Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Overview study on integrated care f... by NIHR CLAHRC West ... 265 views
- Mining User Lifecycles from Online ... by Matthew Rowe 1240 views
- Measuring the Topical Specificity o... by Matthew Rowe 1828 views
- Discover inclusion and exclusion cr... by STARSurg 13379 views
- WWW2014 Overview of W3C Linked Data... by alehors 5673 views
- Improving Inclusion/Exclusion Crite... by Brook White, PMP 1766 views

1,256 views

1,129 views

1,129 views

Published on

How can we detect communities when the social graphs is not available?

No Downloads

Total views

1,256

On SlideShare

0

From Embeds

0

Number of Embeds

97

Shares

0

Downloads

9

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Inﬂuence'based,Network'oblivious,, Community,Detec:on, Nicola,Barbieri, ,Francesco,Bonchi, , Yahoo!,Labs,',Barcelona,,Spain,, {barbieri,bonchi}@yahoo'inc.com , Giuseppe,Manco, , ICAR'CNR,,',Rende,,Italy, ,, manco@icar.cnr.it,
- 2. The,task,of,detec:ng,close&knit+communi.es,of,like&minded, people,in,on'line,social,networks,has,plenty,of,applica:ons,in, marke:ng,and,personaliza:on., If,a,user,responded,posi:vely,to,a, certain,campaign:, TARGET&USERS&IN&THE&SAME& COMMUNITY.& By,homophily,one,can,expect,similar,users,to,be,more,likely,to, be,interested,in,the,same,product,than,random,users., If,more,users,in,the,same,community,adopt,the,same, product,,this,might,eventually,create,a,word&of&mouth+buzz.,
- 3. The,companies,that,would,mostly,beneﬁt,from,knowing,the, structure,of,the,social,network,oSen,do+not+have+access+to+ the+network!, How&can&we&detect&communi<es&when&the&social&graph&is¬& available?& A,company,adver:sing,or,developing,applica:ons,over,an,on'line, social,network,owns,the,log+of+user+ac.vity,that,it,produces., Exploit,the,phenomenon,of,social+contagion+to+detect+ communi.es.,
- 4. Inﬂuence'driven,informa:on,propaga:on., Users,performs,ac:ons,(likes,,purchases,,shares,,tweets),and,those, ac:ons,propagate,across,the,network., A,Propaga.on+model,governs,how,inﬂuence,propagates,across,a, network., Independent+Cascade+Model:++ When,a,node,(v),become,ac:ve,it,is,considered, contagious,and,it,has,a,single,chance,to, ac:vate,each,inac:ve,neighbor,(u),with, probability,pv,u., As,informa:on,spreads,over,social,connec:ons,,,the,network, naturally,shapes,the,process,of,informa:on,diﬀusion.,,
- 5. where Nk = Optimizing Q( P k ⇡k = 1, 0 Modeling maximum likelihood. We assume that each propagation trace is independent from the others, and we adopt (Unobserved))Social)Network) a maximum a-posteriori perspective. That is, we hypothesize Propaga'on)Log) that action probabilities adhere to a mathematical model ⇡k = governed by a set of parameters ⇥.Our,framework,assumes,the, The likelihood of the data given the model parameters ⇥, canexistence,of,an,unobserved+ hence be expressed as: Here, the prop Y Communi'es) mation of L(⇥; D) = social+network,having,a,modular+the P (u|⇥) u2V structure., community no suppressed. Th where P (u|⇥) represents the likelihood to observe u’s behavnumber