Attention Inequality in Social Media

0.44 MB·

by  Linhong Zhu

Attention Inequality in Social Media
LinhongZhuandKristinaLerman
USCInformationSciencesInstitute
4676AdmiraltyWay
MarinaDelRey,CA90292
{linhong,lerman}@isi.edu
Abstract KlamerandVanDalen2002).Researchershavelongspec-
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ulated about the origins of inequalities. Current consen-
1
Socialmediacanbeviewedasasocialsystemwherethecur- sus holds that inequality cannot be explained by variation
0
rencyisattention.Peoplepostcontentandinteractwithoth- in quality of the work or individual talent (Adler 1985;
2
erstoattractattentionandgainnewfollowers.Inthispaper,
Salganik, Dodds, and Watts 2006; Lariviere and Gingras
n weexaminethedistributionofattentionacrossalargesam-
2010). Rather, it is the result of the underlying social pro-
a pleofusersofapopularsocialmediasiteTwitter.Through
cesses,suchasthe“cumulativeadvantage”orthe“rich-get-
J empiricalanalysisofthesedataweconcludethatattentionis
veryunequallydistributed:thetop20%ofTwitterusersown richer”effect,whichbringsgreaterrecognitiontothosewho
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morethan96%ofallfollowers,93%oftheretweets,and93% arealreadydistinguished(Merton1968;Allison,Long,and
2
ofthementions.Weinvestigatethemechanismsthatleadto Kraze1982).

] attention inequality and find that it results from the “rich- Inonlinesocialmediaattentionisalsonon-uniformlydis-
I get-richer”and“poor-get-poorer”dynamicsofattentiondif-
tributed.Someuser-generatedphotosandvideosareviewed,
S
fusion. Namely, users who are “rich” in attention, because
liked, or shared orders of magnitude more times than the
s. they are often mentioned and retweeted, are more likely to rest. Psy’s Gangnam Style video, for example, was viewed
c gainnewfollowers,whilethosewhoare“poor”inattention
over 2 billion times on YouTube, while the average video
[ arelikelytolosefollowers.Wedevelopaphenomenological
modelthatquantifiesattentiondiffusionandnetworkdynam- hasfewerthan100thousandviews.Attentionpaidtopeople
1 ics,andsolveittostudyhowattentioninequalitygrowsover isalsohighlyunequal.AfewTwitterusershavetensofmil-
v timeinadynamicenvironmentofsocialmedia. lionsoffollowers,whilethemajorityhavejustahandfulof
0
followers.Thesepopularusersenjoyanunparalleledadvan-
0
tage:thingstheysayonTwitter,fromproductendorsements
2
Introduction topersonalopinions,areseenbymillions,makingthempo-
7
tentiallyfarmoreinfluentialthanordinaryusers.
0 Inequality is a pervasive social phenomenon. For example,
1. incomeandwealtharenotevenlydistributedinmodernso- In this paper, we carry out quantitative analysis of data
0 ciety,butinstead,concentratedamongfewindividuals.The from the microblogging service Twitter to characterize and
6 richest 20% in the United States own more than 80% of quantify attention inequality in social media. We identify a
1 the country’s total wealth, while the poorest 20% own just set of almost 6,000 users and monitor their activity over a
: afractionofonepercent(NortonandAriely2011).Inequal- periodofseveralmonths.Weshowthatattentionisveryun-
v
ity does not just violate our sense of fairness (Norton and equallydistributed:thetop20%ofTwitterusersinoursam-
i
X Ariely 2011), but it also reduces economic opportunities, ple own 97% of all followers, 93% of retweets, and 93%
r and has been linked to negative social outcomes, such as of mentions. By observing how the monitored users allo-
a poor health (Kawachi and Kennedy 1999) and high crime cate their attention over time, via following, mentioning or
rates(Kelly2000)insocietieswithhighlevelsofinequality. retweeting other users, we are able to study the processes
While economic inequality has been widely discussed re- leadingtoattentioninequality.Weempiricallydemonstrate
cently,inequalitiesarerampantinotherdomains,including that the attention Twitter users receive by being mentioned
science and entertainment, where a few individuals receive or having their posts retweeted by others results in them
disproportionateshareofattentionandthebenefitsitbrings. gainingnewfollowers.Incontrast,userswithfewerfollow-
In science, specifically, attention can be measured by the ersareretweetedlessfrequentlyandaremorelikelytolose
number of citations that papers (and scientists who publish followers.Weconstructaphenomenologicalmodelofatten-
them)receive.Thedistributionofthenumberofcitationsis tiondiffusionwherethesetwoprocesses—mentioningand
extremelyunequal,withsomepapersreceivingthousandsof retweeting — result in changes in the structure of the un-
