Centrality in Social Networks. Conceptual Clarification

25 Pages·1.57 MB·English

by  Freeman L.

laicoS ,skrowteN 1 (1978/79) 215-239 215
reiveslE@ aiouqeS ,.A.S ennasuaL - detnirP ni eht sdnalrehteN
ytilartneC ni Social skrowteN
lautpecnoC Clarification
notniL C. nameerF
Lehigh university *
The intuitive background for measures of structural centrality in social
networks is reviewed aPzd existing measures are evaluated in terms of their
consistency with intuitions and their interpretability.
Three distinct intuitive conceptions of centrality are uncovered and
existing measures are refined to embody these conceptions. Three measures
are developed for each concept, one absolute and one relative measure of the
~entra~~t~o~f ~os~tio~lsi n a network, and one relenting the degree of centrali-
zation of the entire network. The implications of these measures for the
experimental study of small groups is examined.
ehT melborp fo ytilartnec
ehT aedi fo ytilartnec as deilppa ot human noitacinummoc was decudortni
yb salevaB ni 1948. eH was yllacificeps denrecnoc htiw noitacinummoc ni
small spuorg and eh dezisehtopyh a pihsnoitaler neewteb larutcurts -nec
ytilart and ecneulfni ni puorg .sessecorp
ehT tsrif hcraeser noitacilppa fo ytilartnec was edam rednu eht noitcerid
fo salevaB at eht puorG skrowteN ,yrotarobaL ,.T.I.M ni eht etal 1940s. ehT
tsrif seiduts erew detcudnoc yb dloraH ttivaeL )9491( and yendiS htimS
.)05.91( yehT erew detroper yb salevaB )0591( and salevaB and tterraB
591( ,)l and erew tsrif debircsed ni liated yb ttivaeL 591( .)1 esehT stroper all
dedulcnoc that ytilartnec was detaler ot puorg ycneiciffe ni ,gnivlos-melborp
noitpecrep fo pihsredael and eht lanosrep noitcafsitas fo .stnapicitrap
esehT stroper dedivorp eht sutepmi rof a taerg ynam erom stnem~epx~
hguorht eht 1950s and 1960s. erehT erew ,snoisnetxe snoitacifidom and
snoitarobale fo eht lanigiro .T.I.M .ngised As ecnedive ,detalumucca -woh
,reve eht stluser denrut tuo ot eb gnisufnoc and netfo .yrotcidartnoc -muS
gniziram eht latnemirepxe erutaretil ni 1968, ssegruB )8691( dedulcnoc that
eht“ hcraeser has ton decudorp tnetsisnoc and evitalumuc .”stluser -reveN
,sseleht eht stluser od wohs that ytilartnec is tnaveler ot eht yaw spuorg teg
tnemtrapeD* fo laicoS ,snoitaleR ecirP ,llaH gnidliuB $40, ,mehelhteB AP 18015, .A.S.U
dezinagro ot evlos at tsael emos kinds fo .smelborp sweiveR fo this -irepxe
latnem krow evah neeb decudorp yb tnemalF ,6591( 1960, 1963, ,)5691
redluM ,6591( ,)8591 reznalG and resalG ,7591( ,)1691 nehoC ,)4691(
Shaw ,)4691( ssegruB ,)8691( ykswodanS ,)2791( and sregoR and Agarwala-
sregoR ( .)6791
snoitacilppA fo eht tpecnoc fo ,ytilartnec ,revewoh evah ton neeb -noc
denif ot latnemirepxe seiduts fo puorg .gnivlos-melborp nhoC and ttoirraM
)8591( desu eht ytilartnec aedi ni rieht tpmetta ot dnatsrednu lacitilop -etni
noitarg ni eht txetnoc fo eht ytisrevid fo naidnI laicos .efil nI ,tceffe yeht
deksa woh a noitan as egral and suoenegoreteh as aidnI dluoc eb deretsinimda
at all. riehT noisulcnoc was that yreve tcepsa fo naidnI laicos efil was tink
rehtegot yb krowten sretnec that dnuob“ and ”deniwtretni esrevid strands
otni a detanidrooc .erutcurts sttiP )5691( denimaxe eht secneuqesnoc fo
ytilartnec ni noitacinummoc paths rof nabru .tnempoleved eH detcurtsnocer
eht htflewt yrutnec krowten fo revir noitatropsnart ni lartnec Russia ni an
tpmetta ot nialpxe eht ecnenimeerp fo eht nredom ytic fo wocsoM as ti
degreme morf gnoma eht ynam stelmah ni eht .aera ,wocsoM ti denrut ,tuo
was a rojam larutcurts retnec ni eht laveidem Russian noitatropsnart and
noitacinummoc .krowten htoB pmahcuaeB ( )5691 and eiznekcaM )b6691(
derolpxe eht snoitacilpmi fo ytilartnec rof eht ngised fo .snoitazinagro -uaeB
pmahc detseggus that eht ycneiciffe fo a wen noitazinagro gninibmoc owt ro
erom gnitsixe seno dluoc eb dezimitpo yb gnitcennoc eht subunits rehtegot
at rieht tsom lartnec .stniop ,eiznekcaM no eht rehto hand, deugra that eht
pihsnoitaler neewteb lanoitazinagro erutcurts and ycneiciffe dluohs dneped
nopu eht ytixelpmoc fo eht lanoitazinagro task.