of com ior relative to D. As a consequence, the corresponding learning letting some o ˆ problem is ﬁnding the optimal ⇥ that maximizes L(⇥; D). We,assume,that,user,ac:vi:es,are,governed,by,an,underlying, general The Following the standard mixture modeling approach [?], we as explained i stochas.c+diﬀusion+process,over,the,unobserved,social,network., assume that users’ actions can only happen relative to a its robustness community of membership. That is, we assume that a hidden Figure 1: Topic-aware inﬂuence parameters are learnt from arbitrarily Each,user,is,associated,with,a,level,of,membership,and,inﬂuence+ larg binary variable social network following zu,k denotes the membership of user u to the log of past propagations and the PK [?]. These are the community to build the INFLEX index prerequisites k, with the constraints in,each,community., zu,k = 1. Thus, pitfallstoof loca k=1 likely be co that we use to e ciently answer TIM queries. {⇡1 , . . . , ⇡K , ⇥1 , . . . , ⇥K }, where ⇥ can be partitioned into P (⇥) is ma ⇥ represents the propagation model, We,can,model,the,behavior,of,users,by,exploi:ng,the,standard, a of m of these three paperskdeﬁne an inﬂuenceparameter set relative to community k, and same order instead in a recent ⇡k =Barbieri et al. [?] extend the classic work P (zu,k = 1). We can rewrite the likelihood as mixture+modeling+approach:, prior in [?] wo IC model to be topic-aware: the resulting model is named K Topic-aware Independent Cascade (TIC) . of communites YX Barbieri et al. also devise methods to learn,= from a log P (u|⇥k )⇡k , of L(⇥; D) the computatio past propagations, the model parameters, i.e., topic-aware u k=1 without optimi inﬂuence strength for each link and topic-distribution for Stochas:c,Framework,for,Network,Oblivious,CD.,
- 6. u v Following [13] weinﬂuenced delaythe adoption ofto deﬁne adopt to u in threshold a deﬁne ng [13] meter set we adopt a delay threshold i. Similarly we de ⇥)|D; ⇥0 ] + + s.Community'Independent,Cascade,(C'IC)., 2 V |0 of users wh Speciﬁcally, we deﬁne Fi,u = we deﬁne Fi,u V |0 |t ete-Data inﬂuencers. Speciﬁcally,{v =2{v 2 V u (i) = t{v > } Fi,u v (i) }tuas the tset ofusersfailedthe inﬂuencing u over i. potentially can speci (i) as potentially v (i) v (i) (2) } who in set of users who Then, we ⇥k )The Generaliza:on,of,the,IC,Model:,each,new,ac:ve,user,v,exerts,her, + log P hts. + the adoption of i. Similarlysurvives uninfected at set analytical}optimization of that a user we deﬁne d u in log ⇡kinﬂuenced(⇥)in the adoption of the least until time tu ) and hazardthe set u i. Similarly we deﬁne as |t , ↵ ) (modeling instantaneous infections). Y We resort|t (i) explicit (i) > } of usersu who deﬁnitely weights. The analytical optimization of that a user surv to the modeling functions H(t v v,u Learning inﬂuence v 2 V u Fi,uv = {v 2 V |tu (i) tv (i) >)(u|⇥difﬁcult. We resort + (i|u, ⇥modeling functions H(tk t inﬂuence,globally,,with,a,strength,that,depends,on,the,community+ ⇥ u } of users P to the deﬁnitely who ) )·P Q(⇥; 0 n data to(2) u the optimization we can