citations,whilemanyothersareseldomcited(Allison1980; derlyingfollowernetwork.Afterparameterizingthemodel,
wesolveittostudyhowattentionandinequalityevolveover
Copyright(cid:13)c 2015,AssociationfortheAdvancementofArtificial time.Ourmodelproducesa“rich-get-richer”and“poor-get-
Intelligence(www.aaai.org).Allrightsreserved. poorer” dynamic, whereby users with more followers, i.e.,
Twitter celebrities, are mentioned and retweeted more fre-
quently,whichresultsinthemgainingevenmorefollowers. ncy10−2 ncy10−2
e e
Our model is able to identify users who will gain more at- equ10−5 equ10−5
tention,topredicttheevolutionofthefollowernetworkand d fr d fr
e e
attentioninequality. aliz10−10 Seed users aliz10−10 Seed users
There are consequences to high attention inequality, few orm All users orm All users
of which have been explored in the context of social me- N 102 104 106107 N 101 103 105 107
dia.Highinequalityfocusestheattentionoftheonlinecom- Number of followers Number of friends
munityonrelativelyfewusers,allowingthemtodispropor- (a) (b)
tionately benefit from the attention they receive. Compa-
niescompensatesocialmediacelebritiesforendorsingtheir ncy10−2 ncy10−2
potaonorftophfotosoedb.trphueAuecpinlrtteastgoceraunpwcnrtleierdaoeltemncanbhrceeireisisedtnssyt.ao,hTgotietehrbcsiusew.usiOrfi,anrtingrecndinhetriceieqnny.tuawgWraoyelafihetuesyvtseonieddrcraei,ismavlarlevinernmidcydiesoeeoihndovteei,heaseiinttsrahomnrveedsiedutd,waicesvnhoeiesmesrpttsiiahailnytarayger-t Normalized freque11110000−−−−18640 10SA1ell eu1ds0 eu2rsse1r0s3 104 105 Normalized freque11110000−−−−18640 101SAell eu1ds0 eu2rssers103 104 105
Number of status Number of retweets
ofcontentandviewpointstowhichpeopleareexposed.But
(c) (d)
inequalitymayalsohavehiddenbenefits.Wheneverysingle
personhadwatchedthesamevideo,orreadthesamestory,
Figure 1: Distribution of the numbers of (a) followers, (b)
itgivespeopleacommontopicforconversation(andtweet-
friends,(c)statusupdatesposted,and(d)retweetsreceived
ing)andacommonvocabularywithwhichtodiscuss.Thus,
forseedusersandallusers.Welogarithmicallybindatafor
inequalityreinforcessocialidentity.
smoothing.
Ourworkraisesanumberofquestions,includingwhydo
userstolerateextremeattentioninequality?Howdoesitaf-
fect user behavior? Does it reduce their engagement with
edges. To provide more details about statistics of our data,
Twitter, or does it inspire them to be more active in gain-
we analyze the distribution of the numbers of followers,
ingattention?Andnotleast,shouldTwitter,andothersocial
friends,statusupdates(useractivity),andthetimesthatthe
mediasites,dosomethingtoensureattentionismoreequi-
tweets of the seed users, and all users in our data, were
tablydistributed?Wehopethatourworkstimulatesresearch
retweeted. Note that we use ‘status updates’ to refer to the
communitytoaddressthesequestions.
aggregatetweetspostedbyusersinceaccountcreation,and
we use ‘tweets’ to refer to the tweets user posted during
QuantifyingAttentionInequality
a specific time period. These distributions, plotted in Fig-
In this section we quantitatively characterize the attention
ure 1, have a characteristic long-tailed form, where a few
receivedbyusersandcontenttheypostonTwitter.Although
usershaveextremelylargenumbersoffriends,followersor
we focus on Twitter, our definitions of attention generalize
postedtweets,whilemanyusershavefewfriends,followers
toothersocialmediaplatforms.
orpostedtweets.Suchdistributionsaregenerallyassociated
withhighinequality.
DataDescription
Exceptforactivity(numberofpostedtweets)andnumber
We collected data from Twitter over a period from March
ofretweets,thedistributionsforseedandallusersarevery
2014 to Oct 2014 using the following strategies. First, we
similar,whichindicatessuccessofourrandomseeduserse-
randomly selected 5600 seed users whose user ids range
lectionstrategy.Thesmalldifferencesbetweenthedistribu-
from[0,760000000].WethenusedtheTwitterAPItoperi-
tion of number of tweets among seed users and among all
odicallyquerythefriendsofseedusers(i.e.,whomtheseed
users arise because the profile information of seed users is
userswerefollowing),whichresultedinadynamicnetwork
obtained from raw Jason objects of tweets, and thus users
where each edge had a time stamp with its creation and/or
whoneverpostanytweetscannotbeselectedasseedusers
deletiondate.Next,wecollectedprofileinformationforall
inourdatacollection.Duetothesedifferences,thedistribu-
the seed users and their friends, as well as timeline tweets.
tion of number of retweets among seed users, also slightly
The number of tweets gives that user’s activity, or engage-
differsfromthatamongallusers.
