[PDF] Evolution of Retweet Rates in Twitter User Careers: Analysis and

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Evolutionof RetweetRates inT witterUser Careers: AnalysisandModel

Kiran Garimella,

1

Robert West

2 1 MIT 2 EPFL garimell@mit.edu,robert.west@epfl.ch

Abstract

Westudy thee volution ofthenumberofretweetsrecei ved by Twitteruserso ver thecourseoftheir"careers"on the platform. Wefindthat ona verage thenumber ofretweets receivedbyusers tendsto increaseo ver time.This ispartly expectedbeca useuserstendto graduallyaccumulate followers. Normalizing bythe number offollowers,ho wev er,rev eals

that therelati ve,per-followerretweet ratetendstobenon-monotonic, maximizedat a"peak age"after whichit doesnot

increase, ore vendecreases.Wede velop asimplemathematical model ofthe processbehind thisphenomenon, whichassumes a constantlygro wingnumberoffollo wers,each ofwhom loses interesto vertime.Wesho wthat thismodelissuf ficient to explainthenon-monotonic nature ofper -followerretweet rates, withoutan yassumptionsaboutthe qualityof content posted atdif ferenttimes.

1 Introduction

This paperstudies thee volution ofindividualimpactin

the contextofsocial media.W efocus onT witterusers're-tweet rates - the numberof retweetsrecei ved bytheirposts

("tweets") - as aw aytomeasurethe impactand reachof their content. Weaimto understandho wthis measureof impact changes overthecourseof users'"careers" (thetime since theyjoined theplatform). Specifically, weask: Dousers' retweet ratesfollo wcertainpatterns?Ho wdoes auser' sre- tweet ratedepend onthe sizeof th eir audience?And whenin a user'scareeris theircontent retweetedmost? Individualimpact hasbeen previously studiedin contexts other thansocial media,including science(Radicchi and Vespignani2009; Sinatraet al.2016), work (DeNisiand Stevens1981;Barrick andMount 1991),sports (Yuceso y and Barabási2016), andart (Dennis1966; Liuet al.2018).

Measuring impacton socialmedia is,ho wev er, inherentlydifferent,due tothe highlydynamic natureof onlinesocial

networks,which maygro w(and sometimesshrink)byorders of magnitudewithin very shorttimeframes,accompanied by numerous confoundingf actors,includingtemporalchanges in audiencesize (numberof followers), productivity (number of postswritten), experience (ageontheplatform), interests (content ofposts), andconte xt(e xternalevents suchelec- tions). Studyingthe eff ectsofthesefactors,and separating Copyright©2021, Associationfor theAlv ancementof ArtiÞcial

Intelligence (www.aaai.org).Allrightsreserv el.

them fromtrue impact,is nottri vial.F orinstance, weobserve that ona veragetheimpactofinli vilualuse rs,as measu rel by thenumber ofretweets theirconte ntrecei ves, increases overthecourse oftheir careers.Is thislue toa realchange in contentquality ,ormightit simplybe explainel bythe fact that usersaccumulate morefollo werso vertime?or lueto their increasinge xpertiseinusingthe platform?or lueto externale vents? Because ofthe challengingnature ofthe problem,research

into long-termtrenls ofinli vilualimpact onsocialmelia,anl onthe factors thatinßuencethesetrenls, hasbeen limitel

to late.Most researchhas attemptelto unlerstanlthe im- pact ofisolatel pieces ofcontent, e.g., bypredicting retweet counts onT witter(Suhetal. 2010;Figueiredo andAlmeida

2011; Kupavskiietal.2012; Martinet al.2016), videovie w

counts onY ouTube(FigueiredoandAlmeida2011), orimage likecounts onInstagram (Zhanget al.2018). Onthe contrary, the literatureis muchthinner reg ardingthe natureandroleof the impactof individual users.

In orderto make progressinthisdirecti on,we usean

extensivedatasetbuilt fromT witter,whichspans nearlya decade andcontains completecareers ofusers witha wide range ofaudience sizeand experience. We alsoobtainhis-

torical follower-countestimatesof theusers inour datasetfrom theInternet Archiv e.Buildingonthisnov eldataset, we

reconstruct thecareers ofusers - from theirv eryfirst posts until datacollection time - andcharacterize theebbi ngand flowingpatterns ofindi vidualimpact. Our expositionproceedsin threeparts. InSec. 2,an empir- ical analysisof absoluteand relativ e(per -follower)retweet rates overthecourseof Twitter usercareers rev ealsanunex-

pected non-monotonicprogression. InSec. 3,we proposea mathematical modelthat capturesthe crucialnon-monotonic

aspect ofthe empirical datawhilerelyingon two simple, intuitiveassumptionsonly .In Sec.4,wev erifythat these assumptions ares upportedbytheempirical data.Finally ,in Sec. 5,we discussour findingsand sketch futuredirections.