specify theP is still )into= community-basedoptimization (i|u, simplify over i. Then, We reformulate this P⇥ framework k hidden data to simplifyexplicit k a ). inﬂuencers nﬂuencing failed in inﬂuencing u over of Then, k as can ispecify the (u|⇥k ) We reformu k+of+the+targeted+node., be a binary variable such that scenario. The C i. (u|⇥ we w P procedure. (C-Rate) propagation model be such that scenario. The analytical optimization of (⇥) a binary variable Learning EM al- The Community-RateThat is, letthati,u,v survives uninfected at least until time tu) and ha e done by means of inﬂuence weights. u,v a user Y i Q(⇥; ⇥and still difﬁcult. Wewherethe i,u,v (i|u,v ⇥k )functions H(t |t ,thethe iprobability that triggered the adoption as by u, 0) is is characterized by the explicit1modeling represents↵v,u) (modelingand is characterized P, = if assumptions: u v item u, instantaneous infecti + Y to w followingthe set of all possible w of such thatby 2 F + . resort he (u|⇥ )of= assignment ⇥ ) the (i|u, ⇥ ), W denote (3) al randomthe itemP ⇥, · v i,u,v P adoption inﬂuence + (i|u, (u|⇥ P =data to simplify the optimization Psurvives uninfected atand until time• tUser’s inﬂ k k analytical potential of ⇥ ) user We reformulate ), uframeworkP (i|u, ⇥ h of the inﬂuencers + P ) Pk let we cankrewrite the (i|u, ⇥ this relativei,u (i|u, that a · Learning until thatweights. .The askhidden optimization inﬂuencers complete-dataklikelihoodleast (3) a community-b activated +Then, u ) That i k to possible wi,u,v i v 2 Fi,u That is,• User’s be a binaryisvariable such the scenario. The Community-Rate into to. and ) wo steps such convergence: let wi,u,v inﬂuence W limited to that community she belongs (C-Rate) propagation m procedure. bility as functions the “out-of-react” inﬂuencers s memb Q(⇥; ⇥0 ) likelihoodu,i,u,v Wetoif v triggered theexplicit modeling none and is characterized by(modeling instantaneous infecti still difﬁcult. = the adoption of the item by u, of H(tu |tv , ↵ w k as complete-data iseach relative 1 resort toto. That i the that is i likely +to inﬂuence/bev,u ) the following assumptions: by inﬂuen eEM⇥k ) representsWthe probability that some(D, Z,2W, ⇥) = P (D, W|⇥, Z) · P (Z|⇥) · PY for alu,k is, user inﬂuenced (⇥), of (i|u, the inﬂuencers as hiddendenote the ⇥ of)allrepresents theof Wei,ureformulatethat some ofto thea community-b the let F of data rewrite possible wi,u,v suchPthat vprobability this framework the communityk ) bel to simplify the optimization P . • ⇥ = 1 where P+ (i|u, set kthe complete-data likelihood relative to(i|u, User’s)inﬂuence is limited into (1 communit p she Then, u and t (u|⇥k )⇡kactivated we can Pby(i|u, ⇥kof wheresame community,to.while is, the effectis likely to inﬂuence/be inﬂue ⇥, the ,Probability,that,some,of,the, + k members ) the proba- Y Y That the user the ,Probability,that,none,of,the,the,“out'of'v P procedure. inﬂuencers That is, let w be a binary variable such that W as scenario. Community-Rate (C-Rate) potential inﬂuencers activatedP (D,W|⇥, Z) = (i|u, ⇥apk ))z ofthe same + propagation e