Predicting Popularity of Twitter Accounts through the Discovery of Link-Propagating Early Adopters

Transcription

1 Predicting Poplarity of Titter Acconts throgh the Discoery of Link-Propagating Early Adopters Daichi Imamori Gradate School of Informatics, Kyoto Uniersity Sakyo, Kyoto Japan ABSTRACT In this paper, e propose a method of ranking recently created Titter acconts according to their prospectie poplarity. Early detection of ne promising acconts is sefl for trend prediction, iral marketing, ser recommendation, and so on. Ne acconts are, hoeer, difficlt to ealate becase they hae not yet established the reptation they desere, and e cannot apply existing link-based or other poplarity-based accont ealation methods. Or method first finds early adopters, i.e., sers ho often find ne good information sorces earlier than others. Or method then regards ne acconts folloed by good early adopters as promising, een if they do not hae many folloers no. In order to find good early adopters, e estimate the freqency of link propagation from each accont, i.e., ho many times the follo links from the accont hae been copied by its folloers. If the freqency is high, the accont mst be a good early adopter ho often find good information sorces earlier than its folloers. We deelop a method of inferring hich links are created by copying hich links. One important adantage of or method is that or method only ses information that can be easily obtained only by craling neighbors of the target acconts in the crrent Titter graph. We ealated or method by an experiment on Titter data. We chose then-ne acconts from an old snapshot of Titter, compte their ranking by or method, and compare it ith the ranking based on the nmber of folloers the acconts crrently hae. The reslt shos that or method prodces better rankings than arios baseline methods, especially for ery ne acconts that hae only a fe folloers. Keyords micro-blogging; link-propagation; hbs; inflence; link prediction; graph analysis; graph eoltion; graph mining. INTRODUCTION In social media, sch as micro-blogs and social netork serices, sers can easily create ne acconts and qickly start p ne information pblishing channels at lo cost. As a reslt, social media Crrently at CyberZ, Inc. Permission to make digital or hard copies of all or part of this ork for personal or classroom se is granted ithot fee proided that copies are not made or distribted for profit or commercial adantage and that copies bear this notice and the fll citation on the first page. Copyrights for components of this ork oned by others than the athor(s) mst be honored. Abstracting ith credit is permitted. To copy otherise, or repblish, to post on serers or to redistribte to lists, reqires prior specific permission and/or a fee. Reqest permissions from permissions@acm.org. CIKM 6, October 24-28, 206, Indianapolis, IN, USA c 206 Copyright held by the oner/athor(s). Pblication rights licensed to ACM. ISBN /6/0... $5.00 DOI: Keishi Tajima Gradate School of Informatics, Kyoto Uniersity Sakyo, Kyoto Japan tajima@i.kyoto-.ac.jp are highly dynamic orld. Micro-blogs, sch as Titter, are especially dynamic becase they focs more on prompt information dissemination, hile social netork serices, sch as Facebook, focs more on commnication oer long-term relationship. Becase of the dynamicity, ne poplar acconts continally appear and disappear in micro-blogs. Early detection of ne acconts that ill become poplar in ftre is sefl for seeral applications, sch as trend detection, iral marketing, and ser recommendation. Estimation of poplarity of an accont is also sefl for approximating the qality of information it posts. The qality of information is generally difficlt to estimate ithot hman interention. To sole this problem, poplarity-based methods hae been idely sed. There are eb page ealation methods based on the information on their incoming links [9, 5], and similar idea has also been applied to Titter [22]. The sccess of these methods proed that there is high correlation beteen the poplarity and the qality of information. The poplarity-based methods, hoeer, cannot be applied to ne acconts that hae not yet established the poplarity they desere. If e ant to apply the poplarity-based methods to sch acconts, e need to predict the ftre poplarity of them. In this paper, e propose a method of ranking ne Titter acconts according to their prospectie poplarity, in other ords, the nmber of folloers they ill obtain in ftre. The most important factor deciding the ftre poplarity of an accont is, of corse, the qality of information it posts, bt it is difficlt to estimate as explained aboe, and that is one of the reasons hy e ant to predict poplarity instead. We therefore shold explore a method of predicting ftre poplarity of an accont not based on its information qality bt based on its crrent poplarity. Ne acconts, hoeer, sally hae only a small nmber of folloers. Ho to predict ftre poplarity only ith that information is the challenge of the problem e discss in this paper. Becase the nmber of folloers is sally small, e also se the qality of each folloer. It is basically the same approach as many existing link-based qality estimation methods [9, 5, 22]. We focs on a specific type of qality of folloers that is most important for s: hether the link from it implies more links in ftre. In Titter, and in other social media, there are sers that are good at finding ne good information sorces earlier than other sers. We call sch sers early adopters. Early adopters themseles often hae many folloers, and hen an early adopter creates a link to a ne information sorce, many of its folloers imitate it and create links to the information sorce. In other ords, early adopters play the role of hbs in link propagation in social media. Therefore, links from early adopters imply more links in ftre. Folloing the obseration aboe, or method first comptes early adopter scores of the folloers of ne acconts, then comptes ftre poplarity scores of the ne acconts based on them. If a ne