2 Analysisof RetweetRates Over Time

Data.Westart froma large datasetspanning almost10years (2009-2018) andconsisting ofall 2.6billion tweetsposted by themore than600K userswho retweeteda U.S.presiden-

tial orvice-presidential candidateat leastfi ve timesduring Proceedingsofthe Fifteenth InternationalAAAIConferenceonWeba ndSocial Media(ICWSM2021)

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(b) Per-followerretweetrate Figure 1:Ev olutionof(a)retweet counts, (b)per -follower retweet ratesin usercareers (runninga verages ofwidth 5). that period(Garimellaetal. 2018).F oreach user, thedataset contains alltweets they posted.Wefocus oneach user'sorig- inal contentand discardretweets ofothers' content.F oreach tweet, wea lsoobtainedtheset ofusers whoretweeted it and thenumber oftimes itw asretweeted (asof June2018). Furthermore, weobtained theset offollo wersfor eachuser (as ofJune 2018),as wellas timeseries ofhistorical follower counts estimatedvia theInternet Archiv e's WaybackMa- chine, usingGarimella andW est's (2019)method.Reliable historical estimatesof follower countscouldbeobtained for

23K users,who posteda totalof 111Mtweets. Ouranalyses

focus onthese 23Kusers. 1 Retweet countso vertime.Westart withan empiricalanaly- sis ofthe ev olutionofretweetcountsov erthe courseof users' careers. Fig.1a tracksaggre gate retweetcountsatweekly granularity (wherex=1corresponds toa user's firstweekon Twitter).T oobtainthesecurv es,we firstcomputed, foreach user andweek, themean numberof retweetsobtained by the userfor tweetsposted thatweek, andthen computed,for each week,the mediano ver allusers.Wechoseto consider medians ofmeans becauseretweet-count distributions are heavy-tailed,with asmall numberof "viral"tweets being orders ofmagnitude morepopular thanmost tweets(Goel et al.2016). Suchoutliers canmatter alot forindi vi dualusers, so tocapture them,we considerper -usermeans. Wedonot, however,wantweeklyaggre gatestobe ske wedbyoutlier tweets, sowe considerper -weekmediansof per-usermeans. Note thatFig. 1acontains fourcurv es,each corresponding to adif ferentcareer-lengthrange(users whoha ve beenon Twitterfor 100-200weeks, 200-300weeks, etc.).Study- ing userswith diff erentcareerlengthsseparatelyallows us to teaseapart thee volution ofusersfromtheev olutionof the platformand toa void instancesofSimpson'sparadox, which mightensue bymixing usergroups withsystematically differentcharacteristics.

Fig. 1asho wsthatthenumber ofretweets obtainedper

week tendsto grow overthe courseofusercareers(withthe exceptionof afinal dropfor thegroup ofusers whoha ve been onT witterforthelongest time).Moreo ver ,the curves haveano verall concaveshape,withgro wthratesslowing downas timeprogresses. 1 Data availableathttps://github.com/epß- llab/retweet-e volution per weekmay bee xpected,considering thatuserstendto accrue followersov ertime.Toaccountfor thisef fect,we nextconsider anormalized version ofthe retweetrate,where each user'sweeklymean numberof retweetsper tweetis additionally dividedbythe numberof followers theus erhad at thetime. Aggreg atingagainover allusersperweek viathe median givesrisetoFig. 1b.W eobserv ethat forsome user groups thetime seriesof per-follo werretweet rateshavea non-monotonic shapewith asingle peak.This isparticularly the casefor userswho hav ebeen membersofTwitterfor shorter periodsof time(100-300 weeks),whereas forlonger - term members(300-500 weeks),the curves grow uptosome point, fromwhere onthe ystart toplateau.

Evidence fora "peakage" hasbeen foundin awide va-

riety offields suchas poetry, mathematics,and theoretical physics(Adams 1946;Dennis 1966;Lehman 1953).In line with thisliterature, onepotential narrativ efor explainingthe non-monotonicity couldbe thatusers startas "newbies", then learn tobe goodsocial mediausers, andfinally becomeold and losetouch withthe restof thecommunity (Danescu- Niculescu-Mizil etal. 2013).In whatfollo ws,we willsee, however,thatsuchacomple xnarrati ve isnot necessarytoex- plain thedata. Amuch simplermodel suffices togenerate the crucial qualitativeshapeofthe retweets-per-follo wercurv es.

3 Modelof RetweetRates Over Time

Our simplemathematical modelm akes twoassumptions, shownvisually inFig. 2aand tobe verified empiricallylater (Sec. 4).Assumption Istates thata user's audiencegrowsat a constantrate, e.g.,by onefollo werper day.