u react”,inﬂuencers,succeeds,in,ac:va:ng,u, and members (1 k the proba- • Informatio i,u,v poten:al,inﬂuencers,in,ac:va:ng,u, succeeded: P The by membersdifferentv2Fcommunity, while the m of inﬂuence of v k none of ·the “out-of-react” inﬂuencers is marginal on gence: Z) P(Z|⇥) · P(⇥),P (D, Z, W, ⇥) = P (D, W|⇥, Z) · P (Z|⇥) · P (⇥), i,u,k of inﬂuence is marginal on members communit D, W|⇥, i,u of a ↵ P (u|⇥k )⇡k that adoption of the item i by u, and is characterized by the succeeded: v2F v triggered the none of the “out-of-react” inﬂuencers following assumptions: where diff v, k=1wi,u,v = 1 if bility Y Y community. Y Y community. s k tions that w ·z + where (i|u, Information pk (1 w )·z )1 P maximizing all let + denote the ) of 1 possible • (1 Y k ) v z2 Fi,u · P User’sv •to⇥kk is diffusion from the userpvvto)v of contagi the⇥W (i|u, ⇥k set= Eq. 2. wi,u,v suchpthatdiffusion .fromYuserpkvinﬂuence =limited to (1 community she be Y v Y Information )k ) 1 i,u,k•v2F (1 community isthe k-th theby the density within the the vpv ) characterized within v P+ P (D,W|⇥, Z) + (i|u, ⇥ p = = (1 f(tu |tv , ↵ Then, to u,k the prior P i,ui,u,k v2F k relative to likelihood metric pk )zcan rewrite the complete-datacommunity isv characterized byto. That is, the v,k|tis ,v2F),to thetoexpected The parameter v2F (⇥) (1 we both i,u v where ↵ user relatedin the inﬂuence/be inﬂue delay on is 0 +the v ↵v,k As a consequence,the density f(tu Q(⇥; ⇥likely second inﬂuence:the ac high v2Fi,u contribution tov triggers)within community k. The proba YYa Y k w ·z tions k (1 We W as model the former in wayv k We is relatedYthe canby members that the activa- community, consequence de row ofwEq. )·z expected delay on 2specify the complete-data likelihood be rewritten as 2Fi,u · p can to of same while P (i|u, ⇥k ) = (1 where ↵v,k 1 pv then 0 2 k of contagion depends on the time delayOn v.u.the pv ) i,u,k v2F of 3zu,k 2 the basis ow ·zu,kExpecta:on'Maximiza:on,algorithm,to,determine,the,parameters, diff automatic estimation tions that k ) = within B of inﬂuenceThe probability on members of a term P P(Z|⇥) triggers (1 The parameter ↵v,k khas pXk. wi,u,v an P(D, Z, W, ⇥) =)·zu,k W|⇥, Z) (i|u, ⇥·vP(⇥),X X (1 wi,u,v P(D, community ) X is marginal a direct interpretation in 2. v NetRate model 0 log(1 v2F @log u,k Y ⇡k + 1 pk forAsthe i,u the·it a consequence, contribution to Q(⇥; ⇥ ) in the second inﬂuence:Y values )of ↵v,k cause short delays, and Y mmunities. v As latter, of contagion dependsk on the time delay v.u . pv k 7 high 6 u v2Fi,u community. that,maximize, 1 i v2F denote v as strongly inﬂuential6 P (u|⇥k or P (⇥) row of Eq. 2 can be rewritten as (1 pv )5 1 4 k. within hen where the specify the complete-data likelihood through: 4 consequence of the above observations,· we can adap probability P (a) for0given P (D|Z, ⇥) = Y then Theu,kX X v,k has a X i,u,k On thediffusion from the user v + in a way We can X Bspecify 2 ↵complete-data likelihood through: k C to v within i,u • XInformation 2 3zk parameter 3 u,k X Y v2Fin terms thelog(1 pk ) + direct zinterpretationbasis to⌘i,u,v,k )of ofk )Sec. III, byv2F the i,u z 0 alternatives.