翻译以.原文和在同一文件中前

第五章模型的学述信息或是在网络上如何的，更准确的说，是信息或如何到一个获取u对v产生的影响。那么Puv是该模型的关键参数。这些研究聚焦于如何找到一个小的节点集合去激活，并能使得这些节点被激活时，达到最大化。live-arc模型的对应，对其也有一个概率解释。当不在一个具体模在的。例如，一个用户可能会去并评价《遗忘》这部，这仅仅是由于个人网络中最大化》一文以了解的细节。在统一的模型中，简单假设所有影0.01所有邻居节点相同的。显然，这两个假设都是有问题的。0.001。这种模型的想法是，当节点传出的影和0.1对应高的节点。在的级联模型中，我们定义一个边相关的概率(u,v)为puv=1/din(v)，其中din(v)是节点v的入度。与上述两个模型相比，这种模型对节点间的区别有了更大的分辨率：与入度高的节点相比，入度低的节点从他们的邻居节点中可获取更高的。因此IC在早期的一项开拓性工作中，Saito等人关于对IC模型的问题进行了影响概率一个信息扩散的集合是节点集的序列{D(0),D(1),„,D(T)}D(i)∩D(j)=，i≠j。T为t，D(t)D(0)合=D1,Dn我们如何去了解其影响概率。对于一个给定的节点u，令Nout(u)={v|(u，v)∈E}表示其子节点，Nin(u)={v|(u，v)∈E}表示其父节点，从父节点传在集合<Ds(0),Ds(Ts)>中，对节点v∈Ds(t+1)，Nin(v)∩Ds(t)中的一个节点必须已成功激活v。假设对于任意的v∈Ds(t+1)，Nin(v)∩Ds(t)是一个非空集合。给定一个集合，我们始终可以这么认为，所有的节点v∈Ds(t+1)，他们在Ds(t)中没有任*v6+，D2(1*v7+，D2(2*v8v9+t时间时，w的父节点中至少有一个处于激活状态并成功ww的机会，且激活以任意顺序发生在t时刻。令Cs(t)=∪τ≤tDs(τ)表示在t时刻活跃的节点集合，则观察集合Ds关于θ=w可能性由以下式子给出，这是对边(v,w)的直观描述，若wDs(t1)wt时刻处于激状态w；若wDs(t1)wt时刻均未成功激活该节点。这是我们可以通过集合Ds来研究概率pvw的唯一信息。特别的，该情况下(a)vDs(t)且wCs(t)和(b)vDs(t)对影响概率pvw无关。同时观察组集关于θ的联合概率S=*D1,…,Dn并记录其可能性，我们得到下式上面的方程中Ps(t+1)可被替换为Ps由于其激活时间已知且可被表示。如下式所示，w wIC模型，从研究影响概率的问题，对于一个给定的集合，被正式作为̂ŝ*̂vw+ 考虑在时间t对应父节点v变为活跃节点并在t+1时间成功激活或未激活其子节点w的̂vw/̂ŝvw/̂ 这个对数相关的函数关于这些潜在变量的条件分布的期望值，给定集合S=*D1设SS)Svw成功（或失败 ; vDs(t且wCs(t1)，(v,wE的集合。注意到在一般情况下S:|+S;|=n是不必要的，其中n=|S| 去研究影响概率，我们只是反复去评估方程(5.5)和(5.6)题可以通过后处理等方法得到改善，如Mathioudakis等人的研究中指定一个IC模型参数的向量的SPINE方法。具体而言，SPINEIC模的可能性最大化的增加，直到与整个模型有着同样的可能性，有着原始模型的30%到在本节中，我们描述一种对于阈值模型的研究的参数。在Kempe等人2003GTGTLTv所有阈值θv∈(0,1)是随机的。一个节点上主动的邻居节点的由一个单调函数约束，fv：2Nin(v)→,0,1-，其中Nin(v)v所有入邻居的集合。若S⊆Nin(v)v在时tf(S)v在网络中的影响：因此，当f(S)>θvvt+1时2.16联的权重的函数。Goyal等人研究了关于社交网络的随着行动轨迹的输入数据集以研究边的权重的问题。在下文中描述这种方法。G=(V,E,T)表示一个无向的社会图(V,E)T：EΝ，对于每条边表示被形成的边。假设底层的图是无向的简户，行为，时间）⊆V×Α×N代表用户在网络中执行的所有操作的记录集合。元组(ua,t∈actionut进行的操作。为简单起见，假设用户执行任何行动u更早进行。