2 accont is folloed by good early adopters, or method regards the accont as promising, een if it does not hae many folloers no. Or method comptes the early adopter score of an accont based on the freqency of link propagation throgh it in the past, i.e., ho many times its links hae been copied by its folloers. If the freqency is high, the ser mst be a good early adopter ho find good information sorces earlier than its folloers. In Titter, hoeer, the information on hich links ere created by copying a gien link is not immediately aailable. We infer it by sing the folloing types of information: graph strctre among the neighbors of the link, temporal order of link creation, reciprocity of links, and similarity beteen interests of neighbors of the link. One important adantage of or method is that e can easily obtain all the information aboe only by craling the neighbors of the target accont in the crrent Titter graph. We condcted an experiment ith a partial Titter graph that as collected by Li et al. on May 202 []. We rank then-ne acconts in this data set based on or ftre poplarity score, and compared it ith the ranking based on the nmber of non-reciprocal folloers that the acconts later hae as of May 205. The reslt of the experiment shos that or method otperforms arios baseline methods hen e compare the accracy of the hole ranking of all the ne acconts. Or method otperforms baseline methods especially hen e apply them to ery ne acconts that hae only a fe folloers. In addition, the correlation beteen the ranking by or method and those by baseline methods are lo. It sggests that or method and baseline methods are complementary. Or experimental reslts shos that e can actally prodce a een better ranking by linear regression combining or method and some baseline methods. When e compare only the top part of the rankings, a ariation of HITS [9] or PageRank [5] otperforms or method in most cases. It is mainly becase the top part of the rankings incldes many acconts that ere already poplar in the old snapshot. This reslt again shos that or method is particlarly sefl for finding acconts that are not poplar no bt ill be poplar in ftre. 2. RELATED WORK In sociology, there has been extensie research on the behaior of people in the real orld. Some stdies hae shon that the behaior reported in the past research in sociology is also obsered on social media [8, 4, 7, 7]. One of the stdies in sociology has proposed the concept of triadic closre [6]. In short, it says that if A connects to B and B connects to C, A is likely to connect to C. Or method ses this triangle strctre for inferring hich links are created by copying hich links. There hae been some stdies, sch as [3], that se freqent patterns of local graph strctre, hich are called motifs, for analyzing eoltion of social netork. The triadic closre strctre, hich e se in or method, is also a kind of motif. Recently, there hae also been many stdies on link prediction in online social netork. For example, Liben-Noell and Kleinberg [2] proposed a prediction method based on the proximity of nodes in the netork. Zhang et al. [25] proposed a method that estimates the probability of ftre links by inferring latent paths of link propagation in the netork. They estimate ho important each node is as a mediator of link propagation by sing a probabilistic model. Or method is based on a similar concept of early adopters. Their method, hoeer, reqires mltiple snapshots of the netork strctre at different time point. On the other hand, or method only reqires information that can be obtained by craling the neighbors of the target accont in the crrent snapshot of the netork strctre. This is one big adantage of or method. There hae also been many stdies on estimation of the inflential poer of nodes in social netork. For example, Kak et al. [0] compared three indicators, PageRank, the nmber of folloers, and the nmber of reteets, for the estimation of poplarity of Titter acconts. They shoed that there is a discrepancy beteen the nmber of folloers of an accont and the poplarity of teets by the accont. Weng et al. [22] proposed a method for estimating inflential poer of Titter acconts. Their method is based on the nmber of folloers, bt they also consider the interests of the folloers and compte the probability that each teet is actally read by the folloers. These to stdies focs on inflential poer of nodes in information dissemination, hile early adopters in or method are sers that hae inflential poer in link propagation. The discoery of early adopters in online commnity has been discssed in seeral stdies. Bakshy et al. [3] analyzed ho sers adopt ne contents in a social netork in Second Life, and identified early adopters, bt also fond that early adopters in Second Life do not alays hae significant inflence on the other sers. Saez-Trmper et al. [8] proposed a method of identifying early adopters that also hae significant inflence on the others in information netork, sch as Titter, and called sch sers trendsetters. Zhang et al. [24] proposed a method of identifying bloggers that predicted bzzords before they became poplar. These stdies focsed on temporal relationship of sers adoption of contents, sch as hashtags, URLs, and bzzords. Goyal et al. [6] also proposed a method of identifying leaders in online commnities hose actions, e.g., tagging resorces or rating songs, are imitated by many sers. On the other hand, e focsed on the adoption of ne Titter acconts, i.e., the creation of ne follo links, and imitation of them by the folloers. In this paper, e sho that the idea similar to theirs can also be applied to sch a type of actions in order to predict ftre poplarity of ne acconts in Titter. Another contribtion of this paper is to deelop a method of estimating link propagation freqency throgh each ser, the information hich is not immediately aailable in Titter. 3. ESTIMATION OF COPY FREQUENCY In this section, e explain ho or method estimates the freqency of link propagation throgh each ser. As explained in Section, e infer it based on for kinds of information: graph strctre, temporal order of link creation, link reciprocity, and similarity among interests of sers. We first define some notations sed in this paper. Let G(V, E) be the follo graph of Titter, here V is the set of Titter acconts, and E is a set of all follo links among them., E denotes a follo link from an accont to an accont. For V, Friends() denotes the set of acconts folloed by, and Folloers() denotes the set of folloers of. Also let Copy() denote a set of links created by copying a link from. What e ant to estimate is Copy() for each, i.e., the size of Copy(). 3. Graph Strctre The most important factor in or estimation is graph strctre. In this paper, e focs only on link propagation from a ser to its folloers. In other ords, e assme that sers only copy links of their friends (i.e., sers they follo). In Titter, sers often find ne information sorces by brosing the friend lists of their friends, and follo some of them hich seem interesting to them. This kind of practice is not specific to Titter bt rather common to many social media. It is one of key differences beteen social media and other older media, sch as RSS (RDF Site Smmary or Really Simple Syndication) [4], here sers cannot brose other sers sbscription. We think it is one of