ment, level. We tracked the retweet/reply/mention status of
eachtweetfromitsrawJasonobjectreturnedbyTwitter.The
AttentionInequality
number of retweets gives the number of times users have
sharedaparticulartweetwiththeirownfollowers.Aswear- How much attention do Twitter users receive? Certainly, it
gue later, this is an important measure of attention. For the makes sense to use the number of followers as a measure
seed users, we further monitored their temporal profiles by of attention: the more followers you have, the more peo-
querying user profile weekly during the data collection pe- ple will see the messages you post. However, there is also
riod. awidedistributioninuseractivity,andapersonwithamil-
Thedatacollectionprocessresultedin23,831,568tweets lionfollowerswhoneverpostsanythingwillnotreceiveany
andadynamicnetworkwith1,944,383usersand17,869,415 attention.Therefore,weproposetwoadditionalmeasuresof
ityhasGinicoefficientofaround0.8.1 Attentioninequality
100 100
Number (Gini=0.9412) on Twitter is staggering. The vast majority of users do not
Attention (%) 24680000 NNuummbbeerrfmroeeltlwonwetioeentrsss ((GGiinnii==00..99013343)) Activity (%) 24680000 NNuummbbeerrfsrtiaetnudss ((GGinini=i=00..56983391)) raueetnstcegeeranigevtineeomgnaanetgnhyetamanitsettenhtntehtaeibononndt,utaomacmntbidve9irtt9hyo%efotnocfropTimew1nb%idtitsneoertfd?h!euAyWsemfrhosealagtlosetuwhtrefenaaorndfdmruivotshereeesr

0 0 number of tweets they post. Figure 2(b) shows the Lorenz
0 20 40 60 80 100 0 20 40 60 80 100
Sorted population (%) Sorted population (%) curves of these distributions for the seed users in our data.
(a) (b) Resultsindicatethatuserengagementisalsounequallydis-
tributed,withGinicoefficientofg =0.6831forthenumber
Figure2:LorenzcurvesandGinicoefficientsofinequalities of friends and g = 0.5939 for the number of tweets. Al-
of(a)attentionsand(b)activitiesamongseedusers. thoughstillunequal,useractivityismoreequallydistributed
thanattention.
attention—thenumberoftimesauser’spostsareretweeted, DynamicsofInequality
orresharedbyothers,andthenumberoftimesauserismen-
tionedbyothers.Thesemeasuresofattentionarerelated:on
1.00
average, the number of times a user’s post is retweeted or 2014-03-24 2014-08-12 2014-10-14
the number of times a user is mentioned is proportional to
thenumberoffollowerstheuserhas.However,usingtheto- ni 0.80
Gi
talnumberofretweetsallowsustoincludevariationinuser
activity in a measure of received attention; while using the 0.60
totalnumberofmentionsincludeseffectsofcelebrity.Since 0.50
we have the network, profile and activity information only Gini 0.02
for seed users, we analyze how these seeds users allocate R- 0
-0.02
theirattentiontootherusers. followers RT user RT item M user
seed all seed
Inequalityofadistributioncanbecharacterizedusingthe
Lorenz curve, which graphically represents the cumulative
1
2014-03-24 2014-08-12 2014-10-14
distribution function of the empirical probability distribu-
tion. In other words, it plots the cumulative share of value,
e.g., number of followers, as a function of the cumulative 0.8
percentage of the population who have at least that many ni
followers.Perfectequalityisastraighty = x,inwhichthe Gi
bottom x% of people have y% of the followers. However,
0.6
when inequality exists, the Lorenz curve is highly skewed.
Figure2(a)showstheLorenzcurvesforthenumberoffol-
lowers of seed users, the number of times they were men- 0.5
friends status new friends un friends
tioned,andthenumberoftimestheirpostswereretweeted.
Itshowsattentionunequallydistributed:thetop1%ofseed
Figure 3: Change in inequality users over time. Here
users account for approximately 50% of the total retweets,
RTitem indicates the number of re-tweets a tweet that is
60%ofthetotalfollowers,and66%ofthetotalmentions. all
posted by any seed user received from all users in one
Another way to characterize the inequality of a distribu-
week period; while RTuser/Muser denotes the number of
tioniswiththeGinicoefficient(Gini1997).Givenasample seed seed
retweets/mentions a user received from seed users in one
ofdatax ,···x ,theGinicoefficientcanbecalculatedfrom
1 i
weekperiodatthreepointsintime.
thefollowingequation:
(cid:80) (cid:80)
g = i j|xi−xj| (1) Is high inequality simply an artifact of Twitter’s young
2N2x ageandwillcorrectitselfovertimeasusersdistributetheir
attention more equitably? Or is the inequality increasing
Gini coefficient is zero, meaning there is perfect equality,
overtime?Figure3showshowinequalitychangedovertime
when all values in a sample are the same, and it attains a
during data collection period. The plot shows the Gini co-
value one, meaning maximum inequality, when one num-
efficients of different measures of attention at three points
ber in the set is nonzero and the rest are zeros. Gini co-
intime.Theinequalityofthenumberoffollowers,andthe
efficient also gives the area between the Lorenz curve of
number of retweets of a user (RTuser) are increasing over
a distribution and the line of perfect equality. Gini coeffi- seed
time; while attention to content, given by the number of
cientofthenumberoffollowersinFig.2(a)isg = 0.9412,
times different items are retweeted by all users (RTitem)
for the number of mentions it is g = 0.9133, and for the all
fluctuates over time. The attention inequality in terms of
number of retweets it is g = 0.9034. The Gini coefficient
of the much discussed income inequality is less than 0.5 1http://en.wikipedia.org/wiki/List_of_
for the United States, and the much larger wealth inequal- countries_by_distribution_of_wealth
number of mentions, increases over time, but the margin
Table 1: Statistics of classes when users are divided into
is too small to be significant. The plot also shows that the
attention that items posted on Twitter receive (RTitem) is quintiles based on the number of followers they have. The
all numbersrepresenttheaveragenumberoffollowersandthe
somewhatmoreequitablydistributedthantheattentionpaid
averageactivityofusersineachclass.