eroM ,yltnecer leipezC )4791( desu eht tpecnoc ni an tpmetta ot nialpxe
eht snrettap fo noisuffid fo a lacigolonhcet noitavonni ni eht leets .yrtsudni
siH stluser erew ton yllacitsitats ,tnacifingis tub eh dnuof that leets smrif
that erew erom lartnec ni a krowten fo lamrofni mrif-retni snoitacinummoc
,erew ni ,lareneg reilrae sretpoda fo a wen gnitsac .ssecorp ,ytilartneC ti
,demees evag meht a lacigolonhcet .egde
sregoR )4791( deiduts eht ecnegreme fo owt kinds fo ytilartnec ni -retni
lanoitazinagro .snoitaler eH derevocsid that, sseldrager fo eht yaw ni hcihw
ytilartnec was ,dexedni niatrec snoitazinagro dednet ot eb yltnetsisnoc erom
lartnec than .srehto ,revoeroM ti deraeppa that eht ytilartnec fo an -azinagro
noit was elbatciderp ni trap morf its nwo scitsiretcarahc and ni trap morf eht
seitreporp fo eht krowten ni hcihw ti was .deddebme
nI noitidda ot eseht rehtar cimedaca ,snoitartsulli stpecnoc gnitaler ot
,ytilartnec and erom ylralucitrap ot ,ytilartneced evah yltnecer detcartta eht
noitnetta fo elpoep gnikrow ni ytinummoc snoitazinagro and planning.
erehT is a gniworg noissucsid fo eht noitazitarcomed fo yteicos hguorht -ed
noitazilartnec fo noisiced making. l ehT saedi detareneg ni this erutaretil evah
raelc ecnerefer ot larutcurts .ytilartnec
,eeS‘ rof ,elpmaxe nosaraS et al. .)8791(
Centrality ni social networks 2 17
,suhT eht aedi fo ytilartnec is evila and llew and is gnieb dezilibom ni an
reve gninediw egnar fo .snoitacilppa enoyrevE ,seerga ti ,smees that ytilartnec
is an tnatropmi larutcurts etubirtta fo laicos .skrowten llA edecnoc that ti is
detaler ot a hgih eerged ot rehto tnatropmi puorg seitreporp and .sessecorp
tuB ereht susnesnoc .sdne erehT is ylniatrec on ytiminanu no yltcaxe what
ytilartnec is ro no its lautpecnoc ,snoitadnuof and ereht is yrev elttil -eerga
tnem no eht reporp erudecorp rof its .tnemerusaem
revO eht ,sraey a taerg ynam serusaem fo ytilartnec evah neeb .desoporp
ehT tnempoleved fo serusaem dluohs pleh ot yfiralc a tpecnoc yb gniyficeps
its stnenopmoc and rieht .spihsnoitalerretni nI eht esac fo ,ytilartnec -woh
,reve eht etisoppo tceffe smees ot evah neeb .deveihca ehT lareves serusaem
era netfo ylno yleugav detaler ot eht evitiutni saedi yeht troprup ot ,xedni
and ynam era os xelpmoc that ti is tluciffid ro elbissopmi ot revocsid what,
fi ,gnihtyna yeht era .gnirusaem
ehT emit has ,emoc ti dluow ,mees ot ,pots ekat kcots and yrt ot ekam
