Ecological metrics of diversity in understanding social media

0.18 MB·English

by  Chris von Csefalvay

Ecological metrics of diversity in
understanding social media
∗
CHRIS VON CSEFALVAY FRSA
Abstract
Topicaldiscussionnetworks(TDNs)arenetworkscenteredaroundadiscourseconcerningaparticular
5
concept, whetherinreallifeoronline. Thispaperanalogisesthepopulationofsuchnetworksto
1
0 populationsencounteredinmathematicalecology,andseekstoevaluatewhetherthreemetricsof
2 diversityusedinecology-Shannon’sH(cid:48),Simpson’sλandEvar proposedbySmithandWilsongive
valuableinformationaboutthecompositionanddiversityofTDNs.Itconcludesthateachmetrichas
n
a itsparticularuse,andthechoiceofmetricisbestunderstoodinthecontextoftheparticularresearch
J question.

9
2 I. INTRODUCTION bynomeansunknown. Indeed,thesameexpan-

sionwascarriedout,independently,byHerfind-
]
I In analysing a a topical discussion network ahlandHirschman,increatingthemathemati-
S
(TDN),3,6,8 suchastweetsmentioningthesame callyidenticalHerfindahl-Hirschmanindexused
.
s hashtag or same keyword, an important ques- in antitrust law as a proxy of market concen-
c
[ tionfortheunderstandingofinformationflows tration.4,5,7,15,20 This paper asks, principally,

isthesourcediversityamongvariouscontribu- whethersuchanexpansionwouldyieldthesame
1
v tors to the conversation. In other words, what newinsightsthatityieldedineconomics. Inpar-
1 are the ’market shares’ attributable of individ- ticular,itaskswhetherthemetricsusedinmath-
2 ualsources? Istheconversationdominatedbya ematicalecologytranslatetotheratherdifferent
6
smallnumberofcontributorsorisitabalanced fieldofanalysingnetworksinhumaninteraction,
7
0 exchangewithalargenumberofactorsholding wherethenumberof’species’canoftenbequite
. conversationsatrelativeparity? large.
1
0 Thispaperexaminestheutilityofthreemet- Thispaperconsidersaparticularmanifesta-
5 rics - Shannon’s H(cid:48), the Simpson λ metric and tion of TDNs, namely Twitter hashtags. This
1
the E metricproposedbySmithandWilson ispartlyowingtotheethicalissuesinvolvedin
: var
v (1996) - in understanding TDNs in social me- usingaprimafacieclosedsocialnetwork,such
i
X dia. Inparticular,itisattemptingtoascertain asFacebook,intheresearchofoftencontentious
what each of these metrics reveal about infor- politicalissues,aswellasduetotherelativeease
r
a mation flows within a TDN, and to what ex- bywhichthedatanecessaryforsuchresearchis
tentthesemetricscanbeadaptedtoTDNanal- availablefromTwitter’sAPI.Whiletheresults
ysis. In this sense, a TDN is regarded as an do not indicate any prima facie source depen-
analogueofanentireecosystem,witheachcon- dence,furtherresearchusingotherinformation
tributorbeinga distinct’species’. Theircontri- sourcesisclearlywarranted. Insightsfromsuch
butions - whether Facebook posts, tweets, Pin- studies may lead to a better understanding of
terestpinsorInstagrams-areequivalenttoa how political discourse is acted out in the on-
species’sweightinanecosystem. Analogousex- linepublicspaceandhowsuchdiscoursecanbe
pansion of metrics in mathematical ecology is describedinstatisticalterms.
∗
MA(Oxon),BCL(Oxon),FellowoftheRoyalSocietyofArts.Technicalarchitect,RBplc,215BathRd,SloughSL14AA.
E-mail:[e-mail]ispaperaretheauthor’sownanddonotnecessarilyreflect
thepositionsofanyinstitutiontheauthorisaffiliatedwith.
1
II. BACKGROUND -oneversion,whichhecreditstoBrillouin(1960),
isusablewherethetotalnumberofindividuals,
I. Defining metrics i.e. thesizeofV inourcase,aswellastherich-
nessR ofthepopulation,isknown. Inthatcase,
ConsideraTDN,denotedasΘ,withR contribu-
thediversityofapopulationcanbeexpressedas
torsC ···C andasetT ofV elementscompris-
1 R
ibnugtiaolnlscobnytcroibnutrtiibonusto.rLCetntbheedneunmotbeedraosftcno.nWtrei- H= N1 lnN−(cid:88)R lntj (3)
canthenconstructaprobabilityspaceinwhich j=1
Brillouin’sformula,however,doesnotavail
1. thesamplespaceΩcomprisesallcontribu- usinthealltoofrequentcasewherethesizeof
tions,i.e. theentirepopulationisunknownorunknowable.
Ω={ω|ω∈T} (1) In that scenario, the true population diversity
cannot be calculated. It can, however, be esti-
2. the σ-algebra F the set of all subsets in
matedfromasample. Forasampleofrichness
Ω
thesamplespace,i.e. 2 ,and (cid:48)
R,theestimatedpopulationdiversityH isgiven
by
3. the probability measure that a contribu-
tionselectedatrandomwillbefromacon-
R
tributorCj as H(cid:48)=−(cid:88)pjlnpj (4)
j=1
t
P(ω )= j . (2) where p denotestheproportionofspecies j
j i
R
(cid:80) t in the sample. From (2) follows that for a con-
i
i=1 tributor j, the proportion of his tweets can be
representedas
Inthefollowing,threemetricswillbedefined
for any given TDN Θ consisting of R distinct t t
j = j (5)
contributorsandN distinctcontributionswhere R N
(cid:80) t
thenumberofcontributionsbycontributor jis i
i=1
denotedast andtheproportionofthecontribu-
j (cid:48)
Consequently,H canbecalculatedas
tionsbythatuserisdenotedas p .
j
R t t
II. Shannon-Weaver index H(cid:48)=−(cid:88) j ln j (6)
N N
j=1
Measuring the diversity of multispecies popu-
Shannon’sindexisoneofdiversity,i.e. itis
lations using the concept of ’information con-
stronglyrichness-dependentanditmeasurespri-
tent’gainedtractioninthelate1950s,following
marilytheentropicdissimilarityofthepopula-
theworkofClaudeShannonontheinformation
tionratherthantheevendistribution. Aderiva-
entropy of communication.21 Under the ’infor-
tiveoftheindex,knownsometimesasShannon’s
mationcontent’understandingofspeciesdiver- (cid:48)
evennessmetric J exists,whichisdefinedas
sity, the diversity of a multispecies population
isequivalenttotheuncertaintyoffindinganin- − (cid:80)R tj lntj
daniviidnudaivlidofuasplefrcoiemstihwehpeonpurlaantidoonmPly.11s,e1l6e,1c7tinIng J(cid:48)= H(cid:48) = j=1N N (7)
lnR R
other words, the ’information content’ of an in-
dividual within the population depends on the
III. The Simpson lambda index
population’sinformation(orspecies-occurrence)
entropy. TheSimpsonindex,usuallydenotedbyλ,isthe
Pielou(1966)distiguishestwodefinitionsof probability that two entities drawn at random
informationcontentthatwereinitiallyprevalent fromthepopulation,withreplacement,willbe
2
ofthesamespecies.9 Fortheprobabilityspace
(cid:115)
associatedwiththeTDNΘasdescribedabove, R
N− (cid:80) t2
itcanbecalculatedas i=1 i
E (Θ)= (9)
McI N−(cid:112)N
λ=(cid:88)R p2=(cid:88)R (cid:179)tj(cid:180)2 (8) R
j N SmithandWilson(1996)proposeanewmet-
j=1 j=1
ric,E ,whichisintuitivelybasedonthevari-
var
ance of the logarithm of each species’s popula-
In economics, this index is known as the
tion.22 It uses a trigonometric transformation
Herfindahl-HirschmanIndex(HHI),andserves
firstusedbyAlatalo(1981)toreducetheresult
toquantifymarketconcentration. Sinceitsincep-
toavalueinradians.1
tionin1950,ayearafterSimpson’sindependent
decisocloogvye,rtyheofHtherefisnadmahel-cHonircsecphtminanmIantdheexmhaatsicbael- Evar=1−π2arctanR1 i(cid:88)=R1ln(pi)−j(cid:88)=R1ln(Rpj)2
comethegoldstandardproxyformarketpower,
(10)
andformsthebasisoftheUSDepartmentofJus-
SmithandWilsonhaveprovedthismetrictobe
tice’sanalysisofpossibleanticompetitiveeffects
independent of species richness for all values
ofmergers. Theindex"accountsforthenumber
of R, as well as its sensitivity to changing the
offirmsinthemarket,aswellastheconcentra-
abundanceofthemostminorspecies. TheE
var
tion, by inforporating the relative size (that is,
metricwillbeconsideredalongsidetheShannon-
marketshare)ofallfirmsinthemarket".20 Its
Weaver and Simpson complement indices as a
transferabilityindicatesthatthephenomenonit
non-R sensitivemetric.
measures,namelyrelativeconcentration,isnot
specifictothedomainofitsorigin,butratherde-
V. Adaptation of the indices: diver-
scribesaccumulativerelationshipsanddiversity
gences and challenges
invariousdomains.
A primary feature of these indices is that they
IV. The E metric weredevelopedwithclassicalpopulationecology
var
in mind.10,13 Where they have been adapted,
Ithasbeenobservedthatthenumberofdistinct theyhavegenerallyappliedonlytoasmallnum-
species,knownasrichness(representedasR in berofentities-thus,forinstance,inthecontext
theabovemodeloftheTDNΘ),hasasignificant oftheHerfindahl-Hirschmanindex,thetypical
impactondiversitymetrics.1 Therichnessprob- numberofundertakingstoconsiderdoesnotex-
lemscalesaswetransferthemetricsfrompop- ceedafewdozen.15 Assuch,theadaptationto
ulation ecology to social interactions. In many theeconomiccontextwasreasonablyunproblem-
ofthesamplesthatwillbediscussedinourre- atic,asgivennarrowindustrydefinitions,most
search,R willbeinthethousands-indeed,the ofthecontributiontotheHerfindahl-Hirschman
largestsamplesetthatwillbeconsideredhasa index was by a relatively small number of un-
richnessofalmost150,000distinctcontributing dertakings,approximatelyonparwith(oreven
entities. Furthermore,thepreviousindicespri- smallerthan)thenumberofdistinctspeciesin
marily focused on diversity, whereas that only anecologicalstudy. Evenwherethatwasnotthe
deliverspartofthepicture. Evenness,too,plays case, a capping mechanism was implemented,
a significant role in understanding a TDN. To calculatingtheindexfor,conventionally,thetop
take account of evenness, multiple approaches 20ortop50undertakings.2,12,23
havebeenproposed. Onewastoadapttheindex The issue of much higher richness - al-
proposed by McIntosh (1967) for the measure- most 150,000 in our largest sample, from the
mentofspeciesdiversity,14 intoanindexofeven- #gamergatehashtag-meansrichness-sensitive
ness. The index proposed by Pielou would, for metrics, suchasSimpson’sλand H(cid:48), aremore
theTDNΘasdescribedabove,becalculatedas at risk of not reflecting real diversity than the
3
same metrics would be in the relatively con- berofhashtagsselectedfortopicalunambiguity
fineddomainofecologyorcompetitioneconomics, (asinglehashtagorsearchtermshouldretrieve
where the upper bound of richness was much most,ifnotall,ofacommunitybutnotretrieve
smaller. To alleviate any issues with adapta- resultsconcernedwithadifferentmeaningofthe
tion,thepresentresearchemploystwomethod- word),aswellastocovereachtypologicalclass
ologies. First,arichness-insensitivemetricE of’virtualcommunity’proposedbyPorter.19 The
var
isemployedinadditiontotherichness-sensitive samples were collected throughout the period
metrics.22 Second,themetricswerecalculated, fromOctobertoDecember2014.
alongsidetheentiresample,forthetopquintile The12hashtagsultimatelyselectedcomprise
andthetopdecileofcontributors,byproportion arangeofsubjectareas,popularityandsample
ofcontributions. sizes. ThesampleswerestoredinaMongoDBin-
stance and queried by a custom Python script,
III. METHODOLOGY whichthencalculatedthepertinentmetricsus-
ing pandas as a data abstraction. The met-
For the purpose of this research, a sample of rics were calculated by reference to individual
twelveTDNscenteredaroundarangeoftopics Twitterusers’identifiers,ratherthantheiruser
wereexamined. Usingahigh-throughputauto- namesor’screennames’,whicharebothmutable
mated retrieval engine written in Python that whereasuseridentifiersareassignedatthetime
interactswithTwitterusingaRESTfulAPI,sam- ofaccountcreationandaregloballyuniqueboth
plesofvarioussizeswereobtainedfromanum- acrossTwitterusersandacrosstime.
Table1: Hashtagsincludedintheresearch
Hashtag
Name Subjecttype N R p ×106 σp ×106
i i
#auspol Politics 206,040 25,410 39.3544 18.0504
#blacklivesmatter Politics 216,097 101,539 9.8484 36.3133
#cashinin Politics 3,682 1,258 794.9212 2103.0147
#dataviz Professional 5,079 3,236 309.0175 641.6421
#ferguson Politics 354,548 128,800 7.7648 41.1617
#gamergate Entertainment 3,711,580 146,472 6.8273 247.8704
#mtvstars Entertainment 103,400 26,638 37.5406 76.3736
#p2 Politics 88,594 24,085 41.5660 23.4973
#rstats Professional 1,071 645 1,550.4202 2.3789
#startup Professional 36,100 7,550 132.4515 36.0704
#tcot Politics 567,763 85,717 11.6663 66.5921
#uniteblue Politics 59,280 15,496 64.5327 14.4400
Afterdeterminingthesamplesize(N),rich- Themetricswerethenseparatelycalculatedfor
ness(R)andaverage p foreachhashtagsample, twosubpopulations,namelythetopquintileand
i
thethreemetricsthatformthebasisofthisstudy topdecileofusers,respectively,toassistwiththe
havebeencalculatedforeachofthehashtagsus- estimationofeachmetric’ssensitivitytoR.
ingaPythonimplementationofthecalculations.
4
0.015
0.8
9
Shannon entropy (H')6 l00..000150 Evar00..46
3
0.000
1,000 100,000 1,000 100,000 1,000 100,000
Richness Richness Richness
(a)Shannon’sH(cid:48) (b)Simpson’sλ (c)Evar
Figure1: MetricscalculatedforthesampledTDNs. Subsamplesderivedfromthesamehashtag(total,
firstquintileandfirstdecile)areconnectedbylines. ThecoloursidentifytheTDN’sclassification(red
=entertainment,green=politics,blue=professional).
Inaddition,totesteachmetric’sdependence forcomparingbiomeswithsignificantlydi-
on R, thePearsonproduct-momentcorrelation vergentnumbersofspecies. Thisisnotnec-
was computed for each of three subsamples of essarily a drawback, however, where the
each hashtag’s population, namely the whole comparisonrequiresrichnesstobetaken
hashtag,thetopquintileandthetopdecile. The intoaccount. Totheextentthatoneiscon-
(cid:48)
examination of the correlations found that H cerned with the relative likelihood of en-
stronglypositivelycorrelateswithR (r=0.692, counteringdivergentopinions,therichness
95%CI:0.471-0.832)andweaklynegativelycor- of a sample is hardly irrelevant. Indeed,
relatestoeachofλ(r=−0.253,95%CI:−0.553 therichnessofaTDNmightbecomerele-
- 0.059) and E (r=−0.275, 95% CI: −0.553 - vantwhenconsideringsuchquestionsas
var
0.059). Ofthesecorrelations,onlythatbetween whethertheconversationissubjecttothe
R and H(cid:48) isstatisticallysignificantat p<0.05. dominanceofafeworawidelyparticipa-
This indicates that at the sample sizes consid- torymarketplace,evenifitdoesnotallow
ered, both λ and E deliver accurate results comparisonofevennessindependentlyof
var
thatarenotstatisticallysignificantlyinfluenced richness. Itwould,therefore,bepremature
by the richness of the sample. Figure 1 shows todiscarditasuseless-withintherealm
(cid:48)
the relationship between the selected metrics ofsocialmediaanalysis,H couldserveto
andtherichnessofthesample. distinguishlow-participation,nicheTDNs
InagreementwiththeresearchbySmithand fromTDNswhereparticipationiswideand
Wilson(1996),thisexaminationoftheevenness relativelyeven,withthecaveatthatitwill
anddiversitymetricsconcludesthatH(cid:48)issignifi- belesssensitivetounevennessasrichness
cantlyaffectedbytherichnessRinthesample.22 increases.
FollowingthedistinctiondrawnbyPielou(1977),
a metric of diversity ought ideally to measure 2. Withtheexceptionofasingleoutlier(the
onlyoneoftheconstituentcomponentsofdiver- #startuphashtag),Simpson’sλappears
sity,namelyeitherrichnessorevenness,butnot to be the least affected by divergences in
ofboth. Thus,ametricsignificantlyaffectedby richness, both in the case of subsamples
richnesswouldbeanunsuitablemetricofeven- drawn from the same sample and in the
ness,andviceversa.18 Basedonthis,amethod- caseofinter-sampledifferences. Thisindi-
ologicalroleforeachofthemetricsemerges: catesthatλisagoodmetricformeasuring
and comparing evenness and measuring
(cid:48)
1. Shannon’s H is unsuitable for compar- dominanceinawaythatislesssensitive
isonsbetweenTDNsofsignificantlydiver- to R and (quadratically) more sensitive
gentR,justastheywouldnotbesuitable to the existence of entities with a larger
5
shareoftheconversation-inthisinstance, tributetoit? Asthediscussionintheprevious
it excelled at identifying the outlier, the chapterhasshown,eachofthethreemetricscon-
#mtvstarshashtag,whichwasmarkedby sideredinthispaperhasaparticularroleindis-
the strong participation of a few signifi- cerningthediversityandevennessofasample
cant sources (mainly media outlets and derived from a TDN. In mathematical ecology,
celebritybloggers),adistinctionothermet- the concept of diversity, dominance and even-
ricsdidnotpickupon. Assuch,wherethe nessservestounderstandtheinterrelationship
researchquestionseekstoidentifyrelative betweenvariousspeciesofvariousabundances
imbalances between the largest few and each. Do a few species dominate or do a large
theremainderofthesampleandthereby numberofdifferentspeciessharetheresources
pinpointsituationsofunusualdominance, availabletothepopulation? Doindividualsfall
thequadraticamplificationofsuchdomi- intospecieswithrelativelyevenprobabilitiesor
nancebytheλmetricisahelpfulmathe- isthereadistinct’fall-off’? Inthissense,therela-
maticaltool. tivedominancerelationshipsbetweenindividual
species within a population can be categorised
3. The E metric is also relatively unaf-
var andunderstoodbasedpurelyonmathematical
fectedbychangesinR,althoughnottothe
indicators. In the context of a social network,
extentthatSimpson’sλisinmostcases. It
such as a TDN, the issue is slightly different.
isinferiortoλindiscerningdominanceby
For one, the constraining factor is slightly dif-
a small number of highly dominant enti-
ferent. In a TDN, voices compete for a share
ties. Itdoes,however,deliversuperiorper-
of the conversation. Each contribution comes
formanceindiscerningtherelativeeven-
at a cost in terms of time, energy and various
ness between each contributor’s share in
system-providedmaximaofdailyorhourlycon-
the conversation in a way that is largely
tributionrates(e.g. Twitter’smaximumof2,400
sensitivetochangesintheshareofthecon-
messages). Assuch,thecontributionsrepresent
versationheldbythelowest-contributing
notmerelyhowmuchanindividualcontributor’s
contributors. Assuch,aseachsubsample
opinion adds to the whole of the conversation
isexpanded,theexpansionyieldseitheran
butalsoameasureofhisorherexpenditurein
increase or a decrease in E , reflecting
var termsoftimeandtherelativelylimitedresource
theinfluenceofthelessdominantcontrib-
ofdailytweetstotheconversation. Inthissense,
utors’ shares on the index, whereas such
higher p translatesnotonlytoalouder, more
expansion does not affect λ as most of a i
influential voice but also to a more significant
sample’sλisdeterminedbyitsfirstdecile
expenditure of a relatively scarce resource on
(indeed,oftenonlyafractionthereof).
participating in a TDN. Consequently, a TDN
withhighconcentrationindicatesthepresenceof
IV. RESULTS agentswhoareabletoexpendconsiderabletime
andeffortinmakingtheirvoiceheardaswell.
Whatdometricsofconcentrationandevenness
teachusaboutaTDNandthepeoplewhocon-
6
Table2: Diversitymetricsforvarioushashtags
Hashtag Fullsample Topdecile
Name Subjecttype H(cid:48) λ×104 E H(cid:48) λ×104 E
var var
#auspol Politics 8.4148 8.6722 0.3919 5.6269 8.6143 0.6143
#blacklivesmatter Politics 10.6933 1.2001 0.7085 4.6908 1.1666 0.7069
#cashinin Politics 6.1643 63.5415 0.6245 2.8426 61.3470 0.6671
#dataviz Professional 7.6130 16.4942 0.8492 2.3283 15.0661 0.7112
#ferguson Politics 10.5688 2.2599 0.6730 5.2954 2.2405 0.6360
#gamergate Entertainment 8.6397 6.6783 0.3204 7.6901 6.6778 0.2919
#mtvstars Entertainment 6.9324 145.6970 0.7445 3.9151 145.6642 0.5407
#p2 Politics 8.6366 17.3569 0.5978 4.5832 17.2662 0.6249
#rstats Professional 6.0897 47.2607 0.8218 1.7600 39.3447 0.7980
#startup Professional 6.9181 76.6908 0.5778 3.8879 76.4920 0.4257
#tcot Politics 9.3136 8.8127 0.4223 6.2346 8.7970 0.5940
#uniteblue Politics 8.4368 9.8394 0.5576 4.4738 9.6779 0.6784
Thisstudyfocusedontwofundamentalques- tributorsregardlessofsize,E isthe
var
tions. First,aremetricsofpopulationdiversity bestmetric. Astheexpansionshowed,itis
andevennessastheyareusedinpopulationecol- capableofindicatingsmallchangesinthe
ogyusefulmetricsofthediversityofaTDN?Sec- proportionoftheleast-contributingmem-
ond,whatdosuchmetricssayaboutaparticular bers as well, making it suitable for sen-
TDN? sitiveassaysofchangesinapopulation’s
participationrate.
I. Validity of the metrics
II. Interpretation
Asdiscussedabove,eachofthemetricshadtheir
own suitability spectrum. In other words, the Justaseachofthemetricshadaparticularas-
metricsconsideredeachhadaparticularinfor- pectofvalidity, theirinterpretationinthecon-
mationvalue. Assuch, thecrucialissueforre- textofTDNsissubtlydifferentforeach.
searcherswillbetoselecttheappropriatemetric
(cid:48) (cid:48)
fortheresearchquestion. 1. InterpretingH : AhigherH indicatesa
more vibrant conversation, either by in-
1. For research questions where rich-
creasing diversity (fewer dominant par-
(cid:48)
nessisrelevant,Shannon’sH metricis
ticipantsandlowerdominancerates), in-
agoodwaytodifferentiaterichandvibrant
creasing the number of participants or
TDNs from TDNs that are either domi-
both. Fortheeffectivenessassessmentof
natedbyafewloudvoicesorarerelatively
a TND, such as when evaluating the effi-
small.
ciencyofsocialmediainterventionstogen-
2. For near-complete independence eratediscoursearoundaparticulartopic,
from R and a good way to highlight this metric is more valuable than the R-
smallimbalances,theλmetricisasuit- independentmetrics,sincerichnessinand
able indicator that the conversation is ofitselfactsasaproxyofreachandconsti-
dominated by a few prominent partici- tutesanoptimisationgoal.
pants.
2. Interpreting λ: λis anextremely sensi-
3. For the assessment of evenness sensi- tive measure of diversity that is largely
tivetochangesintheshareofallcon- independent of R. It is a good metric to
7
measure evenness and dominance, and [2] Duncan Bailey and Stanley E Boyle. The
very sensitive to even small increases in optimal measure of concentration. Jour-
thedominanceofthelargestfewcontribu- naloftheAmericanStatisticalAssociation,
tors. Assuch,wheretheresearchquestion 66(336):702–706,1971.
seekstoidentifytherelativedominanceof
themostprominentvoices,theλmetricis [3] Axel Bruns and Jean E Burgess. New
methodologiesforresearchingnewsdiscus-
mostappropriate.
sionontwitter. 2011.
3. InterpretingE : Unlikeλ,E excels
var var [4] Stephen Calkins. The new merger guide-
at both ends of the user/frequency distri-
linesandtheherfindahl-hirschmanindex.
bution. Itismoresensitivetophenomena
California Law Review, pages 402–429,
such as the decreasing prominence of al-
1983.
ready less prominent users, a ’silencing’
phenomenon that can indicate a TDN’s [5] Neil B Cohen and Charles A Sullivan.
turn from discourse towards information Herfindahl-hirschman index and the new
distributionwiththeoccasionalcomment antitrustmergerguidelines: Concentrating
fromothercontributors. onconcentration. Tex.L.Rev.,62:453,1983.
[6] Sandra Gonzalez-Bailon, Andreas
V. CONCLUSION Kaltenbrunner, and Rafael E Banchs.
The structure of political discussion net-
Metricsofdiversityinpopulationecologyhave works: a model for the analysis of online
beenusefullyappliedinotherfields,suchascom- deliberation. Journal of Information
petitioneconomics. However,todate,theyhave Technology,25(2):230–243,2010.
notbeenusedinthecontextofanalysingTDNs.
[7] TimothyHHannan. Marketshareinequal-
Thisisnottheleastduetotheapparentdiffer-
ity,thenumberofcompetitors,andthehhi:
ences,suchasthevastlylargerrichnessand,usu-
Anexaminationofbankpricing. Reviewof
ally,individualswithinTDNsamples. Thisstudy
IndustrialOrganization,12(1):23–35,1997.
concludedthatecologicalmetricsofdiversityare
similarly useful in describing various features [8] TimHighfield,LarsKirchhoff,andThomas
ofTDNs,butneedtobeappliedwithintheirdo- Nicolai. Challengesoftrackingtopicaldis-
mains. Theconclusionislimitedontheevidence cussion networks online. Social Science
fromarelativelysmallnumberofhashtags,but ComputerReview,29(3):340–353,2011.
thesamplecanformanyreasonsberegardedas
representative. Furtherresearchandvalidation [9] PaulRHunterandMichaelAGaston. Nu-
ofecologicalmetricsofsocialmediainteractions mericalindexofthediscriminatoryability
on TDNs is certainly required and justified, in oftypingsystems: anapplicationofsimp-
particularwithaviewtoclassificationandclus- son’sindexofdiversity. Journalofclinical
tering of TDNs and cross-correlation of value microbiology,26(11):2465–2466,1988.
rangestoparticularpatternsofcentre-periphery
[10] Anne E Magurran. Measuring biological
distributions(suchascentralitymeasuresfrom
diversity. 2004.
SNA).
[11] AnneEMagurranandAnneEMagurran.
Ecological diversity and its measurement,
REFERENCES
volume168. Springer,1988.
[1] RaunoVAlatalo. Problemsinthemeasure- [12] ChristianMarfels. Bird’seyeviewtomea-
mentofevennessinecology. Oikos,pages sures of concentration, a. Antitrust Bull.,
199–204,1981. 20:485,1975.
8
[13] Norman WH Mason, Kit MacGillivray, [18] EvelynChrisPielou. Mathematicalecology.
John B Steel, and J Bastow Wilson. An Johnwileyetsons,1977.
index of functional diversity. Journal of
VegetationScience,14(4):571–578,2003. [19] Constance Elise Porter. A typology of
virtualcommunities: Amulti-disciplinary
[14] Robert P McIntosh. An index of diversity
foundation for future research. Jour-
andtherelationofcertainconceptstodiver-
nalofComputer-MediatedCommunication,
sity. Ecology,pages392–404,1967.
10(1):00–00,2004.
[15] Richard A Miller. Herfindahl-hirschman
[20] StephenARhoades. Herfindahl-hirschman
index as a market structure variable: An
index,the. Fed.Res.Bull.,79:188,1993.
exposition for antitrust practitioners, the.
AntitrustBull.,27:593,1982.
[21] ClaudeShannon. Collectedpapers. 1993.
[16] EC Pielou. Shannon’s formula as a mea-
[22] Benjamin Smith and J Bastow Wilson. A
sureofspecificdiversity: itsuseandmisuse.
consumer’sguidetoevennessindices.Oikos,
AmericanNaturalist,pages463–465,1966.
pages70–82,1996.
[17] ECJPielou. Themeasurementofdiversity
in different types of biological collections. [23] David S Weinstock. Using the herfindahl
Journaloftheoreticalbiology,13:131–144, indextomeasureconcentration. Antitrust
1966. Bull.,27:285,1982.
9