⇡k + u,k NetRatekmodel ﬁt the scheme p A plugging e two different = ⌘ p (1 u,k @ tribution to Q(⇥; ⇥Z) in the second (1 log pv )2 highY vof ↵v,k causei,u,v,k log2vis+characterizedY the z P(D,W|⇥, )Y 3zu,k delays, andY log(1 3density f(tu |tv , ↵ mation of inﬂuence: v2F values short as a by Y6 i v2F community u,k u k i 6 k 7 k 7 P (u|⇥k ) = S(T |tv (i), ↵v,k )· ten as 41 = (1 i,u Y · 4 Y vwhere ⌘pv )5 where ↵ is Y the to the expected i,u,k v2F pv )consequence denote (1 strongly inﬂuential within k. user v in triggering delay on the a 5 as i,u,v,k is71“responsibility” of i:u62C v2C k 7 latter, it v,k related 6 6 k the u,k i,u i,u,v u,k i,u,v u,k u,k + i,u i,u i,u,v i,u,v u,k u,k + i,u i,u + i,u i,u i,u,k + P (D|Z, ⇥)X X = 1 (1 p ) · Y Y (1 p ) i i
- 7. X basis of ) in the depends inﬂuence: delay the likelihood that values .of v,k On 0 short delays, X Y Y ofiscontagion secondthe user v in triggering= P(wi,u,vv.u1|u, i,↵u,k = causeadoption ) and as a ,; ⇥or“responsibility”X on the timehighthe consideredthemodel to(i),the above observations, we can ada ⌘i,u,v,k (i), ↵v,k ) = z NetRate H(tu (i)|tv ﬁt v,k scheme of Sec. III, u,kpluggin 1, ⇥ ) of B the ↵ the k by log(1 u k ) @log ⇡k + of the community k: S(tp(i)|tv denote v(5) strongly inﬂuential within k. consequence as terms v2C v Learning. Again, instead of directly optimizing the k X X in the context parameterwithin itime(i) T . This approach assumes that Y Y has of i,tu (i) appen ↵iv,kv2F v2Ci,tu interpretation pin i:u2C a direct v . binary observations, likelihood, we the X On the = Q nce: high values i,of ↵v,k⇥0cause basis of the+ above )as a P (u|⇥kwe can adaptintroduce|tthe latentv,k )· variable = S(T v (i), + ↵ = P(wi,u,v = 1|u, i,u between short delays,(1andk time of )the zu,k = 1, H(t (i)|t (i), ↵ ) adoption ) 1 pw u,v,kdependency apk ) u v the ﬁt i,u scheme v,k w2F denoting the fact NetRate modelLearning.the1 instead of Sec. III, by i:u62C abovethat u has been infected by v on i. Th to plugging Again, of directly optimizing thei v2Ci quence denote v asv2C strongly inﬂuential within k. k v The,likelihood,of,an,ac:va:on,can,be,formulated,,by,applying, pv k X likelihood be Finally, optimizing Q(⇥; 0 with adapt the and thelog pkaboveu(i)the inﬂuenced. CIn NetRate [4], Y cani,u,v rewritten by deﬁninghu,ii2D X Q . i,t likelihood,k Y the=⌘1i,u,v,k +the v pk )of observations, P ⇥ )canwerespect to pvthe latent binary variable w Y S(t (i)|t (i), ↵ ) basis of one+ (1 ⌘i,u,v,k ) log(1 Y)Aintroduce yields we pk X (1 1w P (u|⇥kdenoting the fact⌘that uS(Tbeen(i), ↵v,k )· on i. Then, the Y Y Y ) =hu,ii |tv infected by v u v v,k survival+analysis:, by u,k ateLearning. w2Fi,u in Sec.directlythis dependency in modeled P(D, W|Z, ⇥) = to ﬁt Sec. III, byabove · i,u,v,k plugging described instead of II,of optimizing the i:u62C v2C has +model Again, the scheme S(T|tv (i), ↵v,k )zu,k + 2Fi,u i:u2Ci v2Ci,tu (i) v2F i i,u likelihood can mizing Q(⇥; ⇥0wewithY Ypklatent binarypkvariable wY i be rewritten by deﬁning respect