显然，|Au|v|=|Au|+|Av|−|Au&v|。定义5.1行为我们说一个动作a∈A从用户u到v当且仅当：(1)(u,v)∈E；(2)∃(u,a,tu),(v,a,tv)∈actions且tu<tv；(3)T(u,v)≤tu。当这些成立时我们描述为prop(a,u,v,Δt)，其中Δt=tv−tu。u和v必须在边(u,v)形成后按照此顺序执行操作，上述概念描述了图的自然定义5.2图给定一个社会网络和动作记录，每一个动作引起的图PG(a.V(a)，E(a)/定义如下：V(a=*v|∃t:(v,a,t)∈actions+，且当prop(a,uv,成立时有一条边uvE(a)如图5.2中的例子，顶点描绘了一个无向社交网络和从操作记录；边的表示了边形边，由于该边(y,z)的形成是在时间12ya中进行操作的时间10。图5.2无向社交网络，动作记录，及行动代表行动图的形式。这就是我们所拥有的，根据这些数据来了解函数p：E→,0,1×,0,1-对于边(uv)∈EPuvPvuv是网络中的任意节点，S是其当前活跃的邻居集合。特别的，这些SSGT模型的阈值函数fv(S（2.16）来建模。直观的说，fv(·)可视为边的概率的函数*Puv|u∈S+。因为它在最大化的情况下使用，函数fv(·)需要与所用于的最大化模型一致。正如第2章所讨论的，这些fv(·)满足以下特性：(1)单调性：当S⊆T时，fv(S<fv(T)；(2)子模：对S⊆Tx∈V\T，有fv(T∪*x+)−fv(T)<fv(S∪*x+)vSv该概率是相互独立的。这会导致可以证明有如上定义的函数fv(·)是单调的和子模函数（Goyal2010年的相关文章2.17。另外，此功能有很好的可被逐步计算的特性。邻居节点w∉S已经被激活。我们可以这样计算fv(S∪*u+)：vu成为v的邻居。定义了如何使用各种静态模型来估计边概率。为了验证这些模型，我们需要计算给定节5.3.2是否保持静态了智能，起初我们会觉得应该自己也去尝试买一下。但随着时间的推移，这种由朋友在这具体行动的背景下发挥的影响可能会，这是合理的。虽然这个例子很直观，Goyal2010年在一个真实数据集上的一种测量的结果。他们测量的是对Flickr中邻居之间在不同的时间间隔动作的次数与。图5.3显示了邻居之间的动作表现之间的时间间隔行动的数量。分布显示在三个层次粒度：第一个小时内每10分钟，第一周每隔一小时，每星期一个大的延迟行动。在所有情况下，图5.3共动与两个用户执行操作之间的时间差的频率（a第一小时内的每10的情况c）这里p0uvut=tu时刻之后的pt衰减指数。参数p0可以使用任何静态模型包括伯努利或杰卡德模型，无论有或 没有局部的估计。参数τuvuvttt 说，v的激活提供max*fv+≥fv(·)，其中θv的激活阈值。计算max*fv upuvvtuu被激活的时间，τuv是一个S应该只占被影响的邻居，而不仅是活跃的邻居。特别的，如果一个活跃的邻居u刚刚成为无的，我们可以更新概率为连续时间模型不够精确，但有增加的可计算性的优点。我们所的常数概率Puv可以的概率Puv有部分可被估计为12种模型：一个模型可以是静态的或时间相关影响，同样的一切行动都同样受到影响。这是一个现实的假设么？Goyal等人在2010我们可以模拟在上述方程中使用的在离散时间模型和在任何合适的方式中参数τuv的值，一种可能性是把它定义为一个动作从用户u到v所经过的平均时间：Uaainfl(a)的行动，相对于其对于一个社交网络，我们需要研究面小节中模型参数。社交网络可以包都是重要的。为方便、高效处理，我们假设行动操作由动作ID分组，每个组由时间进算法14介绍了一个学习模型的算法，需要的是这是一个一般性质的算法用于学习前面讨论过的所有12个模型。以前过的元组(u,a,tu)且u<v并在时间tv前已存在边(u,v)，我们认为，该行动可能从u到v并更新所有相关参数。当考虑部分图时，我们也更新了它。15146步中需要tvtu≤τuv。为简单算法来验证多个模型的完整细节，读者可参考Goyal等人2010年的相关。1,450,34740,562,923条边的子图。