Novelty and Collective Attention

Transcription

1 ovely and Collecive Aenion Fang Wu and Bernardo A. Huberman Informaion Dynamics Laboraory HP Labs Palo Alo, CA 9434 Absrac The subjec of collecive aenion is cenral o an informaion age where millions of people are inundaed wih daily messages. I is hus of ineres o undersand how aenion o novel iems propagaes and evenually fades among large populaions. We have analyzed he dynamics of collecive aenion among one million users of an ineracive websie digg.com- devoed o housands of novel news sories. The observaions can be described by a dynamical model characerized by a single novely facor. Our measuremens indicae ha novely wihin groups decays wih a srechedexponenial law, suggesing he exisence of a naural ime scale over which aenion fades.

2 The problem of collecive aenion is a he hear of decision making and he spread of ideas, and as such i has been sudied a he individual and small group level by a number of psychologiss [1, 2] economiss [3], and researchers in he area of markeing and adverising [5, 6, 7]. Aenion also affecs he propagaion of informaion in social neworks, deermining he effeciveness of adverising and viral markeing [8]. And while progress on his problem has been made in small laboraory sudies and in he heoreical lieraure of aenion economics [4], i is sill lacking empirical resuls from very large groups in a naural, non-laboraory, seing. To undersand he process underlying aenion in large groups, consider as an example how a news sory spreads among a group of people. When i firs comes ou, he sory caches he aenion of a few ones, who may furher pass i on o ohers if hey find i ineresing enough. If a lo of people sar o pay aenion o his sory, is exposure in he media will coninue o increase. In oher words, a posiive reinforcemen effec ses in such ha he more popular he sory becomes, he faser i spreads. This growh is counerbalanced by he fac ha he novely of a sory ends o fade wih ime and hus he aenion ha people pay o i. Therefore, in considering he dynamics of collecive aenion wo compeing effecs are presen: he growh in he number of people ha aend o a given sory and he habiuaion ha makes he same sory less likely o be aracive as ime goes on. This process becomes more complex in he realisic case of muliple iems or sories appearing a he same ime, for now people also have he choice of which sories o aend wih heir limied aenion. In order o sudy he dynamics of collecive aenion and is relaion o novel inpus in a naural seing, we analyzed he behavioral paerns of one million people

3 ineracing wih a news websie whose conen is solely deermined by is own users. Because people using his websie assign each news sory an explici measure of populariy, we were able o deermine he growh and decay of aenion for housands of news sories and o validae a heoreical model which predics boh he dynamics and he saisical disribuion of sory lifeimes. The websie under sudy, digg.com, is a digial media democracy which allows is users o submi news sories hey discover from he Inerne [9]. A new submission immediaely appears on a reposiory webpage called Upcoming Sories, where oher members can find he sory and, if hey like i, add a digg o i. A so-called digg number is shown nex o each sory's headline, which simply couns how many users have digged he sory in he pas. If a submission fails o receive enough diggs wihin a cerain ime limi, i evenually falls ou of he Upcoming secion, bu if i does earn a criical mass of diggs quickly enough, i becomes popular and jumps o he digg.com fron page. Because he fron page can only display a limied number of sories, old sories evenually ge replaced by newer sories as he page ges consanly updaed. If a sory however, becomes very popular i qualifies as a Top 1 and says on he righ side of he fron page for a very long ime. When a sory firs appears on he fron page i aracs much aenion, causing is digg number,, o build up quickly. Afer a couple of hours is digg rae slows down because of is lack of novely and prominen visibiliy (refleced in he fac ha i moves away from he fron page). Thus he digg number of each sory evenually sauraes o a value,, ha depends on boh is populariy growh and is novely decay. In order o deermine he saisical disribuion of his sauraion number, which corresponds o he number of diggs i has accumulaed hroughou is evoluion, we measured he hisogram of he final diggs of all 29,864 popular sories in he year 26. As can be seen from Fig.

4 1, he disribuion appears o be quie skewed, wih he normal Q-Q plo of log( ) a sraigh line. A Kolmogorov-Smirnov normaliy es of log( ) wih mean and sandard deviaion.6626 yields a p-value of.939, suggesing ha follows a lognormal disribuion. Frequency Sample Quaniles ormal Q Q Plo diggs Theoreical Quaniles Figure 1: (a) The hisogram of he 29,684 diggs in 26, as on January 9, 27. (b) The normal Q-Q plo of log( ). The sraigh line shows ha log( ) follows a normal disribuion wih a slighly longer ail. This is due o digg.com's buil-in reinforcemen mechanism ha favors hose op sories, which can say on he fron page and can be found a many oher places (e.g. popular sories in 3 days and popular sories in 365 days ). I is hen naural o ask wheher, he number of diggs of a popular sory afer a finie ime, also follows a log-normal disribuion. To answer his quesion, we racked he digg numbers of 1,11 sories in January 26 minue by minue. The disribuion of log( ) again obeys a bell shape curve. A Kolmogorov-Smirnov normaliy es of log( 2 hours) wih mean and sandard deviaion.5451 yields a p-value as high as.565, supporing he hypohesis ha also follows a log-normal disribuion.

