Considering clustering measures: third ties, means, and triplets. Binh Phan BI Norwegian Business School. Kenth Engø-Monsen Telenor Group

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1 This file s donloded from he insiuionl reposiory BI Brge - hp://brge.bibsys.no/bi (Open Access) Considering clusering mesures: hird ies, mens, nd riples Binh Phn BI Noregin Business School Kenh Engø-Monsen Telenor Group Øysein D. Fjeldsd BI Noregin Business School This is he uhors finl, cceped nd refereed mnuscrip o he ricle published in Socil Neorks, 5(01): DOI: hp://dx.doi.org/ /j.socne The publisher, Elsevier, llos he uhor o rein righs o pos revised personl version of he ex of he finl journl ricle (o reflec chnges mde in he peer revie process) on your personl or insiuionl ebsie or server for scholrly purposes, incorporing he complee ciion nd ih link o he Digil Objec Idenifier (DOI) of he ricle. (Publisher s policy 011).

2 Considering Clusering Mesures: Third Ties, Mens, nd Triples 1 by Binh Phn, Kenh Engø-Monsen, nd Øysein D. Fjeldsd Absrc Mesures h esime he clusering coefficiens of ego nd overll socil neorks re imporn o socil neork sudies. Exising mesures differ in ho hey define nd esime riple clusering ih implicions for ho neork heoreic properies re refleced. In his pper, e propose novel definiion of riple clusering for eighed nd undireced socil neorks h explicily considers he relive srengh of he ie connecing he o lers of he ego in he riple. We rgue h our proposed definiion beer reflecs heorized effecs of he imporn hird ie in he socil neork lierure. We lso develop ne mehods for esiming riple, locl nd globl clusering. Three differen ypes of mhemicl mens, i.e. rihmeic, geomeric, nd qudric, re used o reflec lernive heoreicl ssumpions concerning he mrginl effec of ie subsiuion. Key ords: socil neork, neork clusering, riple clusering, clusering mesure, nd ie srengh 1. Inroducion Clusering is mjor srucurl propery of socil neorks (Nemn, 00; Uzzi e l., 007). Clusering cn consrin neork cors s ell s provide hem ih neork closure benefis. Locl mesures cpure he clusering of ego neorks, heres globl mesures cpure he clusering of overll socil neorks. Acors re ffeced by boh locl nd globl neork properies (Colemn, 1988; Fleming e l., 007; Snijders e l., 010; Uzzi nd Spiro, 005; Xio nd Tsui, 007). The y clusering is defined nd mesured hs implicions for ho neork heoreic properies re refleced in sudy. Tble 1 presens ell-esblished locl nd globl neork clusering mesures. A number of hese re derived from mesuremen of neork riples. 1 This pper hs been cceped for publicion in Socil Neorks. Deprmen of Sregy nd Logisics, BI Noregin Business School, 0484 Oslo, Nory. Emils:.binh.phn@bi.no nd oysein.fjeldsd@bi.no Telenor Group, Reserch nd Fuure Sudies, Snrøyveien 0, N-11 Fornebu, Nory. Emil: Kenh.Engo- Monsen@elenor.com 1