to Y, (4) C (tu |tv likelihood, ) introducefthe v yields , ↵v,u ) of+ transmission, which D k v= i,u,v nal(u|⇥ ) = pk )A ofS(T user v in triggering Y Y S(t (i)|t (i), ↵ ) Xhu,ii62H(tv2Ci (i), ↵ ) P )likelihood log(1 is the khu,ii k · ⌘ the |tv (i), ↵v,k )·Sv,k + Sv,k Yu Y Y u (i)|tv hu,ii2D Y u,v,k “responsibility” P v,k v denoting the fact u,k ui,u,v,k infectedP v onP(D, W|Z, ⇥) = P that has been by i. Then, the zv,k u,k (5) S(T|tv (i), ↵v,k ) v2C· H(tu (i)|tv (i), ↵v, + n likelihoodv2Fi,u bei:u62thev2Cideﬁning= k: u,k and S i v2Ci,tu (i) u,k . the delay community hu,ii i:u2C of a n the context of Ci v,u . SThe likelihood = hu,ii propagation i,tu (i) with + v,k X2D k v2Ci pk = can rewritten by v,k ,Y (4) + 0 Y hu,ii6 hu,ii2D k v2Ci,tu (i) v + v2F 1 Similar formulations can i,u i,u Y mulatedSi,u,v S= 1|u, Yzu,kstandard survivalv2FYH(tu (i)|t[14], v,k ) wi,u,v zu,k by Y Y i, = u ⇥ )z = P (w v,k + applying S(t1,(i)|tv (i), ↵v,k ) Y analysis v (i), ↵ ,v,k v in triggering v,k user zu,k (5) P P(D, W|Z, ⇥) = kP · YNAMICS H(tu (i)|t (i), ↵v,k ) omitted · S(tu for S(T|tV.(i), ↵v,k ) u,k T EMPORAL DLearning. Again, v instead of directlyhere(i)|tv (i), ↵v,kthe optimizing Modeling,the,probability,that,a,user,survives, lack) of v M ODELING are nitysurvival v,k pvi v2Ci,tu (i) k: v2C the probability of hu,ii u,k and Si:u2CD uX i,u,k . .v,u ) (modeling i,tu (i) v2Ci,tu(i) we introduce the latent binary variable S(t khu,iiv ↵ = |t hu,ii2D k v2C Q hu,ii62 v2F C-IC does not explicitly model temporal dynamics, as it + = likelihood, v2F i,u Y w) +Y 1 = 1, ⇥0 )user Yi,u (1 i,u pkuninfected wi,u,vleast uninfected,at,least,un:l,:me,tu,, theby v on i. Th z w2F that a TEMPORAL DYNAMICS H(t(i)|tv (i),v↵v,kbinary zactivations by employingfactu↵v,k )uu,khas P(D|Z, ⇥) with above component survives on modeling just ) at u,k denotingS(tu(i)|tav t and replacing been infected ) and · the focuses H(tu u (i)|t (i), ↵v,k ) of until time (i),that the hazard we adopt the exponential d M ODELING · likelihood. In the following Learning. Again, instead directly optimizing above i,tu likelihood alterdiscrete-time pv yields functions 0hu,ii2Dv2Clikelihood,propagation model. HereModeling,instantaneous,infec:ons, ↵v,k v,u}, which H(tkuv2Cdynamics, as kwe introduce the latent anbinary tioninfections).v,k exp { |ti,t(i)↵to ) (modeling we present can be variable , ↵wi,u,v ↵ ,(i) v,u it instantaneousrewritten v,k ) = f(tu |t by deﬁning sizing Q(⇥;model with respect . not explicitly ⇥ ) temporal vu and replacing P(D|Z,to characterizeabove component vin theY ⇥) with the the P exploits z ) · S(tmodeling thatlikelihood. delays (i)|t modeling just binary activationsnativeemploying a↵fact u,k timethe following wea community-based ↵v,k v,u } and H(tuu,k, ↵ v, ·by directly denoting (i), v,k ) that u the P (D, W|Z, the = on|ti. distribu- exp hu,ii We reformulate udiffusion optimizingInhas been infected⇥)exponential↵Then, the {S(T |tv (i), ↵v,k )z |tv this process. into adopt by vS(tuY v,kY= u,k of⌘i,u,v,k vthe framework above ning. Again, instead overall + v2F Here propagation model.i,u we likelihood can tion rewritten )by ↵v,k exp { ↵v,k v,u↵v,k . 