5 The log-normal disribuion can be explained by a simple sochasic dynamical model which we now describe. If represens he number of people who know he sory a ime, in he absence of any habiuaion, on average a fracion µ of hose people will furher spread he sory o some of heir friends. Mahemaically his assumpion can be expressed as 1+ X ), where X, X, 1 2 K are posiive i.i.d. random variables = ( 1 wih mean µ and variance σ 2. The requiremen ha X i mus be posiive ensures ha can only grow wih ime. As we have discussed above, his growh in ime is evenually curailed by a decay in novely, which we parameerize by a ime dependen facor r consising of a series of decreasing posiive numbers wih he propery ha r = 1 1 and r as. Wih his addiional parameer, he full sochasic dynamics of sory propagaion is governed by = ( 1+ r X ) 1, where he facor r X acs as a discouned random muliplicaive facor. When X is small (which is he case for small ime seps) we have he following approximae soluion: rs X s rs X s s = 1 = (1 + rs X s ) e = e s= 1 s= 1, (1) where is he iniial populaion ha is aware of he sory. Taking logarihm of boh sides, we obain log log = s= 1 r X s s. (2) The righ hand side of (2) is a discouned sum of random variables, which for r near one (small ime seps) can be shown o be described by a normal disribuion [11]. I hen follows ha for large he probabiliy disribuion of will be approximaely log-normal. Our dynamic model can be furher esed by aking he mean and variance of boh sides of Eq. (2):

6 E(log var(log log log ) ) = s= 1 s= 1 s rs μ μ =. (3) 2 2 r σ σ Hence if our model is correc, a plo of he sample mean of ( log log ) versus he sample variance for each ime, should yield a sraigh line passing hrough he origin wih slope µ/σ 2. One such plo for 1,11 sories colleced in January 27 is shown in Fig. 2. As can be seen, he poins indeed lie on a line wih slope sample mean sample variance Figure 2: Sample mean of log log versus sample variance, for 1,11 sories in January 27. Time uni is one minue. The poins are ploed as follows. For each sory we calculae he quaniy log log, which is he logarihm of is digg number measured minues afer is firs appearance on he fron page, minus he logarihm of is iniial digg number. We collec 1,11 such quaniies for 1,11 sories. We compued heir sample mean y and sample variance x, and mark he poin (x,y). This is for one. We repea he process for =1,2,,144 and plo 144 poins in oal (i.e. 24 hours). They lie roughly on a sraigh line passing hrough he origin wih slope The decay facor r can now be compued explicily from up o a consan scale. Since we have normalized r 1 o 1, we have

7 r = E(log ) E(log 1 ). (4) E(log ) E(log ) 1 The curve of r esimaed from he 1,11 sories in January 27 is shown in Fig. 3(a). As can be seen, r decays very fas in he firs wo o hree hours, and is value becomes less han.3 afer hree hours. Fig. 3(b,c) show ha r decays slower han exponenial and faser han power law. Fig. 3(d) shows ha r can be fi empirically o a.4 sreched exponenial relaxaion or Kohlrausch-Williams-Was law [12]: r ~ e. The halflife τ of r can hen be deermined by solving he equaion τ e = d e d. (5).4 A numerical calculaion gives τ =69 minues, or abou one hour. This characerisic ime is consisen wih he fac ha a sory usually lives on he fron page for a period beween one and wo hours. r log(r ) (a) (b)

8 log(r ) log(r ) log() (c) (d) Figure 3: (a) The decay facor r as a funcion of ime. Time is measured in minues. (b) log(r ) versus. r decays slower han exponenial. (c) log(r ) versus. r decays faser han power law. (d) log(r ) versus.4. The slope is approximaely -.4. The sreched exponenial relaxaion ofen occurs as he resul of muliple characerisic relaxaion ime scales [12, 13]. This is consisen wih he fac ha he decay rae of a sory on digg.com depends on many facors, such as he sory's opic caegory and he ime of day when i appears on he fron page. The measured decay facor r is hus an average of hese various raes and describes he collecive decay of aenion. These measuremens, comprising he dynamics of one million users aending o housands of novel sories, allowed us o deermine he effec of novely on he collecive aenion of very large groups of individuals, while nicely isolaing boh he speed of propagaion of new sories and heir decay. We also showed ha he growh and decay of collecive aenion can be described by a dynamical model characerized by a single novely facor which deermines he naural ime scale over which aenion fades. The

9 exac value of he decay consan depends on he naure of he medium bu is funcional form is universal. These experimens, which complemen large social nework sudies of viral markeing [8] are faciliaed by he availabiliy of websies ha arac millions of users, a fac ha urns he inerne ino an ineresing naural laboraory for esing and discovering he behavioral paerns of large populaions on a very large scale [14]. References [1] Kahneman, D. Aenion and effor. Englewood Cliffs,.J.: Prenice Hall. [2] Pashler, H. E. The psychology of aenion. MIT Press (1998). [3] Camerer, C. The behavioral challenge o economics: undersanding normal people. Paper presened a Federal Reserve of Boson meeing (23). [4] Falkinger, J. Aenion Economies. Forhcoming in Journal of Economic Theory (23). [5] Pieers, F. G. M., Rosbergen, E. and Wedel, M. Visual aenion o repeaed prin adverising: A es of scanpah heory. Journal of Markeing Research 36(4), (1999). [6] Dukas, R. Causes and consequences of limied aenion. Brain Behavior and Evoluion 63, (24). [7] Reis, R. Inaenive Consumers. Journal of Moneary Economics Vol. 53, (26). [8] Lefkovic, J., Adamic, L. and Huberman, B. A. The dynamics of viral markeing. Proceedings of he ACM Conference on Elecronic Commerce (EC 26). [9] How Digg Works. hp:// [1] Privae communicaion wih he digg.com suppor eam.

10 [11] Embrechs, P. and Maejima, M. The cenral limi heorem for summabiliy mehods of i.i.d. random variables. Probabiliy Theory and Relaed Fields, Volume 68, umber 2 (1984). [12] Lindsey, C. P. and Paerson, G. D. Deailed comparison of he William-Was and Cole-Davidson funcions. J. Chem. Phys. 73(7) (198). [13] Frisch, U. and Sornee, D. Exreme deviaions and applicaions. J. Phys. I France (1997). [14] Was, D. A weny firs cenury science. aure 445, pp. 489 ( 27)