3 Inser Tble 1 pproximely here A riple is defined s consising of focl cor (or ego) nd o lers. In riple, here re o ies (T1 nd T) connecing he ego ih is o lers, nd possible hird ie (T) connecing he o lers (Luce nd Perry, 1949; Opshl nd Pnzrs, 009; Ws nd Srogz, 1998). The ies my vry in srengh. See Figure 1. Inser Figure 1 pproximely here The exisence nd properies of hird ie connecing n ego s lers re given much enion in he socil neork lierure (c.f. Bur (199) nd Grnoveer (197)). Hoever, is heoreicl chrcerisics re no fully refleced in he exising clusering mesures. In his pper, e propose definiion of riple clusering for eighed socil neorks h explicily considers he relive srengh of he hird ie nd develop corresponding locl nd globl clusering mesures. We rgue h he proposed definiion nd mesures beer reflec he heoreicl effecs h clusering is inended o cpure. We only consider undireced neorks nd leve he exension o direced neorks for fuure reserch. The clusering coefficien of neork is esimed by ggreging he clusering vlues of ll riples in he neork. A riple is he pproprie uni of nlysis for sudying clusering in neork. The clusering vlue of riple is n indicor of he focl cor s embeddedness, opporuniies nd consrins. Locl clusering mesures ke ino ccoun ll riples in hich he ego is he focl cor (Brr e l., 004; Opshl, 009; Ws nd Srogz, 1998). Globl clusering mesures esime he clusering coefficien of overll socil neorks by ggregion of riples (Opshl nd Pnzrs, 009). 4 Exising clusering mesures differ in ho hey esime he relive clusering vlue of riples. The differences re rooed in he definiions of riple clusering nd he ype of men used o esime riple vlue. Inser Tble pproximely here 4 The belo revie of clusering mesures drs on Engø-Monsen n Cnrigh (011)

4 Exising clusering mesures cn be clssified ino hree min groups, See Tble. The firs binry group defines riple clusering s funcion of riple connecedness (Luce nd Perry, 1949; Ws nd Srogz, 1998). A riple does no hve clusering if T is missing nd vice vers clusered if T is presen. The second group of clusering mesures, define riple clusering s funcion of riple connecedness nd he men of he srengh of ll hree ies in riple herefer denoed s he srengh of T1// (c.f. Grindrod (00), Holme, Min Prk, Kim nd Edling (007), Kln nd Highm (007), Lor nd Mrchiori (00), Zhng nd Horvh (005)). The hird group defines riple clusering s funcion of riple connecedness nd he men of he srenghs of T1 nd T herefer denoed he srengh of T1/ (Brr e l., 004; Opshl, 009; Opshl nd Pnzrs, 009). In summry, he firs group of clusering mesures does no ddress ie srengh. The second group verges he hree ies hen presen, heres he hird group considers only he srengh of T1/. The srengh of T1/ reflecs he ime nd resources h he focl cor hs invesed in he riple, heres he srengh of T ffecs he relive posiion of riple members nd hus he focl cor s neork benefis nd consrins (Bur, 199; Colemn, 1988; Grnoveer, 197; Uzzi, 1996; Uzzi nd Ryon, 00). The ype of mhemicl men used in compuing clusering yields differen mrginl res of subsiuion mong ie srenghs. 5 The geomeric, qudric nd rihmeic mens respecively yield diminishing, incresing nd consn mrginl res of subsiuion mong ie srenghs. Hence, differen ypes of mens cn be used o reflec differen neork heoreic properies e.g. embeddedness, opporuniies, nd consrins (c.f. Bur (001), Uzzi (1996)). Figure illusres he indifference curves for geomeric, rihmeic nd qudric mens. Inser Figure pproximely here We clim hree conribuions o he socil neork lierure. Firs, e offer ne clusering definiions nd mesures h explicily consider he relive srengh of he hird ie in riple. Second, e sho ho differen ypes of mhemicl mens cn be used o reflec imporn neork heoreic differences. Third, e sho by y of nlysis, empiricl esing, nd simulion ho he mesures ffec clusering coefficiens. The second nd hird conribuions imply h 5 Mrginl re of subsiuion mong ie srenghs refer o he re hich he moun of he srengh of one or more ies needs o decrese in response o 1-uni increse of noher ie s srengh so h he join effec of ll hese ies on he socil cor remins he sme.