tousers.
quin-tile #statuses #followers
Inaddition,wecalculateinequalityofuseractivityatdif-
Q 1st 14,086 29,021
ferentpointsintime,whichincludesthenumberoffriends, 1
numberofstatusupdates,numberofunfriends(i.e.,number Q2 2nd 7,184 1,058
oflostfollowers),andnumberofnewfriendsgainedatdif- Q3 3rd 4,053 445
ferenttimeperiods.Theresultsareplottedinthelowerpart
of Figure 3. Note that we use number of status updates to
denotetheaggregatenumberoftweetsusersmadesinceac- and tweeted over 14K status updates on average. Users in
countcreation.Theseresultsindicatethatactivityinequality thesecondquintile,representingthenext20%ofmostpop-
alsoincreasesovertime,whichindicatesthatbothnewcon- ularusers,havealmost1Kfollowersonaverageandposted
tentandnewlinksarecreatedbyasmallsetofsuper-active almost 7K tweets, while users in the third quintile have a
users. Also, activity inequality, while high, is substantially little morethan 400 followers andposted almost 4K times.
lessthanattentioninequality. Usersinthebottomtwoquintilesdidnothavemuchactivity
Thesefindingssuggestthatthereareprocessesthatdrive andwereexcludedfromanalysis.Themainideabehindthis
attention to be concentrated on few individuals at the ex- division is that users within the same quintile will be more
clusionofothers,eventhoughuserengagementisfarmore homogeneousandsimilartoeachother.
equally distributed. In the next section we examine these
questionstoconstructamodelofthedynamicsofattention.
AttentionDiffusionModel ers106 ets105
ow105 we104
TsdthoiofecfihuTaeswlilopmitnteeedoxrifpafloa,atlwitlneoenwdtdiyeoernvnane.melTotiwphcisoasrokmmfaooasddtteeuellsnecotirafospnntaeuritrenweesboqrrtuokhauedlgieythyvntoatilmonutitoichonsenlvianoitea-f Number of foll111000234 mmeeadnian Number of ret1111100000−01231 QQQ123 bbbyyy fffooollllllooowwweeerrrsss
tention of potential new followers. As these subsequently 1 00 101 102 103 1 00 101 102 103
Number of tweets Number of tweets
linktotheoriginalusers,theTwitternetworkgrows,bring-
(a) (b)
ingevenmoreattentiontotheoriginalusers.
One of the best-known models of network evolution is
the BA model (Barabasi and Albert 1999) with preferen- Figure4:Relationbetweenpostingactivityandattention((a)
tialattachmentmechanism.Inthismodel,anewnodelinks numberoffollowers,(b)numberofretweets).Moretweets
to an existing node with probability proportional to its de- auserposted,morefollowersandretweetsauserreceived.
gree.Thismakesitmorelikelyforhighdegreenodestoac-
quirenewlinks,resultingina“richgetricher”phenomenon. Aftercreatingmorehomogeneouspopulations,weexam-
This is a plausible model for the growth of the Twitter fol- ine how the attention users receive leads them to gain, or
lowernetwork:userswhoalreadyhavemanyfollowersare lose, followers. Again, we quantify attention by the num-
morelikelytogetattentionandattractevenmorefollowers. ber of times users are mentioned and the number of times
However,Twitterplatformishighlycomplexanddynamic: theyareretweetedbytheirfollowers.Bothofthesedepend
users post messages, which may then be retweeted by oth- onthenumberoffollowersuserscurrentlyhave.Whilethis
ers.Theseretweetscreateopportunitiesfornewfollowlinks isnotdirectlyunderuser’scontrol,theycantrytoincrease
to be formed (Weng et al. 2013). Users are also mentioned thenumberoffollowersbyfollowingotherusers,andhop-
by others, which also creates opportunities for new follow ing that some of these will result in reciprocal links. Users
links(ZhuandLerman2014).However,linksareoftenbro- can also attempt to increase the number of times they are
ken as users unfollow others (Myers and Leskovec 2014). retweetedbypostingmoretweets,althoughburstsoftweet-
Below we empirically characterize how these factors con- ing activity may cause followers to classify users as spam-
tributetothegrowth—andtheloss—ofattentiononTwit- mersandunfollowthem.Belowweexaminehowthesefac-
ter. torsarecorrelatedwiththeattentionusersreceiveandtheir
Twitter users are heterogeneous, and popular users who probabilitytogainorlosefollowers.
are “wealthy” in attention could behave in a qualitatively
Observation1 (Effect of tweeting) Users who post more
different manner from the less popular users. To partially
tweetsreceivemoreattention.