emos esnes fo eht tpecnoc fo ytilartnec and eht egnar and stimil fo its -netop
lait rof .noitacilppa tahT is yltcaxe eht aim fo eht tneserp .yasse ehT esoprup
ereh is ot yfiralc and evloser emos fo eht lautpecnoc smelborp fo ytilartnec
and ot erolpxe emos fo eht syaw ni hcihw ytilartnec nac eb desu ni -rednu
standing human .spuorg
sihT ,yasse ,neht lliw nigeb htiw an noitanimaxe fo lautpecnoc and
tnemerusaem smelborp htiw tcepser ot .ytilartnec ,ylsuoivbO eht emphasis
ni hcus a noissucsid lliw eb no eht larutcurts seitreporp fo human -inummoc
noitac .skrowten ehT noissucsid fo larutcurts seitreporp fo skrowten is
yltaerg deifilpmis yb ecnerefer ot a wef smret and stpecnoc morf eht yroeht
fo graphs. erofeB gninimaxe ytilartnec as ,hcus ,neht eht txen noitces lliw
edivorp a feirb weiver fo eht tnaveler seitreporp fo hparg .yroeht
ehT lautpecnoc snoitadnuof fo ytilartnec lliw eb derolpxe ni eht gniwollof
owt .snoitces tsriF ew shall ekat up eht ytilartnec tpecnoc as ti srefer ot eht
snoitacol fo snoitisop ro stniop ni ,skrowten and neht enimaxe eht tpecnoc
as ti is deilppa ot eht llarevo erutcurts fo a krowten nekat as a .elohw
lareveS wen serusaem lliw eb decudortni ni an tpmetta ot etacilpxe eht
lautpecnoc noitadnuof fo .ytilartnec nI a ,esnes eht noitcudortni fo wen
serusaem at this egats is .etairporppani ,yllaedI serusaem dluohs worg tuo fo
decnavda laciteroeht ;stroffe yeht dluohs eb denifed ni eht txetnoc fo ticilpxe
ssecorp .sledom erofeB hcus sledom nac eb ,depoleved ,revewoh a niatrec
tnuoma fo lautpecnoc noitacificeps is ;yrassecen eht cisab sretemarap fo eht
melborp must eb tes .nwod ,suhT eht noitcudortni fo serusaem ni eht tneserp
txetnoc must eb dootsrednu ylpmis as a yaw fo gniyfiralc eht ytilartnec -noc
.tpec ehT aim is ton ot kcol“ in” ot yna tros fo etamitlu ytilartnec .erusaem
emoS terms dna stpecnoc fo hparg yroeht
A hparg stsisnoc fo a tes fo points and a tes fo senil ro segde gnitcennoc
pairs fo .stniop A hparg gnitsisnoc fo evif stniop and evif segde is nwohs ni
erugiF 1.
218 notniL C nameerF
erugiF .1 A graph htiw five stniop and five edges.
P3
P2 P4
IP fi P5
nehW owt stniop era yltcerid detcennoc yb an egde yeht era adjacent.
ehT rebmun fo rehto stniop ot hcihw a nevig tniop is tnecajda is dellac eht
degree fo that .tniop nI eht noitartsulli fo erugiF 1, tniop p, has a eerged fo
I and tniop p2 has a eerged fo 3.