3 , the link created by copying the original link, the original link,, imitated one of its friends t 3 t 2 t 2 t 2 t t 3 Figre : Triadic closre prodced as the reslt of s copying of s follo link to. the featres that promoted the groth of social media oer the older media. In addition, in Titter, sers can reteet (i.e., forard) a teet from their friends to their folloers, and hen sers find a teet reteeted by a friend interesting to them, they often create direct follo links to the accont that originally posted the teet. Similar forarding fnctions are fond in many social media. These obserations are the rationale of or assmption that link propagation most often occrs from sers to their folloers. Of corse, there are many other ays for sers to kno ne information sorces, sch as s from friends and the recommendation serice by Titter. Hoeer, or prpose is not necessarily to inclde all of them, bt to choose information that are accrate and sefl for the prediction of ftre poplarity of acconts. In other ords, precision is more important than recall for or prpose. For this reason, e conseratiely focs only on the link propagation from acconts to their folloers, information on hich is expected to be more accrate than information e can obtain for other types of link propagation. Inclsion of other types of link propagation into the model is an interesting direction for ftre research. If e assme that link propagation only occrs from a ser to its folloers, a link created by imitation mst be a part of a triangle consisting of three links: an original link, a link created by copying it, and a link from the ser ho copied the link to the ser hose link as copied. Figre shos an example of sch a triangle. In this example, the ser created a link, by copying a link from its friend, i.e., by copying,. In other ords, a link to propagated from to its folloer. We first collect candidates of links created by imitation by finding triangles of this form. We call sch candidate triangles triadic closres. We define a predicate Strctre(,, ) that determines if,, V form a triadic closre as follos: DEFINITION. Strctre(,, ), the predicate determining if,, V form a triadic closre: Strctre(,, ), E, E, E. We se this constraint on the strctre as the main factor for identifying links created by imitation. We se three other factors (time order of links, link reciprocity, and the similarity beteen sers) as optional factors for frther narroing don the candidates. 3.2 Time Order of Link Creation The first optional factor is time order of link creation. In a triadic closre, the link created by copying mst be neer than the other to links. In the example in Figre,, mst be neer than, and,. Otherise, it mst not be a reslt of copying. The information necessary for checking this constraint can be obtained from the crrent Titter data. Titter API proides fnctions that retrn a list of folloers and a list of friends of a gien ser. These fnctions retrn lists sorted by time hen they became Figre 2: In the triangle at left,, may be a copy of,, bt in the triangle at right,, can neer be. The condition can be determined by the order of, in the friend list of and the order of, in the folloer list of, as shon at middle, here the to sets of and 2 represent their time order. folloers or friends from the neest one to the oldest one. Let idx(, l) denotes the position of in the list l. A predicate representing hether a triadic closre satisfies the necessary temporal condition is then defined as follos: DEFINITION 2. Time(,, ), the predicate representing the condition on temporal order of link creation in a triadic closre consisting of,, V : Time(,, ) idx(, Friends()) < idx(, Friends()) idx(, Folloers()) < idx(, Folloers()). Note that a neer link has a smaller index in these lists. Figre 2 illstrates examples of alid triangle (left) and inalid triangle (right). It also shos ho e can check the condition (middle). The accont mst be neer than in the friend list of (temporal order represented by and 2 ith circles), and mst be neer than in the folloer list of (temporal order represented by and 2 ithot circles). One disadantage of this optional condition is that e need to store the time order of friends of and folloers of. 3.3 Reciprocity of Links The second optional factor, reciprocity of links, is sed for distingishing links to information sorces from the other types of links. Follo links in Titter can be classified into seeral types, sch as links to information sorces and links to personal friends [2, 20]. There is also a practice called folloback. In Titter, some sers follo back to its folloers as an act of cortesy. Among these three types of links, links of the latter to types are sally reciprocal. Personal friends sally link to each other [23, 9], and links created by folloback are alays reciprocal. On the other hand, links to information sorces are sally non-reciprocal nless the information sorce is a type of ser ho follos back to its folloers. Therefore, in Titter, non-reciprocal links are more likely to be links to information sorces than reciprocal links are [23, 9]. For the discoery of early adopters, links to information sorces are important. Therefore, e shold exclde the other types of links from the candidates of links created by imitation. As it is difficlt to flly distingish links to information sorces from the others, e again conseratiely exclde reciprocal links becase it excldes most of the other types of links (hile it also excldes some links to information sorces). Another reason e shold exclde reciprocal links is that links created by copying links from early adopters are sally non-reciprocal.