其中在该图中的多个连接的组成率(TPR)对误判率(FPR)TPRTP/(TPFN)FPR=FP/(FP+TN)是正确例的数量（否定）和FN(FP)是误判例（肯定）的数量。在我们的设定中，我们定义这些量如下。TP为，当至少有一个用户的邻居在该用户之前执行操作的数量，根据模型可以估计它执行的动作的。FP为当用户不进行操作，至少有一个它的邻居执行行动，根据模型估计它执行的动作。同样，TN是当用户不执行的操作和至少一个它的邻居执行的操作情况下的数量，同样以模型估计它不执行该操作。最后，FN是当行该操作。ROC曲线在其比较不同的二元分类模型的有效性是公认的，Provost等人在型是有理的。在ROC曲线中，曲线的峰值越接近点(0,1)其性能越好。比也有些微的优势。我们专注于静态和时间关注模型之间和对用户采取的行动和效果的相对性能。图5.4比较静态和时间关注模型，并清楚地表明，时间关注的模型进行比静态模型要好得多。图5.5和5.6描述的ROC曲线的用户和行动的的点阈值θv，其中所有的节点都设置为相同的阈值。该图证实了较大的用户的和较InformationandPropagationinCHAPTE LearningPropagationMostofthestudiesoninformationpropagationandinfluence izationinsocialnetworksaparameterp.u;v/(orsimplypuv)witharc.u;v/capturingtheinfluenceuexertsoverv.epuv’sarekeyparametersofthepropagationmodel.esestudiesfocusontheproblemofhowtonodesthatareactivatedwhenthepropagationsaturatesis fortheICmodelorarcinfluenceweightsfortheLTmodel.esestudiesmakeseveralassumptions.(1)earcparameterspuvcapturinginfluenceisamountstoakindofclosedworldassumption:othersourcesofinformationareassumedtoeclosedworldassumptionignorestheinfluenceofsuchexternalsources.ingthelimitationsimposedbytheaboveassumptions.BASICEarlyworkininfluence withundirectedgraphsandusethetermarctoindicatedirectededgesinadirectedgraph.1065.LEARNINGPROPAGATIONallitsneighbors.Clearly,bothoftheseassumptionsarequestionable.f0:1ofanodearenotdiscriminated.Forinstance,allneighborsofanodemaybeinfluencedwithprobability0:001.eideaofthismodelisthatnodeswhose(outgoing)influenceis0:001canarc.u;v/aspuvDinvwhereinvisthein-degreeofnodev.ismodelhasagreaterresolutionthantheabovetwomodelsindistinguishingbetweennodes:nodeswithalowin-degreethewayinwhichinfluenceprobabilitiesarecomputed.eprobabilitiesarecomputedfromthenetworkstructureratherthananyevidenceofpastinfluenceamongnodes.ICproblem.Let