1 Then, an k k yields which enables toand = introduce with the above componentf(tu the, (C-Rate) propagation model p pv we hood,vreplacing P(D|Z, ⇥) Community-Rate|tv[0, T], thei,u,vis to explicitly },hu,ii62D k v2Ci binary in ↵v,k scenario.time+ topresent an observation window (4) w idea The the latentalter- be variable = deﬁning Given , the ing that exploits S delays v,k characterize X XX Yv ↵v,k X ↵ i. ing process. In thatv,k hasSbeen thethe exponentialuthev ,Y )atYexpeach↵v,k v,u } and H(tu |tzu,k) /) = Ylog ⇡k the factBy,adop:ng,the,exponen.al+distribu.on,as,density,for,the, u + weby infected by v|t1on v,k Then, the user adopted Q(⇥; ,⇥0Y u,k model the following assumptions: likelihoodS(t distribu- = Y of time which { ,k is ion characterized adopt P likelihood. the following P (D, W|Z, ⇥)↵= . Then, P H(tu (i)|t (i), u,k S(T |tv (i), ·↵ ) v,k condi:onal,transmission,likelihood,and,by,introducing,hidden,vv2Ci ↵v,k each item, or }, u,k . hood can |tbe↵rewrittenidea {deﬁningthe which enables the considered adoption v,k byis to v,k v,u likelihood that hu,ii tion f(twindowand ↵v,k = u,k hu,ii62D k bservation u u,k v,k ) T],Sv,kexp ↵ explicitly , hu,ii v , [0, = the (4) k hu,ii62DX v2Ci + Xthat Xshe belongs X X hu,ii2D k v2Ci,tu (i) X X v2F did userv2Fi,u within ,time )T. Y approach is community S(t• v User’s which↵each not }happenH(tu |tvQ(⇥; ⇥0 )=ThistheY ⇡assumes ↵v,k / Yz ui,u ↵v,k time Yinﬂuenceand limited to v,k v,u kelihood|tof,variables,for,modeling,the,iden:ty,of,the,inﬂuencer,,we,obtain:, logu,k the ) = atexp { Y Y adopted + u,k log ktime of the u,k v ↵v,k · S(t i,u,v,k u,k z u,k there is a dependency W|Z,1⇥) = T EMPORAL DYNAMICS · ↵v,k ) the adoption H(tu (i)|tv (i), ↵v,k )wi,u,v zu,k(i)|tv⌘(i), ↵v,k ) ↵v,k M ODELING That considered S(T |tv (i), between ↵v,k .likelihood that the is, theadoption is likely to inﬂuence/be inﬂuenced u Then, or the to. user of the u,k hu,ii6 hu,ii2D k v2Ci,tu (i) hu,ii u,k . hu,ii62D k inﬂuencer and the onehu,ii2Dinﬂuenced. In NetRate2D k v2Ci [4], v2Ci X X v2Ci,t X k X X u (i) X X en i,u explicitly model temporal dynamics, as it and replacing P (D|Z, ⇥) with X X not by X Y Y the same community, while the effect above component 2F within time T. This approach assumes that Y log ⇡ previously described in Sec.↵II, this dependency in modeled by u,k log ↵z 0 the ⌘ Q(⇥; between the adoption time of the u,k v,k ) / members of u,k u,v ↵v,k , u,k u,k k i,u,v zu,k ependency⇥just binary activations H(tu (i)|tv (i),+ a )w· S(tu (i)|ti,u,v,k ↵v,k ) v,k odeling · by employing↵v,k likelihood. In(i), following we adopt the exponential di v the v NAMICS aIn NetRate [4], i conditional v2C hu,ii2D k hu,ii2D k u,k D d the oneof theinﬂuencehu,ii6is kpresent f(tu |tv , ↵v,uX members of a differentv2Ci,tu (i) of hu,ii2D k Herei,twe marginal on of transmission, which inﬂuenced.v2C 2(i) likelihood an alter- ) X v2Ci,tu (i) propagation model. X uon the delay X u |tv , ↵v,k ) = ↵v,k exp { ↵v,k v,u }, which e X X depends tion a ral