4 information sorce, non-reciprocal copier,, either reciprocal or non-reciprocal early adopter either reciprocal or non-reciprocal Figre 3: Exclsion of reciprocal links from candidates. mltiple candidates of the original link n a link created by imitation Figre 4: Follo link, is part of many triadic closres and it is not obios ho in,..., n as imitated by. We define a predicate representing the non-reciprocity condition for a candidate link in a triadic closre by the formla belo: DEFINITION 3. Noec(,, ), a predicate representing noeciprocity condition for a triadic closre consisting of,, : Noec(,, ) Folloer(). Figre 3 illstrates this constraint. In a triadic closre shon in Figre 3, the link beteen and mst be non-reciprocal, hile the other to links may be either reciprocal or non-reciprocal. We expect that e can distingish triadic closres corresponding to a circle of friends and those corresponding to imitation of early adopters by sing this constraint. Or experimental reslt, hich ill be shon in Section 7, shos that e can actally improe the precision by sing this constraint. In this paper, e simply exclde reciprocal links from the candidates, bt it is also possible to gie some smaller eights to triadic closres that hae reciprocal links beteen and. 3.4 Similarity beteen Interests of Users Een if e find a triadic closre and the links in it satisfy the conditions aboe, the candidate link in it may not actally be a copy of the link in that triadic closre. If the candidate link is also a component of many other triadic closres, it may be a copy of another link in another triadic closre. Figre 4 illstrates sch a sitation. In this example, the follo link from the ser to the ser is a part of many triadic closres, and it is not obios ho in,..., n as actally imitated by. When e hae sch mltiple candidates, instead of selecting one of them as the original link, e assign each of them the probability that it is really the imitated one. The simplest ay to assign the probability is to assign eqal probability to all the candidates. We also designed and tested a method that assigns probability that is proportional to the similarity beteen interests of related sers. This eighting scheme is based on an assmption that link propagation is more likely to occr hen the interests of related sers are similar to each other. In Titter, arios sers ith arios interests pblish or collect information. Early adopters mst also hae some specific interests, and each early adopter mst be good at finding ne sefl information sorces only on those specific topics. Similarly, sers imitating early adopters also hae some specific interests, and they are more likely to imitate early adopters hose interests are similar to theirs. Or eighting scheme comptes eights gien to each candidate based on these assmptions. For example, sppose e hae the graph shon in Figre 4. If the interests of i and are similar, is likely to be an information sorce on a topic for hich i is a good early adopter. Similarly, if the interests of and i are similar, is more likely to imitate i than other j hose interests are not similar to that of. We measre similarity beteen interests of to sers by the similarity of their friend lists. Users folloing similar information sorces mst hae similar interests. Similarity beteen, V, denoted by Sim(, ), is defined as follos: DEFINITION 4. Similarity beteen interests of and : Sim(, ) = Friends() Friends() Friends() Friends(). The details of ho to assign eighted probability to candidates is explained in Section Ptting Together We hae explained for kinds of information e se: graph strctre, time order of link creation, reciprocity of links, and similarity of friend lists of sers. Notice that all of them can be obtained easily only by craling the neighbors of the target ne acconts in the crrent Titter graph. This is one important adantage of or method as explained before. In Titter, link propagation to folloers is especially likely to occr hen sers hae receied interesting messages reteeted by their friends. Or method, hoeer, does not se the information on reteeting becase it reqires monitoring of the teet stream, and e old lose the adantage of or method mentioned aboe. No e explain ho e compte Copy(), imitation freqency of a ser, by sing these for kinds of information. We se graph strctre as the main factor, and se the three other factors as optional factors. We tested all eight combinations of the three optional factors in or experiment, and the reslt shos that the link reciprocity are highly sefl in most cases, bt time order of links and similarity beteen sers are not sefl in most cases. We first estimate P, (), the probability that a link, as created by copying a s link,, by the formla belo: DEFINITION 5. P, (), the probability that the link, E is a copy of, E: here p, () P, () = x V p, (x) p, (x) =Strctre(x,, ) Time(x,, ) Noec(x,, ) Sim(x, ) Sim(, x). The formla aboe corresponds to the case here e se all three optional factors. When e do not se some of them, e simply remoe terms corresponding to them from the formla of p, (x). In this paper, e defined Strctre(x,, ), Time(x,, ), and Noec(,, ) as Boolean predicates. Their ales are interpreted as or 0 in the formla aboe. They are sed to gie the score 0 to candidates that do not satisfy the corresponding conditions. Hoeer, it is easy to generalize these predicates to fnctions that take some eight ales that e ant to gie to candidates. For example, e can define Noec(,, ) as a fnction that gies