neviG an deredronu riap fo ,stniop ,ip( ,)ip hcae is reachable morf eht
rehto fi and ylno fi ereht stsixe a htap - a ecneuqes fo eno ro erom ,segde
,iP( ,>%I (Pa, ,)bP ,bP( dP . . . ,ZP( ,>iP gninnigeb at ,ip PerhaPs gnissaP hguorht
etaidemretni gniknil ,stniop pa, pb, pC, . . . . ,Zp and gnidne at .jp A path that
snigeb and sdne at eht emas tniop is dellac a .Y&KJC nI erugiF ,1 eht path
,2p( AP kk AP ,+b >,nI si a .elcyc
nehW yreve tniop is elbahcaer morf yna rehto tniop eht hparg is dellac
connected. ehT hparg nwohs ni erugiF 1, rof ,elpmaxe is .detcennoc
detaicossA htiw hcae path is a distance lauqe ot eht rebmun fo segde ni
that path, ehT tsetrohs paths gniknil a nevig riap fo stniop era dellac -oeg
desics. erugiF 1 shows yltcaxe owt paths morf tniop p1 ot tniop ps; eno
hguorht stniop p2, p3 and p4, and eno hguorht stniop p2 and p4. ecniS eht
tsrif fo eseht paths has a ecnatsid fo 4 and eht dnoces a ecnatsid fo 3, eht
dnoces path is a .cisedoeg stnioP gnillaf no eht ylno cisedoeg ro no all -oeg
scised gniknil a nevig riap fo stniop era said ot stand neewteb eht dne .stniop
ynA noitacinummoc krowten nac eb detneserper yb a .hparg hcaE -isop
noit ni eht krowten sdnopserroc ot a tniop ni eht hparg and hcae -irtemmys
lac noita~inunrmoc knil sdnopserroc ot an egde ro enil that stcennoc a riap
fo .stniop
tnioP net ytilart
hguohtlA ti has reven neeb ylticilpxe ,detats eno lareneg evitiutni emeht
smees ot evah nur hguorht all eht reilrae gnikniht tuoba tniop ytilartnec ni
laicos :skrowten eht tniop at eht retnec fo a star ro eht hub fo a ,leehw
ekil that nwohs ni erugiF 2, is eht tsom lartnec elbissop .noitisop A nosrep
detacol ni eht retnec fo a star is yllasrevinu assumed ot eb yllarutcurts erom
lartnec than yna rehto nosrep ni yna rehto noitisop ni yna rehto krowten fo
ralimis .ezis
On eht ecaf fo ,ti this noitiutni smees ot eb larutan .hguone ehT retnec fo
a star seod raeppa ot eb ni emos tros fo laiceps noitisop htiw tcepser ot eht
Centrality ni social networks 219
llarevo .erutcurts ehT melborp is, ,revewoh ot enimreted eht yaw ro syaw ni
hcihw hcus a noitisop is yllarutcurts .euqinu
suoiverP stpmetta ot elpparg htiw this melborp evah emoc up htiw eerht
tcnitsid larutcurts seitreporp that era yleuqinu dessessop yb eht retnec fo a
star. tahT noitisop has eht mumixam elbissop ;eerged ti sllaf no eht -oeg
scised neewteb eht tsegral elbissop rebmun fo rehto stniop and, ecnis ti is
detacol at eht muminim ecnatsid morf all rehto ,stniop ti is yllamixam esolc
ot .meht
erugiF 2. A star or leehw htiw five points.
P2
PI P4
P3
+
ecniS eseht era all larutcurts seitreporp fo eht retnec fo a star, yeht -moc
etep as eht gninifed ytreporp fo .ytilartnec llA serusaem evah neeb desab
erom ro less yltcerid no eno ro rehtona fo .meht ,revoeroM hcae erusaem
is detaicossa htiw emos tros fo evitiutni basis ro elanoitar rof its nwo -rap
ralucit larutcurts .ytreporp esehT era ,desab ,ylsuoirav no eht ,ygolohcysp
scitilop ro ymonoce fo human .noitacinummoc
ehT tselpmis and spahrep eht tsom ylevitiutni suoivbo noitpecnoc is that
tniop ytilartnec is emos noitcnuf fo eht eerged fo a .tniop ehT eerged fo a
,tniop ,ip is ylpmis eht tnuoc fo eht rebmun fo rehto ,stniop ip i( f ,)i that
era tnecajda ot ti and htiw hcihw ti is, ,erofereht ni tcerid .tcatnoc ehT
lartnec ,tniop p3, ni erugiF 2, is tnecajda ot ruof rehto ;stniop its eerged is
.ruof nI a tniop-evif ,hparg yna tniop nac eb tnecajda ylno ot eht ruof