Assumption II

states thateach follower' sretweetratef(t)(the expected number ofretweets pertime unit)decays asa power law: f(t)=ct .(1)

The numberF(t)of retweetsat timetis thenthe sumof

the retweetsrecei vedfromallfollowers accumulatedby that time:F(t)=P t ⌧=1 f(⌧). Ane xampleisshown inFig. 2a:the magenta andgreen vertical linesmarkspecificpoints inthe focal user'scareer;the intersectionsof thev erticallines with the followers'retweet-ratecurv esare indicatedbymagenta and greendots. Bysumming thev aluesof themagenta orthe green dots,we obtainthe values indicated bytherespectiv e dots showninFig. 2b.By repeatingthis procedurefor each time step,we obtainthe fullcurv eof Fig.2b. ThefunctionF is increasingand concav e(Fig.2b),sincefis decreasingand positive.Theretweet rateper follower attime tis obtained viaG(t)=F(t)/t(without lossof generality, assumingone newfollo werpertimestep). Assho wnin Fig.2c, Gis a monotonically decreasingfunction. Empirical analoguesof FandGwere displayedin Fig.1. While theshape ofthe theoreticalF(Fig. 2b)is approxi- mately mirroredby empiricaldata (Fig.1a), thereis adis- crepancyfor

G, whichis monotonicallydecreasing inthe

model (Fig.2c), but notsoinempirical data(Fig. 1b).The discrepancyis resolved bylettingusersstart notwith zero butwith s>0followers.As wewill seel ater(Fig. 3a),this is inline withempirical data.An initialfollo wercount s>0 could potentiallybec ausedby theplatformrecommending 1065

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s t s = 0 s = 4 s = 6s = 8s = 10s = 12 (d) Figure 2:Model ofretweet rates.(a) Retweetrates f(t)of individualfollo wers(onecurve perfollo wer,shiftedalong t- axis) decayas apo werla w.(b)A user'sretweetcountF(t)at time tis obtainedby summingthe retweetsfrom allfollo wers at timet. (c-d)The relativ eretweetrateperfollower ,G s (t), is obtainedby dividing theabsolutenumberof followers by s+t, wheresis theinitial numberof followers. newcomersto otherusers moreaggressi vely rightafter join- ing, sothe ywouldquickly accumulateacertainnumber of followersimmediately inthe beginning. With s0initial followers,the relativ enumberofretweetsperfollo weris G s (t)=F(t)/(s+t), whichhas aninternal maximumfor s>0(Fig. 2d)and capturesthe essenceof theempirical curvesof Fig.1b. The curvesshown inFig.2a-dwere obtainedempiricall y for aspecific setof parameters( ↵=0.8,c=1). Inthe gen- eral case,the modelis moreeasily analyzedwhen assuming continuous, ratherthan discrete,time:

F(t)=Z

t 0 Z t 0 c d ⌧(3) c ↵1⇣ ↵1) ↵1) ,(4) for↵6=1. 2 the case↵1, where R t 0 f(⌧)d⌧=R t 0 c d ⌧=•would diverge. In thecontinuous case,Fhas thesame increasingconca ve shape asin thediscrete case(Fig. 2b).The shapeof G s (the retweets perfollo werover time)isanalyzedindetailin Ap- 2 The case↵=1needs tobe handledseparately: here,F(t)=

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pendix A, 3 where wesho wthatG s attains aunique internal maximum ata timet >0if andonly ifs>0(as inthe discrete case;Fig. 2d).

Tosummarize, thenon-monotonic curves indicatinga

tiveassk etchedearlier (inexperiencedyouth,age ofmaximal alignment withcommunity norms,losing touchwith newer members inold age).Rather ,the ycanbee xplainedwitha much simplermodel containingjust two ingredients:(1) a constant influxof new followers,(2)each ofwhomloses interest overtimeaccordingto apo werla w. Tobe clear, wedonotclaim that ourm odelfully captures all thedynamics ofretweet rates.Rather ,we hav eshownthat the seeminglycomple x,non-monotonicev olutionof retweets per followerdirectlyfollo wsfrom twointuitiv eassumptions, which aresho wntoapproximatelyhold inempirical datain the followingsection. As aside note,we remarkthat, for↵>1,F(t), thenum- ber ofretweets pertime unit,is upper-bounded bya constant c 1 ↵1 . Thatis, for↵>1, themodel predictsan inherent barrier onthe numberof retweetsa usercan getin asingle time unit,ev enifheraudiencegrows ata constantrate: the drop-offof interestw ouldbe toosteeptobe compensatedby the influxof new followers.Later, wewillobserve empiri- cally (Fig.4) thatthe empiricale xponents↵are wellbelo w1, which theoreticallyimplies apotentially unboundednumber of retweetsper timeunit.