5 choice of clusering mesures in socil neork sudies should be mde coningen on he neork heoreic properies under considerion. The reminder of he pper is orgnized s follos. The second secion nlyzes he clusering properies of riples, rids, ego neorks nd overll socil neorks, nd proposes ne definiion of riple clusering. The hird secion refines he proposed mehod for esiming riple clusering vlues, by using hree differen ypes of mhemicl men nd develops he corresponding specific locl nd globl clusering mesures. In he fourh secion, e provide nlysis nd empiricl ess. The fifh secion oulines he prmeers of he proposed mesure in Bernoulli rndom grph. The finl secion discusses conribuions, limiions nd implicions for reserch.. Clusering properies of riples, rids, ego neorks nd socil neorks An undireced socil neork includes se of socil cors nd se of undireced relions (ies) h connec he cors (Bocclei e l., 006). Ties vry in srengh frequenly mesured by he inensiy of inercions beeen pirs of cors (Brr e l., 004; Grnoveer, 197; Nemn, 001). The srengh of ie reflecs he moun of ime nd resources h he cors hve invesed in i. Belo e revie common clusering properies nd mesuremens using smll illusrive eighed neork of five cors (A, B, C, D, nd E). In his neork, depiced by Figure b, he srengh of ies is denoed on he verices of he grph. Inser Figure pproximely here A socil neork conins number of ego neorks, i.e. neorks consising of focl node (ego) nd he nodes direcly conneced o i. In Figure, here re 5 ego neorks. Wihin n ego neork, here re number of rids subses of hree cors nd possible ies mong hem (Wssermn nd Fus, 1994: 19). For exmple, in he ego neork in hich B is he ego, here re 6 rids, specificlly, A-B-C, B-C-E, A-B-D, A-B-E, B-C-D, nd B-D-E. Beeen-rid differences: Trids vry in heir degree of clusering. Trid clusering definiions incorpore connecedness mong cors ihin rid s ell s he srenghs of ies connecing hem. Among hese six rids, he rids A-B-D, A-B-E, B-C-D, nd B-C-E re incomplee nd hve no clusering since ie is missing, heres he rids A-B-C nd B-D-E re complee nd clusered since ll heir hree cors re conneced o ech oher. Beeen he o complee rids, A-B-C is 4

6 more clusered hn B-D-E becuse he verge srengh of he ies in he former is greer hn in he verge srengh of he ler. Wihin-rid difference: Inside rid, he hree cors differ in he moun of ime nd resources h hey hve invesed in he rid. For exmple, in he A-B-C rid, A nd C hve invesed more ime nd resources, nd re herefore more embedded in he rid hn B. Such ihin rid srucurl differences re cpured by riples, hich consis of focl cor, o lers, he ies (T1 nd T), connecing he focl cor ih he o lers nd possible ie (T) connecing he o lers (Opshl nd Pnzrs, 009). A complee rid consiues hree closed riples ih ech of he rid s members being he focl cor in one of he riples. An incomplee rid ih only o ies consiues n open riple in hich he cor connecing he o lers is he focl cor nd T is missing. For insnce, he complee rid A-B-C consiues he hree closed riples, A-B-C, B-A- C nd A-C-B (he middle node is he focl cor of he riple). The incomplee rid A-B-D consiues one open riple A-B-D in hich B is he focl cor nd he ie beeen A nd D, T, is missing. We define riple clusering s funcion of hree fcors: (i) riple connecedness, (ii) he srengh of T1/, nd (iii) he relive srengh of T o he srengh of T1/. We discuss he role of ech fcor belo. i. Triple connecedness differenies closed riples from open riples. An open riple, here T is missing, hs no clusering nd crees no neork closure benefis for he focl cor. The focl cor is no embedded in he riple. Insed, in n open riple, he focl cor occupies he brokering posiion, is relively poerful in he rid, nd cn obin brokering benefis by plying he o lers gins one noher or being he hird pry (Bur, 199, 005; Cook nd Emerson, 1978; Emerson, 196). By conrs, ll closed riples re clusered nd cn cree neork closure benefis for heir focl cors. ii. The srengh of T1/ is n indicor of he focl cors invesmen of ime nd resources in he riple nd hence he focl cor s embeddedness. In he complee rid A-B-C, A nd C hve invesed more ime nd resources, nd re herefore more embedded in he rid hn B. This is indiced by he srengh of he closed B-A-C nd A-C-B riples T1/s being greer hn he srengh of T1/ of he closed A-B-C riple. Hence, he B-A-C nd A-C-B riples re more clusered hn he A-B-C riple. iii. The focl cor is lso ffeced by he relive posiion of he focl cor in relion o is o lers, of hich he relive srengh of T o he srengh of T1/ is n indicor. The srengh of T being greer hn T1/ indices h he o lers hve invesed more ime nd resources in 5