gniniamer ,stniop os eht mumixam eerged fo yna tniop is .ruof ,suhT p3 is
an elpmaxe fo a tniop fo mumixam eerged ni a tniop-evif .hparg
Shaw ( )4591 decudortni eht aedi fo using eerged as an xedni fo tniop
,ytilartnec and gnola htiw rehto sretirw xuehcuaF( and icivocsoM ;0691
eiznekcaM 1964, 1966a; leipezC ;4791 nenimeiN 1973, ;4791 sregoR )4791
ohw deviecnoc fo tniop ytilartnec ni this ,yaw eh yltnerappa dnuof ti os -ni
ylevitiut gnilaeppa that eh did ton rehtob ot discuss ro etarobale its -noc
lautpec snoitadnuof at all. roF eseht ,sretirw ytilartnec means .eerged
With tcepser ot ,noitacinummoc a tniop htiw ylevitaler hgih eerged is
wohemos “in eht kciht fo things”. eW nac ,etaluceps ,erofereht that sretirw
ohw evah denifed tniop ytilartnec ni smret fo eerged era gnidnopser ot eht
ytilibisiv ro eht laitnetop rof ytivitca ni noitacinummoc fo hcus .stniop
As eht ssecorp fo noitacinummoc seog no ni a laicos ,krowten a nosrep
ohw is ni a noitisop that stimrep tcerid tcatnoc htiw ynam srehto dluohs
nigeb ot ees flesmih and eb nees yb esoht srehto as a rojam lennahc fo -rofni
220 notniL C. nameerF
.noitam nI emos esnes eh is a lacof tniop fo ,noitacinummoc at tsael htiw
tcepser ot eht srehto htiw mohw eh is ni ,tcatnoc and eh is ylekil ot poleved
a esnes fo gnieb ni eht maertsniam fo noitamrofni wolf ni eht .krowten
At eht etisoppo emertxe is a tniop fo wol .eerged ehT tnapucco fo hcus a
noitisop is ylekil ot emoc ot ees flesmih and ot eb nees yb srehto as -irep
.larehp siH noitisop setalosi mih morf tcerid tnemevlovni htiw tsom fo eht
srehto ni eht krowten and stuc mih ffo morf evitca noitapicitrap ni eht -no
gniog noitacinummoc .ssecorp
serusaeM fo tniop ytilartnec desab yleritne ro ni trap no eht eerged ro
ycnecajda fo a tniop evah neeb depoleved yb Shaw ,)4591( xuehcuaF and
icivocsoM ,)0691( nosirraG ,)0691( eiznekcaM ,)a6691( sttiP ,)5.691( sregoR
,)4791( leipezC ,)4791( nenimeiN ,3791( )4791 and inatijaK and amayuraM
.)6791( ,yletanutrofnU eseht desab-eerged serusaem era netfo ylirassecennu
.detacilpmoc
ehT erusaem decudortni yb Shaw ,)45.91( rof ,elpmaxe was yltnerappa
desab no laciripme evruc ;gnittif ti is yletulosba elbigilletninu morf yna
laciteroeht evitcepsrep .revetahw esohT depoleved yb nosirraG ( )0691 and
sttiP )5691( ydobme a yllaitnetop gnitseretni noitazilareneg fo eht eerged
.noiton yehT esu eht aedi fo eht eerged fo a tniop at secnatsid retaerg than
.eno ,suhT yeht nigeb ot teg at eht aedi fo eht htped ot hcihw a tniop is
nevow otni eht latot bew fo .noitacinummoc ,htoB ,revewoh era dednuofnoc
yb eht tcaf that yeht era snoitcnuf ton ylno fo hparg ,ezis tub fo eht tsegnol
cisedoeg ni eht hparg revo hcihw yeht era .detaluclac nI ,ecneuqesnoc rehtien
sedivorp a tcerid xedni fo tniop ytilartnec that nac eb desu ot erapmoc
stniop ni graphs fo gnireffid sezis ro gnissessop gnireffid tsegnol .scisedoeg
htoB xuehcuaF and icivocsoM )0691( and eiznekcaM )a6691( nageb htiw
eerged as eht gniylrednu larutcurts tpecnoc rof gnitcurtsnoc rieht ,serusaem
tub htob dehctiws ot a nrecnoc htiw lacitsitats snoitubirtsid fo seicneuqerf
fo .ytivitca ,htoB ,erofereht dne pu htiw an xedni desab no gnihtemos rehto
than larutcurts .seitreporp
ehT erusaem desoporp yb sregoR )4791( is elpmis tub ti is dengised rof
laiceps snoitacilppa erehw seitinutroppo rof ycnecajda era .detcirtser
s’leipezC )4791( is a elpmis evitaler erusaem tub ti is detcirtser ot a -rap