D.i/jthelengthoftheepisode.Attimet,thenodesinD.t/eactiveforthefirsttime.³nodesD.0/areactivetostartwithandcanbethoughtofasseedsthateffectedtheepisode.Eachepisodeoffersomeevidenceoftheextenttowhichinfluenceofanactivenodesucceededsetofsuchepisodes

Df1;:::;Dn,ghowcanwelearntheinfluenceprobabilities?Foranodeu,letNout.u/Dfvj.u;v/2EgdenoteitschildrenandletNin.u/Dfvj.v;u/2Egnoteitsparents.⁴Influence(forperformingactions)flowsfromnodestotheirchildren.²Gruhletal.[2004]wasanearlierworkonthisproblem,butweprefertodiscussthemorerigorousapproachofSaitoetal.[2008]here.³Inpreviouschapters,weusenotationSttodenotetheactivesetattimet.Conceptually,itissimilartothenotationD.t/here.However,technicallyStisarandomsetgeneratedbyaparticularinfluencediffusionmodelgivensomeseedsetS0,whileD.t/isthesetofobservedactivenodesattimet.us,weusedifferentnotationsforthem.

5.2.IC 58

Ds.tC1/,oneofthenodesin/

musthavesucceededinactivatingv.Observethatwecanassumewithoutlossofgeneral-itythatNin.v/\Ds.t/isnon-empty,foreveryv2Ds.tC1/.Givenanepisodewherethisisnottrue,wecanalwaysconsiderthatallnodesv2Ds.tC1/whichdonothaveanypar-entinDs.t/startanewepisodewherethosenodesaredefinedtobeactiveattime0.Fig-ure5.1showsanexamplesocialnetworkandatablesummarizingthetimesatwhichdifferentactivated.Wecanmodelthisinformationintheformofoneormoreepisodes.E.g.,wecandefinetwoepisodes:D1.0/Dfv1g,D1.1/Dfv2;v3g,D1.2/Dfv4;v5gandD2.0/Dfv6g,D2.1/Dfv7g,D2.2/Dfv8;v9g.Alternatively,wecanmodelthisinformationasoneepisodeby v1;6,fD.1/

inrelationtothedefinitionsoftheseGivenanepisodeDs,theprobabilitythatanode esactiveattimetC1iswPs.tC1/D1 .1—pvw wv2Ninwstr三andhappenattimestept.LetCs.t/D Ds.r/denotethesetofnodesthatareactiver三tinanepisodeDs.enthelikelihoodofobservingtheepisodeDs,w.r.t.0DfpvwgisgivenL.0IDs/D

s

wPs.tCw

s

.1—pvw tD0w2Ds.t tD0v2Nout.t/w2Nout.v/nCs.teintuitionisthatforanarc.v;w/,ifw2Ds.tC1/,weknowthatoneoftheparentswthatwasactiveattimetmusthavesucceededwhereasifw…Ds.tC1/,weknowthat 5.LEARNINGPROPAGATIONofthelikelihoodofobservingthesetofepisodesSDfD1;:::;Dngsimultaneouslyw.r.t.0takinglogofthelikelihood,we nTs-L.0/

X logL.0IDs/

logPs.tCwXw 2Ds.t

log.1—pvw

st/w2Nout.v/nCs.twForanodew,thetimeitbecameactiveinanepisodeDsisknown.Lettsbethiswus,intheequationabove,Ps.tC1/canberecedsimplybyPssincethetimeofwisknownandcanbesuppressed.isYwPsD1w

.1—pvw v2Ds.s-1\Ninus,theproblemoflearninginfluenceprobabilitiesaccordingtotheICmodelfromagivensetofepisodesisformalizedasthatoflearning0 izesEq.(5.3).Sincethis difficulttooptimize ytically,Saitoetal.[2008]solvethisbytakingrecoursetoExpectation correspondingtoaparentvthatbecameactiveattimetsucceedingorfailingtoactivatewwchildrenwattimetC1.eactivationattemptsucceedswithprobabilityOvw=POsandfailsthecurrentestimateoftheparameters0OisgivenbywwnTs- X