4 ValidationofModel Assumptions

Weno wverifythat themodel'sassumptions holdempirically . Assumption I:Constant audiencegr owth. Fig. 3asho ws that themedian numberof followers indeedgro wsapproxi- mately ata constantrate. We alsosee that,empirically,users tend tostart off withanonzeronumber s>0of followers, which isthe con ditionforwhichtheper-followerretweet rate G s (t)is non-monotonicin ourmodel (Fig.2d). Non-mono- tonic empiricalcurv es(Fig.1b)are thusin linewith theshape predicted bythe model.

Assumption II:Decay off ollowers' interest.

Tov erifythe

assumption thateach follower' sretweetratef(t)decays as a powerlaw ,weproceedasfollows. For eachweek t 0 in a useru'scareer ,considerallfollo werswho retweetedufor the 3 Appenlices availableonlineathttp://arxi v.or g/abs/2103.107Ii1066

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c0.0 0.40.8 α Figure 4:Empirically estimatedparameters (power -law ex- ponent ↵and multiplicativefactorc) offollo wers'retweet ratesf(t)=ct , asfunctions ofthe weekin thefocal user's career whenthe followers beganto followthefocaluser. first timethat weekand trackthem forthe restof u'scareer , computing thesefollo wers'meannumbersof retweetsof u's tweets foreach subsequentweek t 0 t 0 , andaggre gating weekly overallusersuvia themedian. Theresult issho wn in Fig.3b (forusers whoha ve beenon Twitterforatleast

400 weeks;for clarity's sake,curves areshownfor onlyfour

starting weekst 0 , butinpractice wecomputed curves forall t 0 ). Weobserve thatdecreasingpower laws ct (solid lines in Fig.3b) fitfollo wers'interest well.Irrespective ofwhen a followerstarts retweeting,t heirinterest inthefollowed user drops offwithtime. Fig.4, whichtracks ↵andcovertime, showsthat thedrop-of fin interestbecomeslesssteep ov er time (asindicated bythe decreasingv aluesof ↵).

5 Discussion

Summary.Our empiricalanalysis (Sec.2) ofthe "careers" of thousands ofT witterusers,startingwith eachuser' sv eryfirst tweet, revealedthatthenumber ofretweets auser receiv es per weekgro wssteadily,at adecreasingspeed(Fig. 1a). The growthispartly dueusers' tendency toaccrue followers overtime(Fig. 3a);the decreasingspeed, dueto followers' tendencyto loseinterest ov ertime (Fig.3b). Accounting forthe growing numberoffollowers bydi vid- ing weeklyretweet countsby thenumber offollo wersat the giventimere veals partlynon-monotoniccurvesofretweets per follower,especiallyformore recentusers (Fig.1b). One might betempted toe xplainthis shapebypositing aprototyp- ical userlife cycle, whereusersstartof funacquainted with the normsof theplatform, subsequentlyproduce contentthat is graduallybecoming moreattracti ve toothers,untilthey reach apeak ageafter whichthe ycease tobe "intune"with the restof thecommunity ,resulting indecreasingretweet rates perfollo werfromthereon. By formulatinga mathematicalmodel (Sec.3), wesho w that sucha complex narrativeis notrequiredinorderto explainthe non-monotonicshape ofretweets-per -follower curves.Rather ,wecanreproduce thequalitati ve shapeof the curvesby makingonly two simple,intuiti veassumptions, both ofwhich werev erifiedto holdempiricallyinthe data (Sec. 4).The assumptionsstate that(1) usersg ainfollo wers at asteady rate,and (2)the followers loseinterest inthe fol- loweduser' scontentaccordingto apo werla w. We emphasize that itis notour goalto prov ethat themodel capturesallthe dynamics thatdri veretweetrates - infact, thereare proba- bly manyfactors andfacetsthat remainunmodeled. Rather, we showthatthe simple modelis sufficienttoe xplaink ey aspects ofthe data.

Further Twitterdatasets.

The aboveanalyseswereper -

formed ona Twitter datasetthatisparticularly wellsuited for studying entireuser careers,as itcontains alltweets posted by apredetermined setof users.Collecting thisda tasetw as demanding, asit requiredcustom-made crawli ngtools, due to thefact thatTwitter's APIonly providesaccesstoeach user's3,200 mostrecent tweets(Garimella andW est2019). Although ouruser sampleis notrandom, but biasedto wards politically interestedusers (Garimellaet al.2018), wecon- sider thistopical bias tobelessse vere thanthe biasthat wouldquotesdbs_dbs14.pdfusesText_20

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