ralucit dnik fo lacirtemmysnon ycnedneped .noitaler tahT decudortni yb
inatijaK and amayuraM )6791( is detcurtsnoc as an esrevni fo eerged rof
ycnetsisnoc ni rieht ;noitavired ti is elbanosaer yllacitamehtam tub awkward
rof .noitatupmoc ylnO nenimeiN )4791( has decudortni a ,elpmis larutan
and yltcefrep lareneg erusaem fo ytilartnec desab nopu .eerged
s’nenimeiN )4791( erusaem is eht tnuoc fo eht eerged ro rebmun fo
seicnecajda rof a ,tniop :kp
E
C,(P,) = ,iP4 )kP
=i
1
erehw
,ip(U )kp = 1 fi and ylno fi ip and pk era detcennoc yb a enil
0 esiwrehto
sA hcus ti si a dra~ofthgiarts xedni fo eht tnetxe ot hcihw kp si a sucof fO
.ytivitca ),p(nC si egral fi tniop kp si tnecajda ,ot ro ni tcerid tcatnoc ,htiw a
egral rebmun fo rehto ,stniop dna llams fi kp sdnet ot eb tuc ffo morf hcus
tcerid .tcatnoc )~p(nC = 0 rof a tniop taht si yllatot detalosi morf tcatnoc
htiw yna rehto .tniop
ehT edutingam fo )kp(nC si yltrap a noitcnuf fo eht ezis fo eht krowten
no hcihw ti si .detaluclac roF emos snoitacilppa siht si .tnavelerri sA a
erusaem fo eht reehs tnuoma fo ytivitca fo a tniop hcus na etulosba tnuoc fo
eerged si .lufesu tuB rof rehto snoitacilppa ti thgim eb elbarised ot evah a
erusaem taht si tnednepedni fo krowten .ezis oT erapmoc eht evitaler
ytilartnec fo stniop morf tnereffid ,shparg rof ,elpmaxe ew deen a erusaem
morf hcihw eht tceffe fo krowten ezis sah neeb .devomer
A nevig ,tniop >kp nac at tsom eb tnecajda ot n -- X rehto stniop ni a
.hparg ehT mumixam fo ,)_clp(nC ,erofereht si ap - .I nehT
si eht nu~~oporp fo rehto stniop taht are tnecajda ot .kp tI yam eb desu
revenehw a desab-eerged evitaler erusaem fo tniop ytilartnec sekam esnes ni
eht txetnoc fo na .noitacilppa
)kpfnc dna ,)kp(bC ,neht are serusaem fO tniop desab
Structural Centrality
no eht eerged fo tniop .kp ehT fo a tniop si deweiv as tnatropmi as
degree
na xedni fo sti laitnetop ~u~~~~~~u~~~~oc .~tiviica
The dnoces weiv fo tniop ytilartnec si desab nopu eht ycneuqerf htiw
hcihw a tniop sllaf neewteb sriap fo rehto stniop no eht tsetrohs ro cisedoeg
shtap gnitcennoc .meht sihT aedi yam eb detartsulli yb gnirrefer kcab ot
erugiF 2 erehw eht ,retnec p 3r swohs mumixam .ssenneewteb ehT net -oeg
scised ni erugiF 2 are nwohs ni erugiF 3. ruoF fo meht wohs secnatsid fo
;eno eseht tcennoc 3p htiw hcae fo eht rehto .stniop ehT rehto xis all wohs
shtgnel fo owt dna at1 evlovni 3p as a tniop-dim neewteb eht .srehto ,suhT
3p sllaf neewteb rehto stniop no xis fo eht net scisedoeg ni siht .hparg ecniS
ruof scisedoeg tsum be desu“‘ ”pu ni gnitcennoc eht lartnec tniop htiw hcae
fo eht ,srehto xis si eht mu~xam ssenneewteb tnuoc elbissop ni a -evif
tniop .hparg
htoB salevaB (1948) dna wahS (1954) detseggus taht nehw a nosrep si
yllacigetarts detacol no eht noitacinummoc shtap gniknil sriap fo ,srehto taht
nosrep si .lartnec A nosrep ni hcus a noitisop nac ecneulfni eht puorg yb
gnidlohhtiw ro gnitrotsid noitamrofni ni .noissimsnart lebmihS ( 1953)
desserts eht ytilibisnopser fO snosrep gniypucco hcus snoitisop rof eht -niam
ecnanet fo snoitacinummoc dna nhoC dna ttoirraM (1958) dezisahpme rieht
laitnetop as srotanidrooc fo puorg .sessecorp
sseldrageR fo ,sisahpme a tniop taht sllaf no eht noitacinummoc shtap
neewteb rehto stniop stibihxe a laitnetop rof lortnoc fo rieht _noitacinulnmoc
tI si siht laitnetop rof lortnoc taht senifed eht ytilartnec fo eseht .stniop
222 notniL .C nameerF
erugiF 3. The ten geodesics from the graph fo erugiF 2.