Ov

vw/

O

vw

—sD1tD0st/w2Noutvs.t

X log.1—pvww2Nout.v/nCs.tLetSS)bethesubsetofepisodesinSforwhichtheactivationattemptbya vonitschildwsucceeded(resp.,failed).Moreprecisely,Sisthesetofepisodesinv2Ds.t/andw2Ds.tC1/,where.v;w/2E.Similarly,Sisthesetofepisodesin

SjCS

5.3.THRESHOLD Ovw O vwDS NoticethatingeneralitisnotnecessarythatSjCSjDn,wherenDjSjisthe Tosummarize,givenasetofepisodeseachconsistingofsetsofnodesthatactivatedatismethodwasproposedbySaitoetal.[2008]fortheICmodelandshowntoworkwelloverarealsmallnetworkdatasetconsistingofapproxima y12,000nodesand80,000arcs.eitusestheEMrequiresparallelizationand/oradifferentmethodfortheestimation.esecondproblemcanbealleviatedbyapost-processingmethodsuchasSPINE[Mathioudakisetal.,2011],whichsparsifiesthevectorofparametersinanICmodel.Concrey,SPINEfirstcomputesasub-setofthearcsintheoriginalmodeltocreateanICmodelhavingnon-zerolikelihoodforthedata.Intheirexperiments,ongraphshavinguptoNext,thealgorithmgreedilyaddsthearcsthatleadtothe umincreaseoflikelihoodoftocreatetheoriginalICmodel.THRESHOLDInthissection,wedescribeamethodforlearninginfluenceparametersw.r.t.anunderlyingthresh-oldmodel.Intheirseminalpaper,Kempeetal.[2003]introducedageneralizedthreshold(GT)propagationmodelasageneralizationofbothICandLTmodels.emethodwediscussinthissectionisbasedontheGTmodel.IntheGTmodel,asintheLTmodel,everynodevchoosesthreshold0v2Œ0;1]atrandom.enetinfluenceoftheactiveneighborsonanodeisbyamonotonefunctionfvW2N.v/!Œ0;1],whereNin.v/isthesetofallin-neighborsofv.IfS巳Nin.v/isthesetofin-neighborsofvactiveattimet,thenfScapturestheirnetinfluenceonv:accordingly, esactiveattimeCt1iff.S三0v(Definition2.16).Letpuvbeweightassociatedwitharc.u;v/.Clearly,f.-/isafunctionoftheweightsassociatedwiththe 5.LEARNINGPROPAGATIONtheirmethodnext.Technically,theparametersassociatedwitharcsintheLTmodelareweights.However,forconvenience,wewillrefertoarcweightsasarcprobabilities.emainobservationisthatprobabilities.Aswewillsee,thetimesatwhichactionsareperformedyakeyroleinthedenoteanundirectedsocialgraph.V;E/alongwithafunctionTWE!thatforeverymethoddiscussedeasilyextendstodirectedgraphs.LetAbeauniverseofactions.enetwork.Atuple.u;a;t/2actionsmeansuseruperformedactionaattimet.Forsimplicity,assumeauserperformsanyactionatmostonce.DenotebyAu;Au&v,anduvthesetsofactionsperformedbyu,byuandv,byuorv,respectively,andletAu2vbethesetofactionsthatwerebyuearlier.Clearly,vjDjAujCjAvj—jAu&vj.Definition5.1Actionpropagation.Wesaythatanactiona2Apropagatesfromuserutoviff:i).u;v2E;(ii)9.u;atuva;tv/2actionswithtu<tv;and(iii)T.u;v/tu.Whenthisholdswewriteprop.au;vL1t/whereL1tDtvtu.Bothuandvmusthaveperformedtheactioninthatorderandafterthearc.u;v/isformed.eabovenotionleadstoanaturalnotionofapropagationgraph.Definition5.2Propagationgraph.Givenasocialnetworkandanactionlog,eachactionain-actionsg;thereisanarcuA!vinE.a/wheneverprop.a;u;v;L1t/holds.areimpossible.Infact,apropagationgraphcanbeviewedasasetofepisodesinthesenseoflog.eedgelabelsindicatethetimesatwhichtheedgeswereformed.erearethreedistinctFigure5.2,bottom.Noticethateventhoughthereisanedgebetweenyandzinthenetwork,thereisnoarcbetweenthemintheactionpropagationgraphforactiona,sincetheedge.y;z/wasformed(attime12)afteruseryperformedactiona(attime10).Inthisway,theactionlog5.3.THRESHOLD x63y63ybzbzc7xcxa7yazaxx pvuwithbothdirectionsofanedge.u;v/2STATICborsareactivatedaftertheybecamev’sneighbors.RecallthattheGTmodelpositsthatthenetusingthethresholdfunctionvSoftheGTmodel(seeDefinition2.16).Intuitively,vcanberegardedasafunctionofthearcprobabilitiesfpuvju2Sg.Sinceitismeanttobeusedinthecontextofinfluence ization,thefunctionfv.•needstobeconsistentwiththepropagationmodelsusedfor ization.AsdiscussedinChapter2,thesepropagationvtonicity:vSfv.T/wheneverS�T;and(ii)submodularity:forS�Tandx2VnT