dynamics, as it and modeled by logP (D|Z, ⇥) with of f (tabove component in the v,u replacing ↵v,k . Thezlikelihood thepropagationu,v ↵v,k , scribed in Sec. + this dependency in ⌘i,u,v,k characterize the II, u,k ng that community. S(tu (i)|t by applying ) u,k S(tu analysis u,k exploits time delays to exp { ↵v,k be } and H(t different den [14], ns by employing a can ·be formulated v (i), ↵v,kstandard survival|tv , ↵v,k ) = 1 Similar formulations canv,uobtained by adopting u |tv , ↵v Modeling,temporal,dynamics,with,C'Rate.,
- 8. Evalua:on,on,Synthe:c,Data., We,use,a,generator,of,benchmark, graphs[1],,which,generates,directed+ unweighted+graphs,with,possibly+ overlapping+communi.es., • Number,of,nodes,=,1000;, • Average,in'degree,=,10;, • Maximum,in'degree,=,150;,, • Min/max,of,the,community,sizes,=,50/750., ,The,four,networks,diﬀer,on,the,percentage,μ, of,overlapping,memberships., • Propaga:on,cascades,are,generated, according,to,the,Net'Rate,propaga:on, model., • The,transmission,rate,for,each,link,is, sampled,from,a,Gamma,distribu:on, (shape=2,,scale=0.3)., ) µ = 0.001 (b) µ = 0.01 TABLE I: Statistics for the synthetic data: four networks corresponding to four values of µ as in Figure ??. # of communities (K) avg # of adoptions avg trace length avg % of communities traversed by a trace S1 9 56k 38 17% S2 7 59k 38 24% S3 11 82k 54 24% S4 6 370k 256 82% The strength of each link is determined by considering both the outdegree (out ) of the source and the indegree (in ) of [1],A.,Lancichineh,and,S.,Fortunato.,Benchmarks,for,tes:ng,community,detec:on,algorithms,on,directed,and,weighted, · · graphs,with,overlapping,communi:es.,Physical,Review,E,,80,,2009., the destination:
- 9. Results., Baseline+Models+ • Based,on,network,reconstruc:on, (assuming,a,dense,graph):, • Inference,for,the,IC,Model[2];, • Net'Rate[3];, • Communi:es,are,detected,by, applying,METIS[4],on,the, reconstructed,graph., • Mul:nomial,EM, [2],K.,Saito,,R.,Nakano,,and,M.,Kimura,,Predic:on,of,informa:on,diﬀusion,probabili:es,for,independent,cascade,model., KES’08., [3],M.,Gomez'Rodriguez,,D.,Balduzzi,,B.,Schölkopf.,Uncovering,the,Temporal,Dynamics,of,Diﬀusion,Networks.,ICML,2011., [4],G.,Karypis,and,V.,Kumar,,A,fast,and,high,quality,mul:level,scheme,for,par::oning,irregular,graph.;,SIAM,Journal,on, Scien:ﬁc,Compu:ng,,vol.,20,,no.,1,,pp.,359–392,,1999.,
- 10. Evalua:on,on,real,data., TwiEer+data+ • • • • Number,of,nodes,=,28,185;, Number,of,links,=,1,636,4511;, Number,of,propaga:ons,(urls),=,8,541;,, Tweets,=,516,412., TABLE II: Summary of the evaluation on real data. Communities Community size (min/max/median) QG Conductance Internal Density Cut Ratio Time (mins) C-IC 20 C-Rate 64 156/3651/1319 97/1758/328 0.3274 0, 5849 0, 031 0, 001 105 0.2424 0, 6791 0, 051 0.0009 122 Internal,density,is,an,order,of,magnitude,higher,than,the,density,of, the,whole,graph,(0.0041)., Modularity,and,the,diagonal,block,structure,of,the,incidence,matrix,, conﬁrm,the,existence,of,a,good,community,structure.,
- 11. THANKS!

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment