilityp=0.5,teachingproba- ventions,forinfluenceprobabilityp=0.5,teachingproba-
bilityϕ=0.5,assuminguniformdistribution. bilityϕ=0.5,assuminguniformdistribution.