[f eserequirementsshouldnotbeconfusedwith1125.LEARNINGPROPAGATIONels.eaboverequirementsarepurelylocal:wemerelyrequirethatthesetfunctionfvsatisfysoareindependentofeachother.isleadsto⁵YvSD1 .1— Itcanbeshownthatfv.•definedaboveismonotoneandsubmodular(seeGoyaletal.activeneighborsofv.SupposewehavealreadycomputedvSandthatanewneighborw62S eactive.Wecancomputefv.S[fug/asYfv.S[fug/D1—.1—pwv/ .1—DvSC.1—vSwvatis,fv.S[fwg/canbecomputedsolelyintermsofvSand

WeremarkthattheaboveisjustonewayofdefiningvSthatguaranteestheaboveallowsthelearningofarcprobabilitiesaslongasanysuchdefinitionofvSisavailable.WecanmodelinfluenceusingaBernoullidistribution.Anactiveneighboruofvhasafixedprobabilityofinfluencingvandactivateit.EachattemptatactivationisaBernoullitrial.eumlikelihoodestimate(MLE)ofthesuccessprobabilityisjusttheratioofthenumberofactionsthatpropagatedfromutovoverthenumberofactionsperformedbyu,

definedasthenumberofitemscommonbetweenthesetsoverthosethatbelongtoeitherset.Adaptedtooursetting,thisyields

⁵Technically,vSdefinedinEquation(5.7)isathresholdfunctionintheGTmodel.However,accordingtotheGTmodelvisactivatedwhenfv.S/：：：0vwhere0visdrawnuniformlyatrandomfromŒ0;1],whichimpliesthattheprobabilitythatvisactivatedisalsofv.S/.us,wesometimesalsorefertovSasthe(joint)probabilitythatv’sactivein-neighborssuccessfullyactivatev.5.3.THRESHOLD arcprobabilityof.u;v/.iscanbeaddressedbyconsideringthatthe“credit”formakingvperformanactionaisequallysharedbyallneighborsuofvwhichperformedactionabeforevdidbutafterubecameaneighborofv.Moreprecisely,letSbethesetofallneighborsofv,regardlessofwhentheybecamev’sneighbors.Define

.a/D

Pa2AP

havediscussedhowtoestimateindividualarcprobabilitiesusingavarietyofstaticmodels.Inordertovalidatethesemodels,weneedtocalculatethejointprobabilityofasetofactiveneighborsonagivennode.iscanbedoneusingEquation(5.7).Animportantissueinvalidationisverylargedatasets.Wewillreturntothisissuewhendiscussingvalidation.asmartphonefromfriendswhoboughtandreviewedit,atfirstwefeeltheurgetotryitoutFigure5.3,takenfromGoyaletal.[2010],showsthenumberofpropagatedactionsagainstthetimeelapsedbetweentheperformanceoftheactionbetweentheneighbors.edistributionshownatthreelevelsofgranularity:every10minduringthefirsthour,everyhourduringthecanclearlyobservethatagreatmajorityofactionspropagatedwithinashorttimeelapseandconfirmingourintuitionthatinfluencedecayswithtime.numberofedges(in114numberofedges(intimedifference(intervalof10