P2
I. I_+
P3
PI
2.
--p3
PI
3.
4p-p-
PI P3
4.
P3
PI
-I
IPe
P3
6.
P
I P4
P3
I
P2
7. P3
SP
.8
R--
P3
IP3
9.
P5
P4
P3
IO. r P5
hguohtlA Shaw )4591( dedulcni ssenneewteb stnuoc ni a xelpmoc -iripme
yllac desab erusaem fo ,ytilartnec eh did ton poleved a erusaem fo -neewteb
ness. tceriD serusaem erew depoleved yltnednepedni yb essinohtnA 791( )1
and nameerF .)7791(
gninimreteD ssenneewteb is elpmis and drawrofthgiarts nehw ylno eno
cisedoeg stcennoc hcae riap fo ,stniop as ni eht elpmaxe .evoba ,erehT eht
lartnec tniop nac erom ro less yletelpmoc lortnoc noitacinummoc neewteb
pairs fo .srehto tuB nehw ereht era lareves scisedoeg gnitcennoc a riap fo
,stniop eht noitautis semoceb erom .detacilpmoc A tniop that sllaf no emos
tub ton all fo eht scisedoeg gnitcennoc a riap fo srehto has a erom detimil
laitnetop rof .lortnoc
nI eht hparg fo erugiF 4, ereht era owt scisedoeg gniknil ip htiw p3,
eno EJ~U p2 and eno via p4. ,suhT rehtien p2 ron p4 is yltcirts neewteb p, and
p3 and rehtien nac lortnoc rieht .noitacinummoc ,htoB ,revewoh evah emos
laitnetop rof .lortnoc
e Centrality ni social networks 223
erugiF 4. A graph htiw four points and five edges.
PI
P2 4P
3P
sihT tros fo laitrap ssenneewteb nac eb denifed ni smret fo ,seitilibaborp
fI ew assume owt stniop ip and ip era tnereffidni htiw tcepser ot hcihw
fo lareves evitanretla scisedoeg seirrac rieht ,snoitacinummoc eht -aborp
ytilib fo using yna eno is
1
gij
erehw jig is eht rebmun fo scisedoeg gniknil ip and .ip ehT laitnetop fo tniop
pk rof lortnoc fo noitamrofni passing neewteb ip and ip neht yam eb denifed
as eht ytilibaborp that pk sllaf no a ylmodnar detceles cisedoeg gnitcennoc
ip and .ip fI
)kp(iig = eht rebmun fo scisedoeg gniknil ip and ip that niatnoc pk
neht
bij(Pk) = i X gij(Pk>
= gij(Pk>
gij
is eht ytilibaborp ew ;kees ti is eht ytilibaborp that tniop pk sllaf no a -nar
ylmod detceles cisedoeg gniknil ip htiw .ip
oT enimreted eht llarevo ytilartnec fo a tniop pk, ew sum its laitrap
ssenneewteb seulav rof all deredronu pairs fo stniop erehw i # j # :k
CB (Pk) = 5 i bij(Pk>
i<i
erehw 1y is eht rebmun fo stniop ni eht .hparg
ehT sum )kp(Bc is an xedni fo eht llarevO laitrap ssenneewteb fo tniop pk.