5 smaller eights to triadic closres here, is reciprocal. Sch generalization is an interesting direction for ftre research. On the other hand, e defined Sim(x, ) and Sim(, x) as sch fnctions that gie proper eights to mltiple candidates. Sim sally takes ery small ales, bt they are normalized becase of the denominator of the formla defining P, (). We then estimate Copy() by the formla belo: DEFINITION 6. CF (), the expected ale of Copy() : CF () = P, ()., E We estimate the expected ale of the nmber of times as imitated by smming p the probability that each candidate link is a copy of the link of. By sing this CF (), e compte early adopter score of each accont, and e also compte ftre poplarity score of each accont based on the early adopter scores of its folloers. 4. EARLY ADOPTER SCORE: E We first define early adopter scores of acconts by sing CF () in this section. Becase e assme that sers copy links only form their friends (in other ords, becase of Strctre(x,, )), CF () takes its maximm ale hen all folloers of copied all friend links of. Therefore, 0 CF () Folloers() Friends(). We then define I(), the imitation ratio of V, as follos. DEFINITION 7. I(), the imitation ratio of : I() = CF () Folloers() Friends(). When the denominator is 0, e let I() = 0. I() approximates the probability that a link of is imitated by its folloer. It can also be regarded as a ariation of clstering coefficient of nodes in directed graphs ith special restrictions on the direction of edges. Based on I(), e define the early adopter score of. We define it in to ays, and compare their performance by the experiment later. Both definitions try to estimate the expected nmber of link propagation throgh, bt they are based on different assmptions. The first definition is based on the folloing assmption. Sppose a ne information sorce is nely folloed by an early adopter. We then expect that each of the folloer of ill follo independently in the probability I(). Therefore, the expected nmber of ne follo links created by imitating is Folloer() I(). Hoeer, e predict ftre poplarity of an accont based on the crrent snapshot. Een if a recently created accont is folloed by an accont in the snapshot, if most folloers of already hae links to in the snapshot, e cannot expect that many sers ill nely follo by imitating,. With inclding this factor in the comptation, e define E (, ), the first ariation of an early adopter score of ith respect to, as follos. DEFINITION 8. E (, ), the early adopter score of ith respect to (ariation ): E (, ) = I() Folloers() \ Folloers(). This corresponds to the expected increase of the nmber of folloers of throgh. The second definition of the early adopter score of is based on the folloing assmption. Sppose an information sorce is folloed by an early adopter in the crrent snapshot. Some of the folloers of already hae links to. The other folloers of are not likely to follo from no becase they hae not done so ntil no. Hoeer, the ne folloers that ill obtain from no ill follo by imitating in the probability I(). The nmber of folloers that ill obtain from no are nknon, and e simply assme that it is a constant n for any. Under this assmption, the expected increase of the nmber of folloers of throgh is n I(). Becase e se early adopter scores for compting ranking scores of acconts, e can ignore the constant n, and e define the second ariation of the early adopter score as follos. DEFINITION 9. E 2(, ), the early adopter score of (ariation 2): E 2 (, ) = I(). The second parameter of E 2 (, ) is sed only for the compatibility ith the first ariation E (, ), and is not actally sed in this second ariation. There is another ay to interpret E 2 (, ). I() represents ho good is as an early adopter. If a ne accont is folloed by a good early adopter, e can expect that the qality of is high, and therefore, e can expect that it ill hae many folloers in ftre, no matter these ne folloers old find throgh or not. Therefore, e can simply se I() for compting ftre poplarity scores of acconts folloed by. 5. FUTURE POPULARITY SCORE: F By sing the early adopter score defined aboe, e next define the ftre poplarity score of acconts in this section. We se this score for ranking ne acconts based on its prospectie poplarity. The simplest ay to define it old be to sm p the early adopter scores of all the folloers of : DEFINITION 0. Fi Σ (), sm-based ftre poplarity score of (simple definition): Fi Σ () = E i (, ) Folloers() here i is either or 2. This simple definition, hoeer, has a problem hen e se E. E (, ) represents the expected increase of the folloers of throgh, and if and 2 hae some common folloers, simply smming p E (, ) and E ( 2, ) old doble-conts those common folloers. Therefore, e shold define F Σ () in the folloing ay: DEFINITION. F Σ (), sm-based ftre poplarity score of (reised definition for E (, )): F Σ () = P ( p(,, )) Folloers(Folloers()) Folloers() here P (e) is the probability of the eent e and p(,, ) is the eent that copies the link,. That is, e sm p the probability that a folloer of some folloer of ill follo by imitating any of the folloers of. We compte this probability by assming that eents p(, i, ) and p(, j, ) are independent for i j and P (p(,, )) = I(). According to or preliminary experiment, hoeer, the performance of F Σ () in this definition and that of the preios simpler definition hae no significant difference. Therefore, e se the preios simpler definition for both F Σ () and F Σ 2 (). A disadantage of these sm-based definitions of ftre poplarity scores is that it basically gies higher scores to acconts ith many folloers. Or prpose is to find ne promising acconts