controlforheterogeneity,wesplitusersintoclassesaccord-
ingtothenumberoffollowerstheyhave.Forsimplicity,we We examine the relationship between posting activity,
usefiveclasses,correspondingtoquintilesinthenumberof measured by the number of status updates users tweet, and
followers.Table1reportsstatisticsofthetopthreequintiles. the number of followers they have. Figure 4(a) confirm an
Usersinthefirstquintile,representingthe20%oftheusers earlierfinding(Hodas,Kooti,andLerman2013)thatactiv-
with most followers, have over 29K followers on average ityandnumberoffollowersarepositivelycorrelated.Post-
mber of followers110035 mmeeadnian mber of retweets01..1255x 10mm4eeadnian of new followers111000−−−321 QQQ123 bbbyyy fffooollllllooowwweeerrrsss of new followers111000−−−321 QQQ123 bbbyyy fffooollllllooowwweeerrrsss
Nu100 Nu 0 ob ob
100101N1u0m2b1e0r3 o1f 0fr4ie1n0d5s106107 1Numb2er of f3riends4x 1055 Pr10−4 100 1000 60000 Pr10− 4 100 1000 60000
(a) (b) Number of retweets Number of mentions
(a) (b)
Figure5:Relationbetweenfollowingactivityandattention
((a)numberoffollowers,(b)numberofretweets). Figure 6: Probability of gaining new followers vs (a) the
numberofretweetsand(b)mentionsusersreceive.
ingactivityalsoaffectshowmuchattentionuser’spostsre- −2
10
ceive, which we measure by the number of times the posts s
r
e
areretweeted.Figure4(b)showsaclearpositivecorrelation w
between activity and number of retweets for the first three o
userquintiles.Sinceeachretweetincreasesuser’svisibility oll
bybringinghisnameintothefeedsoftheretweetinguser’s w f10−3
e
followers,italsoincreasesthelikelihoodofgainingnewfol- n
lowers(Wengetal.2013;ZhuandLerman2014).Wedonot of
showtheplotthatevaluatestherelationshipbetweenposting b
o
activityandmentionssincethatcorrelationisrelativelylow Pr10−4
0.0259. 0 1 2 3 4 5 6 7
10 10 10 10 10 10 10 10
Observation2 (Effect of following) The following others Number of followers
doesnotnecessarilyincreaseattention.
Next,weexaminewhetherbyfollowingmorepeople,users Figure7:Probabilityofgainingfollowersvsthenumberof
canincreasetheattentiontheyreceive.Ingeneral,thereisa currentfollowersusershave.
positivecorrelationbetweenthenumberoffriendsusersfol-
lowandthenumberoffollowerstheyhave(Figure5(a)).The
Figure6(b)showstherelationshipbetweenthenumberof
exception to this trend are the few outliers who have fewer
mentions users receive and the probability of gaining new
friends than followers. Most of these users represent orga-
followers. Interestingly, mentions serve to attract new fol-
nizationalaccounts,suchasYouTubewithonly851friends
lowers mainly for the most popular users, i.e., those with
but47.6millionfollowers,orcelebrities,suchasLadyGaga
mostfollowers.
with133thousandfriendsbut43.6millionfollowers.How-
ever,asshowninFigure5(b),thereisnostrongcorrelation Observation4 (The Matthew effect) The more attention
between the number of friends and the number of retweets usershave,themoreattentiontheywillreceive.Thelessat-
user receive. Similarly, the correlation between following tentionusershave,theeasieritistoloseattention.
activityandnumberofmentionsisonly0.0124.Therefore,
Thefeedbackbetweenusers’popularity,measuredbythe
following more people does not translate into increased at-
number of followers they have, and the attention they re-
tention.
ceiveviaretweetsandmentions,leadstoa“rich-get-richer”
Observation3 (Effect of attention) The more attention
phenomenon.Figure7empiricallyvalidatesthepositivecor-
users receive, via retweets and mentions, the more follow-
relationbetweenthemeanprobabilityofacquiringnewfol-
ersuserswillgain.
lowerandthenumberoffollowersusersalreadyhave.Note
Aspreviouslyobserved,eachretweet(Wengetal.2013) that we log-binned the data to improve statistics; however,
andmention(ZhuandLerman2014)byfollowersincreases sincetherearefewuserswithverylargenumbersoffollow-
the attention the user receives by bringing his or her name ers,therearelargefluctuationsintheextremerightportion
intothefeedsofotherusers.Theseothersmaysubsequently oftheplotduetopoorstatistics.Morefollowerswill,inturn,
decidetofollowthementionedorretweeteduser,increasing create more attention and even more followers. Overall, it
thenumberoffollowersheorshehasasaresult.Figure6(a) appearsthatMattheweffect(Merton1968)alsooperatesin
teststhisideabyplottingthemeanprobabilityofacquiring socialmedia:“attentionbreedsattention.”
a new follower as a function of retweet count among the In addition to gaining followers, users can also lose fol-
firstthreeuserquintiles.Generally,thereisastrongpositive lowersinthedynamicTwitternetwork(MyersandLeskovec
correlationbetweenthenumberofretweetsandprobability 2014).Infact,weobservednearlyasmanynewlinkscreated
of getting new followers. Moreover, the correlation among asdestroyed.Figure8plotsnumberofnewfollowlinkscre-
userswithlargestnumberoffollowersisevenstronger(Fig- ated and the number of follow links broken by seed users
ure6(a)). duringeachtimeperiod.Bothcreatedlinksandbrokenlinks
x105
1.6 new links Table2:Notationsandexplanations.