revenehW pk sllaf no eht ylno cisedoeg gnitcennoc a riap fo ,stniop )kp(ac
is desaercni yb 1. nehW ereht era evitanretla scisedoeg )kp(nc sworg ni -orp
noitrop ot eht ycneuqerf fo ecnerrucco fo pk gnoma esoht .sevitanretla
gnitacoL scisedoeg and gnitnuoc emoceb tluciffid htiw egral .skrowten
xirtaM sdohtem rof htob fo eseht tasks, ,revewoh era deliated yb yraraH
224 Lirltorl C. Freernm
tc .lu :5691( -431 .)141 and eseht sdohtem timrep eht tnempoleved fo a
elpmis retupmoc margorp ot etaluclac .),p(,C
ekiL ,),p(,C )kp(BC is tnedneped no eht ezis fo eht krowten revo hcihw ti
is .detaluclac And again, rof emos ,snoitacilppa a erusaem tuohtiw this
noitatimil dluow eb .lufesu What is dedeen is a erusaem that is evitaler ot its
mumixam eulav ni smret fo eht rebmun fo stniop ni its .krowten
nameerF )7791( devorp that eht mumixam eulav nekat yb C~(pk) is
deveihca ylno yb eht lartnec tniop ni a star. tI is
112 - 1r3 + 2
2
,eroferehT eht evitaler ytilartnec fo yna tniop ni a hparg yam eb desserpxe
as a ,oitar
seulaV fo )kp(’C yam eb derapmoc neewteb graphs. A star ro ,leehw rof
,elpmaxe fo yna ezis lliw evah a retnec tniop htiw )kp(bC = ;1 all rehto
stniop lliw dleiy ),p(iC = 0.
htoB ),p(,C and )kp(hC yam eb denimreted rof yna cirtemmys hparg
rehtehw detcennoc ro .ton yehT era serusaem fo tniop ytilartnec desab no
eht larutcurts etubirtta fo eht betwecrrness fo tniop pk. ssenneewteB is
lufesu as an xedni fo eht laitnetop fo a tniop rof cmtrol fo .noitacinummoc
ehT driht evitiutni noitpecnoc fo tniop ytilartnec is desab nopu eht eerged
ot hcihw a tniop is esolc ot all rehto stniop ni eht .hparg nI erugiF 2, rof
,elpmaxe tniop p3 is at a ecnatsid fo eno morf hcae fo eht ruof rehto .stniop
hcaE fo eht ,srehto ,revewoh is at a ecnatsid fo eno ylno morf p3, and at a
ecnatsid fo owt morf hcae fo eht gniniamer .stniop tnioP ,jp ,erofereht is
tsesolc ot all rehto .stniop As a rettam fo ,tcaf ecnis eht muminim ecnatsid
neewteb a riap fo stniop is ,eno p3 is as esolc ot its ruof srobhgien as yna
tniop dluoc eb ni yna tniop-evif .hparg
sihT driht weiv fo tniop ytilartnec is osla detaler ot lortnoc fo -acinummoc
noit tub ni a tahwemos tnereffid .yaw ,ereH a tniop is deweiv as lartnec ot
eht tnetxe that ti nac diova eht lortnoc laitnetop fo .srehto gnidroccA ot
salevaB ,)0591( a lartnec-non noitisop is eno that “must yaler segassem
hguorht . . . .”srehto ,suhT as ttivaeL 591( )1 ,detseggus a lartnec noitisop is
eno that is ton tnedneped nopu srehto as seiraidemretni ro ”sreyaler“ fo
.segassem nI ,tcaf this aedi was os hcum a trap fo s’ttivaeL gnikniht that eh
dednet ot esu eht sdrow ytilartnec and ecnednepedni .ylbaegnahcretni
ehT ecnednepedni fo a tniop is denimreted yb its ssenesolc ot all rehto
stniop ni eht .hparg nI erugiF 1, rof ,elpmaxe tniop p2 is ni tcerid hcuot htiw
eerht rehto ,stniop ,lp p3 and p4. eH must pass segassem hguorht p4, -woh
,reve ni redro ot hcaer ps. ,suhT zp sdneped nopu ylno eno reyaler ot -moc
etacinum htiw enoyreve ni eht .krowten On eht rehto hand, p1 sdeen p2 ot
etacinummoc htiw p3 ro p4 and htob p2 and p4 ot hcaer ps. oT hcaer -yreve
,eno ,neht p, must dneped nopu p2 eerht semit and p4 ecno - ruof stca fo