6 een hen they do not hae many folloers no. Therefore, if a ne accont has only a fe folloers, bt all the folloers are ery good early adopters, e ant to assign a high score to it. Another ay to define ftre poplarity scores ith emphasis on sch an aspect is to se g-index [5] instead of sm in the folloing ay. DEFINITION 2. F g i (), ftre poplarity score of based on g-index: F g i () = RG({Ei(, ) Folloers()}) here i is either or 2 and RG(S) is a fnction that comptes the rational g-index [2] of the set of real nmbers S. F g i () is a rational g-index of the set of the early adopter scores of the folloers of. Gien a set S of ales, its g-index can be compted by the folloing procedre. First e make a list L by sorting ales in S in decreasing order. Let L[i] be the i-th ale in L. We then find a maximm g that satisfies g 2 c i g L[i], here c is a parameter. Sch a g is the g-index of S. G-index of a set S is affected only by largest ales in S. G-index only takes natral nmbers, bt rational g-index is an extension of g-index to rational nmbers [2]. The methods explained aboe compte the ftre poplarity score of an accont based on the early adopter scores of its direct folloers. We can easily extend this method to a recrsie method based on arios infection models. As explained before, E (, ) represents the expected nmber of link propagation from to its folloers and E 2 (, ) represents the probability that links are propagated from to its folloers. We can interpret them as the propagation probability of a disease, and can rn some algorithms that predict ho many sers ill be infected starting from a gien infected ser. We tested sch recrsie ersions of or method by sing some simple algorithms, bt sch a recrsie method did not improe the performance of or method in or experiment. We ill inestigate this problem in or ftre research. 6. TWO ALGORITHMS We hae deeloped to algorithms to compte CF (), hich is a core part of the comptation of E i (, ) and F i (). The first one comptes CF () only for a gien accont, and the second one comptes CF () of all acconts in the gien graph. When e only ant to compte the ftre poplarity score of specific acconts, e only need to compte early adopter scores of their folloers. The first algorithm is sefl for sch a case. We omit the formal description, bt it first collects all candidate triadic closres simply by retrieing all the friends of the folloers of the gien accont, and checking if they are also friends of. If e assme folloer lists and friend lists are stored in hash tables, its time complexity is in O(d 2 ) here d is the aerage degree of the graph. For each fond triadic closre consisting of,,, e also need to collect other candidates of the imitated ser,,..., n (see Figre 4). It can be done by compting Folloers() Friends(), and its complexity is in O(d). Therefore, the total complexity of the comptation of CF () of the gien is in O(d 3 ). Becase an accont has d folloers in aerage, e can compte the ftre poplarity score of an accont in O(d 4 ). The second algorithm comptes CF () of all acconts in the graph in a similar ay as the formla in Definition 6. It examines each follo link in the graph one by one. For each link, e collect candidate links that can be the original of the link, and gie the oner of each link the probability that it is the original. At each accont, these gien probabilities are accmlated. By smming p all these probability ales gien to a ser, e obtain the ale of CF (). This algorithm is shon in Algorithm. Algorithm Compting CF () of All Nodes CF () := 0 for all V for all, E do U := Folloers() Friends() s := 0 for all U do p[] := p, () s := s + p[] end for for all U do CF () = CF () + p[]/s end for end for retrn CF () for all V This algorithm ealates p, () for O(md) times here m is the nmber of edges in the graph and d is the degree of nodes. When e se no optional factors, e actally do not need to compte p, () in Algorithm becase it alays retrns gien that, E and Folloers() Friends(). Therefore, the time complexity of this algorithm is in O(md) in that case, if e assme Folloers() and/or Friends() are stored in a hash table. Een if e inclde the factor Time and Noec in p, (), e can simply skip the loop for, that does not satisfy these conditions, and the complexity of the algorithm is still in O(md). When e inclde the factor Sim, the comptation of Sim inside the loop is in O(d), so the oerall complexity is in O(md 2 ). The first algorithm, hich rns in O(d 4 ) for each accont, is far faster hen e ant to compte ftre poplarity scores only for a small nmber of acconts, bt according to or experiment, the latter algorithm, hich rns in O(md) for the entire graph, is faster een hen e compte early adopter scores for slightly less than a thosand of nodes. 7. EXPERIMENT In this section, e ealate or method by the experiment on a Titter data set. We first explain the data set sed in or experiment and the procedre of or experiment. After that, e ill explain the baseline methods ith hich e compared or methods. Finally, e sho and discss the reslts of the experiment. 7. Data Set We se the snapshot of a part of Titter follo graph created by Li et al. in May 20 []. This data set as prodced by random craling of follo links starting from randomly selected 00,000 sers. In this graph, V = 2, 604, 65 and E = 284, 885, 00. Let D (V, E) denote this graph. We extracted all acconts in D that ere ithin to eeks, three eeks, and for eeks from its creation date, and that had at least 0 folloers, 20 folloers, and 30 folloers at the time of D. Let T 2 0, T 2 20, T 2 30, T 3 0, T 3 20, T 3 30, T 4 0, T 4 20, T 4 30 denote these data sets. Therefore, T x 30 T x 20 T x 0 and T 2 x T 3 x T 4 x. Their size is shon at the top of Table. 7.2 Procedre of Experiment We rn or experiment in the folloing procedre:. For all acconts in the data set T x y, e estimated their ftre poplarity both by or methods and by arios baseline methods, and prodce lists of the acconts sorted in the order of their estimated ftre poplarity.