ers 1.4 broken links Variable Description
s
of u 1.2 pi numberoftweetsmadebyuseri
er f numberoffollowersofi
b i
m 1
Nu ri numberofretweetsofi’spostsbyfollowers
m numbermentionsofibyfollowers
0.8 i
Jul 22-Aug 05Aug 06-Aug 2A0ug 21-Sep 0S4ep 05-Sep 1S9ep 20-Oct 04Oct 05-Oct 19Oct 20-Nov 12 PPff+− pprroobbaabbiilliittyyooffgloasiinnigngfoflollolwoweresrs
Figure8:Totalnumberofnewlinksandbrokenlinksmade ever, that tweeting burstiness does not necessarily increase
byseedusersinhalfmonths. theprobabilityofeithergainingnewfollowersorlosingfol-
lowers. The correlation between burstiness and the proba-
bility of gaining new followers for the top 3 quin-tiles is
s 1 s100 0.0797, -0.085 and 0.1596 respectively; while the correla-
er mean er mean
ow median ow median tionbetweenburstinessandtheprobabilityoflosingfollow-
oll oll ers for the top 3 quin-tiles is 0.0765, 0.0451, and -0.0811.
n f0.5 n f10−1
u u Therefore, for users who have low attention, burstiness of
of of
b b tweeting might help gain new followers. In addition, for
o o
Pr 100 0101102103104105106107 Pr10−120 0 101 102 103 104 105 umsiegrhstwcahuosheaivnefomrmucahtioantteonvteirolno,adbuarnsdtinloessesfoofllpoowstervsi.sibility
Number of followers Number of retweets
(a) (b) Collectively these observations suggest that both prongs
oftheMattheweffectoperatetochangethenumberoffol-
lowers and increase the inequality of attention: those with
Figure9: Thecurrentattention((a)numberoffollowers,(b)
most followers also acquire new followers at a faster rate,
numberofretweets)againstthemeanprobabilityoffollow-
andthosewiththeleastfollowersalsolosethematafaster
erstheylost.Hereweuselogarithmicbinsize.
rate. Since the number of followers is directly related to
howmuchattentionusersreceive,thoserichinattentionget
contributesignificantlytothedynamicsofthenetwork.The richer,andthosepoorinattentiongetpoorer.However,users
averageofnewfriendsfollowedperuserinatwo-weekpe- cancontroltosomeextenthowmuchattentiontheyreceive,
riod is around 20.87, and average of unfollowed users per at least via retweeting, by increasing their posting activity
person is around 19.04. These statistics show that broken withoutworryingthattheywilllosefollowersasaresult.
linksoccurasfrequentasnewlinksinsocialmedia.
ModelDescription
Wholosesfollowersandwhy?AsshowninFigure9(a),
userswithmanyfollowers(>1,000)arefarlesslikelytobe Attentiondiffusionprovidesamechanismfortheevolution
unfollowedthanuserswithfewfollowers(<100).Unfollow of the Twitter follower network. Below we introduce a dy-
probability does not vary by posting activity: for the same namicmodelwherethelikelihoodofforminganewfollow
number of followers, users who post many status updates link to a user, or breaking an existing link, depends on the
are not more likely to be unfollowed than users who post amount of attention the user receives. Thus, the main vari-
fewstatusupdates. ables used by the model include the number of followers
Figure 9(b) plots the probability of losing followers as a usershave(fi),thenumberoftimestheytweet(pi),andthe
function of the number of times user’s posts are retweeted. numberoftimestheyarementioned(mi)ortheirpostsare
Clearly, the more times the users are retweeted, the less retweeted(ri).ThesevariablesaresummarizedinTable2.
likelytheyaretolosefollowers,perhapsbecauseothersfind According to the observations of the previous section,
theirtweetsvaluable. the average probability of gaining new followers P+ is a
f
Wealsostudiedwhetherburstsoftweetingactivityresult function of the number of times the user is mentioned or
inalossoffollowers.Toomanytweetsappearingallatonce retweeted.Figures6(a)and6(b)suggestthatinourdataset
infollowers’feedscouldbeinterpretedasspammingbehav- thisfunctioncanbeapproximatedwith
ior,resultinginfollowersbreakingthelink.WeadoptaKur-
tosis metric to evaluate the burstiness of tweeting. Specifi- Pf+ ∼w1rα+w2mβ, (2)
cally,theKurtosismetricisdefinedasfollows:
wherew controlsthecontributionofretweetstotheforma-
1
Kurt(u)= E[(Pu−Pu)]4 tionofnewlinks,andw2 controlsthecontributionofmen-
tionstonewlinks.
(E[(P −P )2])2
u u Similarly,Figure9suggeststhatforourdata,theaverage
where P is a random variable that denotes the number of probabilityofunfollowingP−canbeapproximatedwith:
u f
tweets posted by u at each time period, and P is the av-
u
erage number of tweets per time period. We found, how- P− ∼w r−θ. (3)
f 3
−3
y10
Table3:Modelparametersestimatedbyregression. c −4
n10
Parameter Userquin-tile e −5
u10
αβ 00..Q6836145 1.00Q1.0245 01..Q4144381 ed freq111000−−−876 GSirmouunladti otrnuth
z10−9
θ 0.129 -0.730 -0.020 ali10−10
w 0.00215 0.0 0.00006 m −11
1 r10
o
w2 0.00038 0.00030 0.0 N 0 1 2 3 4 5 6 7 8
w 0.00836 -0.00135 -300.0 10 10 10 10 10 10 10 10 10
3
C 8546 754 -9 Number of followers
RMSE 52800 406 334
Figure 10: Distribution of the predicted and actual number
offollowersofseedusers.