7 he rid hn he focl cor, nd focus on ech oher more hn on he focl cor. Hence, he focl cor fces significn disdvnges. By conrs, if T1/ is greer hn T, he focl cor is poenilly more poerful hn he o lers in he riple, bu he poer is consrined by he exisence of he connecion beeen he o lers (Bur, 199; Grindrod, 00; Holme e l., 007; Onnel e l., 005). Tking ino ccoun he relive srengh of T o T1/ differenies he proposed mesure from he definiions of he second nd hird groups (given in Tble ) of eighed clusering mesures. The second group of clusering mesures define riple clusering s funcion of (i) ie connecedness nd (ii) he srenghs of T1//. There re o issues ih his riple clusering definiion. Firs, i does no discrimine beeen riples here he focl cors hve invesed differen mouns of ime nd resources in he rid. For exmple, in he A-B-C rid, A-B-C, B-A-C, nd A-C-B, ould be considered equivlen even if A nd C hve invesed more ime nd resources in he rid hn B. Moreover, he srengh of T nd he srengh of T1/ ply he sme role in compuing he clusering of he riple nd hence heir respecive effecs on he focl cor re ssumed o be similr. This is unsisfcory, becuse lhough T does no hve direc effec on he focl cor, i ffecs he focl cor indirecly by deermining he relive posiion of he focl cor vis-à-vis he o lers. Our riple clusering definiion reflecs our suggesion for lering he role of he hird ie in he second group s definiion in ys h cpure he indirec effec of his ie on he focl cor. The hird group of clusering mesures define riple clusering s funcion of (i) riple connecedness, (ii) he srengh of T1/. This definiion does no ccoun for he indirec effec of he hird ie. Our definiion exends he hird group s definiion o include he relive srengh of T o T1/.. A Ne riple vlue esimion mehod nd ne clusering mesures.1. Triple vlue esimion mehod ih hree differen ypes of mhemicl men In his secion e develop ne mehod for esiming riple clusering ih hree differen ypes of men bsed on our proposed riple clusering definiion discussed bove. As shon in Figure, he geomeric, rihmeic nd qudric mens hve differen mrginl res of subsiuion mong ie srenghs. The geomeric men yields diminishing re of subsiuion mong ie srenghs. The geomeric men emphsizes he role of ek ies in he neork. In conrs, he qudric men yields n incresing re of subsiuion. I emphsizes he role of srong ies in he neork. The rihmeic men yields consn mrginl res of subsiuion mong ie srenghs, nd herefore considers he vlue of ie being liner funcion of is srengh. Hence, 6