7 2. We sed the nmber of their non-reciprocal folloers as of May 205, hich e denote FW 205 (), as the tre ftre poplarity of the information sorces, and prodce a list of the acconts sorted in that order. 3. We compare the list prodced by each estimation method and the list based on FW 205 (). For the comparison, e sed Spearman s rank correlation coefficient (ρ) and the normalized discont cmlatie gain (ndcg). Spearman s ρ reflects the accracy of the hole ranking, hile ndcg only reflects the accracy of the top part of the ranking. 7.3 Tested Proposed Methods In Section 4, e shoed to definitions of early adopter scores, E (, ) and E 2(, ), and e also shoed to ays to calclate ftre poplarity scores, Fi Σ and F g i. We also hae three optional factors in the comptation of CF (), and there are eight combinations of them. In total, e hae 32 combinations of them and e compared them in or experiment. In this paper, hoeer, e omit the reslt of the methods that se temporal order of links becase it did not improe the accracy of or method, and also becase of the space limitation. In the folloing, let r denote the optional factor of link reciprocity, and let s denote the optional factor of similarity beteen sers. For example, F2 Σ denotes or method that ses E 2 (, ) and Fi Σ (), and only ses the link reciprocity option. The parameter c for g-index as hand-tned to the folloing ales in each case. E : 50000, E + r: 00000, E + s: 50000, E + r, s: 50000, E 2:, E 2 + r: 0, E 2 + s:, E 2 + r, s: Baseline Methods We next explain the baseline methods e compared ith or method, and also explain their parameters. Folloers (FW): It measres the ftre poplarity of ne acconts by the nmber of their crrent folloers in May 20. Noeciprocal Folloers (FW ): It measres the ftre poplarity of ne acconts by the nmber of their non-reciprocal folloers in May 20. As explained in Section 3.3, non-reciprocal follo links are likely to be links to information sorces. Friends (FR): It measres it by the nmber of friends in May 20. Noeciprocal Friends (FR ): It measres it by the nmber of their non-reciprocal friends in May 20. HITS: It comptes athority scores and hb scores of acconts [9], and e se the athority score as the indicator of ftre poplarity. In this experiment, e set the nmber of iterations to 0, ith hich the scores sfficiently conerged. Noeciprocal HITS (HITS ): The same as HITS, bt it comptes athority scores and hb scores on the graph consisting only of non-reciprocal links. The nmber of iterations is 0, ith hich the scores sfficiently conerged. PageRank (PR): It estimates the ftre poplarity by sing PageRank score [5]. In this experiment, e set the damping factor d = 0.9, hich prodced the best reslts among 0., 0.2,..., 0.9, and e set nmber of iterations to 00, ith hich the scores sfficiently conerged. Noeciprocal PageRank (PR ): The same as PageRank, bt it comptes PageRank scores on the graph consisting only of noeciprocal links. We set the damping factor d = 0.9, hich prodced the best reslts among 0., 0.2,..., 0.9, and e set the nmber of iterations to 00, ith hich the scores sfficiently conerged. Adamic/Adar (AD Σ, AD µ): It estimates the ftre poplarity of by estimating the probability of ne links to from other nodes based on Adamic/Adar index []. Gien an accont, e collect all its friends, and also all the folloers of those friends. Then nmber of acconts e+0 e+03 e+05 e triads nmber of acconts e+0 e+03 e+05 e ratio of triads Figre 5: Distribtion of CF () (left) and I() (right) plotted in log-log graphs. e compte Adamic/Adar index for and all these folloers ith regarding their common friends as the common items. The ordinary Adamic/Adar sms all the obtained index ales, bt e compared both smmation (AD Σ ) and mean (AD µ ). 7.5 Reslt and Discssion We first sho the distribtion of the early adopter scores of all acconts in D. Figre 5 shos the distribtion of CF () (left) and I() (right), plotted in log-log graphs. Is is knon that the nmber of folloers in Titter follos the poer la, and CF () can be larger hen has more folloers (and more friends), bt distribtion of CF () is more skeed than the poer la distribtion. On the other hand, the distribtion of the imitation ratio I() has a peak in the middle, except for a higher peak at the left, hich incldes many acconts hose I() = 0. We next compte the correlation beteen the ranking based on FW 205 and the ranking by each method. The left half of Table lists Spearman s ρ ales beteen them. In each colmn, the best scores among the baseline methods and the best scores among the 6 ariations of or method are shon in bold fonts. In addition, the best scores among the both of them are nderlined. Among the baseline methods, FW and FR achieed higher correlation than those ithot reciprocal links. On the contrary, HITS and PR ithot reciprocal links achieed higher correlation than those ith reciprocal links. Among them, HITS as the best, and PR follos. These methods hae higher correlation for the data set inclding acconts ith more folloers. On the other hand, AD methods hae negatie correlation, and AD µ has srprisingly high negatie correlation, hich means it is a good index for predicting ftre poplarity of acconts. It also has higher correlation for the data set inclding acconts ith more folloers. Hoeer, or method, especially F2 Σ, achiees een higher correlation except for to cases, T20 2 and T30, 2 here or method is otperformed by AD µ. These to data sets inclde acconts that hae obtained many folloers (more than 20 or 30) ithin a short time (2 eeks). This means that AD µ orks better for acconts that started to get poplar soon after the creation. We performed some error-analysis, and the reslt shos that the main factor loering the accracy of all the methods is the existence of many acconts that had some folloers in D bt are inactie or deleted as of 205. Therefore, e also created data sets ˆT that only inclde acconts that are actie in 205. The right half of Table shos the reslt on these data sets. For ˆT, all methods achiee higher accracy than for T, i.e., the data set inclding inactie sers. This sggest that e shold combine or method ith some method that predict if a gien accont ill last long or not. Or method again otperforms baselines. Notice that F2 Σ otperforms the baseline methods for all ˆT althogh their scores are not in bold fonts for ˆT 0, x here F2 Σ is otperformed by F g 2.