Theequationsaboveallowustospecifyhowthenumber
of followers of a user i changes over time. Let f be a dy-
i newfollowers,aremuchhighforusersinthefirstquin-tile
namic variable representing the number of followers user i
(Q ) than the other quin-tiles. This shows that users with
hasatagiventime.Thentherateatwhichuserigainsnew 1
most followers, are more likely to be mentioned by oth-
followersis:
ersandconsequentlyfollowed.Themost-followedusersare
df+ likelytobecelebrities.Whentheyarementioned,andpeo-
i =(N −f )(w rα+w mβ) (4)
dt i 1 i 2 i plewhodonotyetfollowthemseetheirname,theymayrec-
ognize the name and decide to follow the celebrity in turn.
whereN isthesizeoftheuserpopulation,andN −f de-
i For the more ordinary users in the third quin-tile Q , men-
notesthesetofavailablepotentialfollowers,i.e.,userswho 3
tions do not lead to any new followers. For them, attention
donotyetfollowuseri.
doesnottranslateintonewfollowers.Inaddition,forusers
WerewritetheEq.4as
inthefirstquin-tile,thenumberoffollowerslostdecaysex-
df+ ponentiallywiththenumberofretweets,indicatingtheyare
dit =bifi−biN (5) muchlesslikelytolosefollowersiftheypostgoodcontent,
evenlotsofit.Incontrast,forusersinthesecondandthird
wherebi =−w1riα−w2mβi.ByintegratingEq.5,weobtain: quintileQ2andQ3,thenumberoffollowerslostdecayssub-
linearly with the number of retweets. In other words, they
fi+(t)=Cebit−biNt (6) are more likely to lose followers than first quin-tile users,
evenwhentheypostcontentthatreceivessimilarlevelofat-
Similarly, the rate at which user i loses followers is de-
tention.Theattention-richusershaveanadvantageoverthe
finedas
df− restoftheusers:theyaccruenewfollowersmoreeasilyand
i =f w r−θ (7) don’tlosethemasmuchastherestoftheusers.
dt i 3 i
ByintegratingEq.7,weobtain: SolvingtheModel
fi−(t)=C2ew3ri−θt (8) B(Tyabsloelv3i)n,gwtehceamnpordeedlic(Ettqh.e9n)uwmibtherthoeffloelalronweedrspaeraacmhesteeerds
CombiningEq.6withEq.8andusingthefactorthatfi(t)= user in our data has at any time after training time period.
fi(0) when t = 0, we have C2 = C. Then the dynamic Here we set the initial condition, that is, f(0), as the num-
numberoffollowerscanbecomputedas: ber of followers each seed user had on July 20 2014. Fig-
ure 10 compares the predicted distribution of followers of
fi(t)=f(0)+Cebit−Cew3ri−θt−biNt (9) seedusersonOctober14,2014totheactualnumberoffol-
lowerstheyhaveonthatdate(groundtruth).Thisvalidates
ParameterEstimation
thattheproposedmodelcancorrectlyreplicatethedistribu-
Themodelisparameterizedbyconstantsα,β,θ,C,w ,w , tionofattentiontousers.
1 2
and w . We estimate these parameters using regression on Next,weexaminehowwelltheproposedmodelpredicts
3
data between March 20 to July 20. Since different popula- dynamicsoffollowerinequality.Figure11comparesthein-
tionsofusers(indifferentquin-tiles)mightbecharacterized equality of the predicted and actual number of followers at
bydifferentparameters,inthefollowing,weseparateusers threedifferentpointsintime.Theattentiondiffusionmodel
by quin-tile, and estimate model parameters separately for results in high follower inequality, with values of the Gini
eachpopulation.Wesettheunitoftimeintervalasfourdays. coefficient around 0.94. In addition, in the attention diffu-
TheresultsofestimatedparametersarereportedinTable3. sionmodel,inequalitygrowsovertimeasattentionbecomes
Table 3 indicates that different user populations attract more concentrated on a small number of users. The trends
newfollowersindifferentways.Forexample,thevaluesof usingtheproposedattentionmodelareclosetotheempiri-
parametersβ andw ,whichcontrolhowmentionsgenerate calgroundtruth.
2
1.0 2014-03-24 2014-08-12 2014-10-14 to use it to model the attention to messages being tweeted.
Besides,resultsinFigure11showthattheattentionuserre-
0.8 ceivedintermsofthenumberofretweetsandthenumberof
Gini mentions,maynotsimplybemodeledbyEq.10andEq.11.
Oneofthefuturedirectionistomodelattentiondiffusionin
0.6
terms of number of retweets and number of mentions with
0.5 bothnetworkandcontentinformation.
ni 10-2
R-Gi 10-3 RelatedWorks
10-4 Our work touches on topics from a variety of fields. We
FL FL tweet RT RT M M
GT model GT GT model GT model
brieflyreviewsomeoftherelevantliterature.
Figure 11: The Gini inequality of predicted number of fol- Inequality
lowersFL versusthatofempiricalgroundtruthnum-
model Thestudyofeconomicinequalityhasalonghistorybegin-
ber of followers FL . In addition, the Gini of predicted
GT ning with the quantitative study of personal wealth distri-
numberofretweetsRT andpredictednumberofmen-
model bution by Vilfredo Pareto (Pareto 1896). In this work, the
tions M is compared with that of empirical number
modeL empirical data analysis showed that the tail of wealth dis-
ofretweetsRT andempiricalnumberofmentionsM .