2 numberofedges(in00 numberofedges(intimedifference(intervalin numberofnumberofedges(in0 timedifference(intervalinthecasesinwhichthetimedifferenceislessthanonehour,i.e.,thecasesin(a));and(c)therestofthedatasetwithweeklygranularityCONTINUOUSTIMEMotivatedbytheaboveobservations,wecanmodelarcprobabilityasacontinuousdecayingfunctionoftime: 0—t—puvDpuveruv Here,p0isthe umstrengthofu’sinfluenceonv,realizedimmediayafteruperformstheaction,i.e.,attDtuandptuvdecaysexponentiallythereafter.eparameterp0canbeesti-matedusinganyofthestaticmodels—BernoulliorJaccard—withorwithoutpartialcredits.eparameterruvcanbethoughtofasameanlifetime,correspondingtotheexpectedtimeelapsebetweenthetimeuperformsanactionandwhenvfollowssuit,andcanbeestimatedasPAv/-P

5.3.THRESHOLD

Wecannowestimatethejointprobabilityofinfluenceofasetofactiveneighborsonanodevasthefollowingtime-basedthresholdfunction:Yft.S/D1 .1-pt v v. v

t·g0,where0istheactivationthresholdofv.Calculatingmaxfft·/grequiresthatwe ft·/everytimeanewneighborisactivated.Sinceft·/isacontinuousfunctionoftime,theseagivennodebecomputableincrementally.isisaddressednext.ingthatanewlyactivenoderemainscontagious,i.e.,capableofactivatingitsneighbors,overafixedintervaloftime.Oncethatwindowelapses,itlosesitscontagiousness,i.e.,itcannotacti-ofactivatingitsneighborvoverthetimeintervalu;tuCruv],wheretuisthetimeu activeandruvisaparameter.isleadstoanimportantissuethatSshouldonlyaccountnon-contagious,wecanupdatetheprobabilitytof.Snfwg/DfvS-pwv v1-Itismoreaccuratethanstaticmodelsbyexplicitlytakingtheeffectoftieintoaccount.Itislessclosethissectionbynotingthattheconstantprobabilitypuvcanbeestimatedusinganyofthewithpartialcreditscanbeestimatedas puv

w2Nin.v/I.0<tv.a/—tw.a/<rwv/ jAu1165.LEARNINGPROPAGATIONInthesetwosections,wehavediscussed12modelsinall:amodelcanbestaticortimeawareandatimeawaremodelmaybecontinuoustimeordiscretetime.econstant(incasestatic)orthe andalsodiscussthevalidationofmodels.Beforethat,weturntoabasicquestion.experimentalresults[Goyaletal.,2010]thathelpusapproachthesequestionstatively.erecanbeatleastthreereasonsforauserperforminganaction.Shemaybeinfluencedbyhersocialnetworkcontacts.HowcanweUserswhoareinitiatorsofactionsandwhoaremoreinfluencedbyexternalfactorsareinfl.v/Djfaj9v;LltWprop.a;u;v;Llt/&0：Llt：ruvj actionpropagatingfromuserutov:

AAv/-forinfl.u/mayexhibitahighdegreeofbeinginfluencedbytheirneighborscomparedtothosewithalowinfluenceabilityscore.isevidenceofinfluencepropagationfromtherest:infl.a/Djfuj9v;LltWprop.a;u;v;Llt/&0：Llt：ruvj jU5.3.THRESHOLD Algorithm14Learning–PhaseInput:GraphGD.V;e/andaction foreachuseruW.u;a;tu/2current_table&&.u;v/2Etu iftu<tv end end