8 data set T0 4 T0 3 T0 2 T20 4 T20 3 T20 2 T30 4 T30 3 T30 2 ˆT 0 4 ˆT 0 3 ˆT 0 2 ˆT 20 4 ˆT 20 3 ˆT 20 2 data size FW FW FR FR HITS HITS PR PR AD Σ AD µ F Σ r s r s g r s r s F 2 Σ r s r s g r s r s PR E LR Table : Spearman s ρ beteen FW 205 and each method. FW to AD µ are baseline methods, F and F 2 are or methods. T denotes data sets inclding both actie and non-actie sers. ˆT denotes data sets only inclding actie sers. For each data set, the best scores among the baseline methods and the best scores among 6 ariations of or method are shon in bold fonts. The best scores among both of them are also nderlined. A ariation of or method F2 Σ otperforms the baseline methods except for T20 2 and T30. 2 Notice that F2 Σ otperforms the baseline methods for all ˆT althogh their scores are not in bold fonts for ˆT 0 x simply becase F g 2 achieed een better reslts for these data sets. PR E 2 is a recrsie extension of F2 Σ, hich did not improe F2 Σ. LR is a linear regression combining HITS, HITS, AD µ, F Σ, and F2 Σ, hich achieed the best accracy for all the data sets. Table also shos the reslts of PR E 2 hich is a simple recrsie extension of F2 Σ. We omit the details of this method, bt it did not improe the performance of F2 Σ. Figre 6 shos scatter diagrams beteen FW 205 and three methods that shoed high correlation, i.e., HITS, AD µ, F2 Σ, for the data set T0 2 (left) and T20 4 (right). In the scatter diagrams for HITS, there are horizontal ros of points near the bottom. They are acconts that had no non-reciprocal folloers in D. HITS and PageRank applied to the graph ithot non-reciprocal links achiee high ρ ales, bt they hae this problem. The diagrams in Figre 6 sho that the accracy of each method is not eqal in the top, middle, and bottom parts of rankings. When e directly se these rankings, the top part of the entire ranking is sally the most important. On the other hand, hen e select ne acconts related to a gien topic (e.g., by sing keyord qeries on teets or profiles), and only sho them to sers, they may be in the middle or other parts in the entire ranking, so the top part of the entire ranking is not necessarily the only important part. For ealating the performance in applications here only the top part is important, e compared baseline methods and or methods by ndcg. The ndcg is a measre of ranking qality, here accracy in the top part is more important than that in the loer part. ndcg@k is a measre that comptes ndcg only for top k in the ranking. We calclated ndcg@k of each method ith arios k ales for seeral data sets. Table 2 shos the reslt. The best scores among the baselines and the best scores among or methods are shon in bold fonts. The best scores among both of them are also nderlined. Among the baselines, HITS, PR, PR, and AD Σ achieed best scores for some cases. In this comparison, or method cold not otperform baselines in most cases. It is mainly becase topranked acconts ere already poplar at 2,3,4 eeks after the creation, and or method is not as good as some baselines for acconts that are already poplar no. Hoeer, e emphasize again that the top part of the entire ranking is not the only important part hen e select some acconts and only sho them to sers. We also calclated the correlation beteen or methods and baseline methods. Table 3 shos the reslt for T0. 4 The reslt shos that there are ery lo correlation beteen the good baseline methods, sch as HITS and AD µ, and or best methods. This sggest that e can achiee better performance by combining these methods. Folloing this obseration, e tested linear regression combining the best baseline methods and or to methods, i.e., HITS, HITS, AD µ, F Σ, and F2 Σ. We learned eight parameters for ranking positions of acconts by each inclded method so that they fit the positions of the acconts in the correct ranking, and ealated the reslt by 0-fold alidation. Table 4 shos the eight parameters e obtained. Both F Σ and F2 Σ ere gien high β ales, hich means they hae high contribtion to the reslt. All p ales are small enogh, hich shos

9 data set T0 4 T0 3 T0 FW FW FR FR HITS HITS PR PR AD Σ AD µ F 2 Σ r s r,s g r s r,s Table 2: ndcg@k by each method here gain is FW 205. Each colmn shos ales for a data set and k shon at the top. Bold entries are highest scores among the baselines and among the ariations of or method. The best scores among both are nderlined. FW FW FR FR HITS HITS PR PR AD Σ AD µ F2 Σ F2 Σ F 2 Σ(s) F 2 Σ (r, s) FW FW FR FR HITS HITS PR PR AD Σ AD µ F2 Σ F2 Σ F2 Σ (s) F2 Σ (r, s).00 Table 3: Spearman s ρ beteen each method for T 4 0. Or best methods and the best baseline methods hae lo correlation. Method β p (Intercept) < HITS HITS AD µ < F Σ r F2 Σ r Table 4: Weights gien to component methods in linear regression for T 4 0. Both F Σ r and F Σ 2 r hae high contribtion. this reslt is statistically reliable. The accracy of this combined method is shon at the line LR at the bottom of Table. This method achieed the best accracy in all cases. 8. CONCLUSION In this paper, e proposed a method of ranking ne Titter acconts according to their prospectie ftre poplarity. Or method first finds early adopters, ho are good at finding ne sefl information sorces earlier than others. Een if a ne accont crrently has only a fe folloers, if the folloers are good early adopters, e expect the ne accont ill hae many folloers in ftre. We find early adopters based on the freqency of link propagation throgh them, i.e., ho often their follo links are copied by their folloers. We also deeloped a method that estimates the freqency of link propagation throgh each ser by sing for factors: graph strctre, temporal order of creation of links, reciprocity of links, and similarity beteen interests of sers. We ealated the performance of or method by creating a ranking of ne Titter acconts based on ftre poplarity estimated by or method, and comparing it ith the ranking based on the nmber of folloers they actally obtain later. The comparison based on Spearman s ρ shos that or method otperforms arios baseline methods in most cases. Or method is especially good for sers that ere not poplar at the time of prediction. On the other hand, or method as otperformed by some baselines in the comparison based on ndcg. It is mainly becase the top part of the rankings inclde many acconts that ere already poplar at the time of prediction. Becase the ranking by or method and the rankings by the best baseline methods hae ery lo correlation, e also tested linear regression combining or method and the best baseline methods. It achieed the best accracy for all the data sets.