GT GT tributionfollowsapower-law.Morerecentanalysisshowed
Since number of re-tweets is related to posting activity, we
that wealth inequality (in the U.S.) is not only large (Nor-
alsoreporttheinequalityofpostingactivityduringoneweek
ton and Ariely 2011) but increasing with time (Piketty and
intervaltweet .
GT Saez 2014). A number of models (Champernowne 1953;
Scafetta, Picozzi, and West 2004; Silva and Yakovenko
2005)havebeensuggestedtounderstandthefeaturesofem-
We can also use the attention diffusion model to predict
pirical wealth distributions and relate them to appropriate
inequalityofattentionviaretweetsovertime.Wemodelthe
mechanisms.Forexample,BanerjeeandYakovenko(Baner-
number of retweets users receive is a function of the num-
jeeandYakovenko2010)adoptedtheFokker-Planckequa-
ber of followers they have and their tweeting activity, and
tion model for the income distribution, which leads to an
thenpredictthenumberofretweetsusingregression.Specif-
approximatedexponentialforsmallandmid-rangeincomes,
ically,thenumberofretweetsauserreceived,canbeapprox-
andasapower-lawforthehighestincomes.
imatelymodeledas:
Inequalityexistseverywhereonline(Wilkinson2008).On
r =afb+cpd (10) theWorldWideWeb,80%ofthebrowsingtrafficgoesto5%
i i i ofthehostnames(McCurley2007).Thebrowsingdistribu-
We can then predict the number of retweets users get at
tionisalsohighlyskewedamongURLs,sitesanddomains.
any time by using their predicted number of followers and
Fortunato et al. (Fortunato, Flammini, and Menczer 2006)
their observed tweeting activity at that time. As shown in
suggests that search engines bias user traffic by their page
Figure 11, the Gini coefficient of the predicted number of
rankingstrategies.Theyobservedahighcorrelationbetween
retweets rises over time; however, it is much less than the
click probability and hit rank. Therefore, the ranking pro-
groundtruthvalue.ThepossiblereasonisthatinEq.10,the
cesshasapotentialtoskewbrowsingattentiontothosetop
number of retweets is directly related to tweeting activity .
rankedURLs.
Since tweeting activity has much lower inequality, the in-
Recently, with the advance of social networks, the stud-
equalityofnumberofpredictedretweetsisalsolowerthan
ies of socio-economic inequalities have sparked a renewed
thegroundtruthvalue.
interests from researchers. Specifically, much research has
Similar to above process, we model the number of men-
focusedonabetterunderstandingofthenatureandoriginof
tionsusersreceivedasafunctionofnumberoffollowers:
inequalitiesfromuserbehaviorandnetworkaspects.Among
m =aebfi (11) them, Fuchs and Thurner (Fuchs and Thurner 2014) ana-
i
lyzed data from the virtual economy of the massive multi-
WithEq.11,wepredictthenumberofmentionsatthree player online game, to find the explanations for origins of
different time points and report its Gini coefficient in Fig- wealth inequality in terms of behavior and network. Their
ure 11. The Gini coefficient of the predicted value, is very resultssuggestedthatwealthyplayershavehighin-andout-
close to the Gini coefficient of number of followers and degree in the trade network; while in contrast, players that
thegroundtruth.However,themarginaldifferencesbetween are not socially well-connected are significantly poorer on
two different time points in the ground truth, are relatively average. In addition to wealth inequality, Chatterjee (Chat-
smallerthanthoseintheprediction.ThisisduetoinEq.11, terjee 2014) have also studied the city size inequality, firm
numberoffollowersistheonlydependantfactorofnumber inequality, voting inequality, opinion inequality and their
ofmentions. origins from a statistical perspective. In this work, we aim
Inconclusion,theproposedmodeldescribesattentiondif- tostudytheattentioninequalitiesinsocialmediaandtoun-
fusion among users in a network and can successfully pre- derstandsomepossiblereasonsthatleadtotheinequalities.
dictthenumberoffuturefollowerseachuserwillhaveand Inaddition,wearealsointerestedinunderstandinghowin-
theresultingattentioninequality.However,itisunclearhow equalitieschangesovertime.
NetworkGrowth model,attentioninequalitygrowsintime.
Our work leaves many questions unanswered. What are
Nodeletal.(NoelandNyhan2011)analyzedthe“unfriend-
the benefits of inequality? Is it due to easier to deal with
ing”behaviorusinglongitudinalsocialnetworkdata.Their
information overload if users only have to pay attention to
resultsindicatethatifthereisanon-negligiblefriendshipat-
popularitems?Whatarethedisadvantagesofinequality?Is
tritionovertime,homophilyinfriendshipretentioninduces
it associated with less diversity in content and viewpoints
significant upward bias to peer influence. Kwak (Kwak,
online?Whatmotivatespeopletobeengagedandcontribute
Chun,andMoon2011)studiedthedynamicsofunfollowing
tosocialmediawhentheydonotreceiveanyattention?We
in Twitter and showed that the reciprocity of the relation-
plantoaddresssomeofthesequestionsinfuturework.
ships,thedurationofarelationship,thefollowee’sinforma-
tiveness,andtheoverlapoftherelationships,affectthedeci-
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