8 differen ypes of men should be considered hen esiming riple clusering vlues nd clusering coefficiens for neorks being exmined from differen heoreicl properies. To mjor heoreicl pproches o he sudy of cor benefis from socil neorks posule conrry effecs of ego-neork clusering nd scribe differen effecs o ie srengh ((Bur, 199, 001; Colemn, 1988; Grnoveer, 197). From n opporuniy perspecive, socil cors obin nonredundn informion nd resources, s ell s brokering benefis from sprse neorks in hich ek ies ply n imporn role (Bur, 199; Grnoveer, 1978; Grnoveer, 197). The geomeric men ccenues he impornce of ek ies in neork. In conrs, from n embeddedness perspecive, socil cors obin rus nd relible suppor from srong ies nd closure (Colemn, 1988; Xio nd Tsui, 007). The qudric men diminishes he role of ek ies in neork. Reserch on smll orld properies of lrge scle neorks hs shon complemenry benefis in neork opogrphies h combine embeddedness nd sprseness (Schilling nd Phelps, 007; Uzzi nd Spiro, 005). In he belo developmen of our proposed mesures, e sr ih he geomeric men nd hen exend i o he rihmeic nd qudric mens. The proposed mehod for esiming riple clusering bsed on geomeric men is: V = (1) i,,,, here, i is he focl cor,,,, nd, denoe he eighs (or srenghs) of he ies connecing he focl cor ih he lers, nd he ie connecing he o lers in Triple. Furher, V i, denoes he relive clusering vlue of Triple in hich i is he focl cor. The fcor, 1, in Equion (1) reflecs he combined role of he ies T1/ in compuing he clusering of he riple. The fcor, 1,, plys o roles in he formul. Firs, i cpures riple connecedness h differenies open riples from closed riples. This fcor is equl o 0 for ll open riples nd greer hn 0 for ll closed riples. Second, i llos reling he srengh of T o he srengh of T1/. Specificlly, if he srengh of T,,, is greer hn he geomericlly verged srengh of T1/,, 1,, hen he produc 1,,,, is greer hn he single fcor, 1,. By conrs, if, is eker hn, 1,, hen he produc of, 1,,, is smller hn he produc of, 1,. If is equl o, 1,, hen he produc of, 1,,, is idenicl o, 1,. Summing up: using 7

9 equion (1), he riple clusering vlue increses disproporionely ih he rio of he srengh of T o T1/. Inuiively, i ould cpure h, ll else equl, srenghening of T disproporionely increses he consrin on he focl cor. Descripively, by king ino ccoun he relive srengh of T compred o he srengh of T1/, his riple esimion mehod is ble o differenie mong closed riples in complee rid here he focl cors hve invesed differen mouns of ime nd resources in he rid. See Appendix A for he corresponding mehod for esiming riple clusering vlues bsed on rihmeic nd qudric mens... Ne Locl Clusering Mesure Locl clusering mesures consider riples in hich he ego is he focl cor (Brr e l., 004; Luce nd Perry, 1949; Nemn, 00; Opshl, 009). For insnce, in he illusrive neork in Figure d, he ego neork in hich B is he ego hs 6 rids A-B-C, A-B-D, A-B-E, B-C-D, B-C- E, nd B-D-E. Among hese rids, he o complee rids A-B-C nd B-D-E consiue 6 closed riples, B-A-C, A-B-C, A-C-B, B-D-E, D-B-E nd B-E-D, heres he four oher incomplee rids consiue 4 open riples, A-B-D, A-B-E, C-B-D nd D-B-E. The exising locl clusering mesures esime he clusering coefficien of his ego neork ih n ggreged clusering vlue of ll riples in hich he ego, B, is he focl cor, specificlly, A-B-C, D-B-E, A-B-D, A-B-E, C-B-D nd D-B-E. In n ego neork, complee rid consiss of hree closed riples. In hese riples, he ego is he focl cor nd in he o oher riples, he ego s lers re he focl cors. The exclusion of riples, from n ego neork, in hich he ego is no he focl cor, ignores heir specific T effecs in he clculion of ego neork clusering coefficiens. Therefore, e propose locl clusering mesures hich ke ino ccoun ll riples in he ego neork. The formul of he proposed locl clusering mesure using geomeric men is s follos: c i = c,,,,,, o, () Where, c i denoes he clusering coefficiens of he ego neork of cor i. denoes riple. c nd o in he denominor denoe closed nd open riples respecively. In he numeror, ll open riples hve no clusering vlue. The numeror is hus he ggreged clusering vlue of ll closed riples. The denominor is normlizion fcor hich ensures h he clusering coefficien vries beeen 0 nd 1. I is combinion of he ggreged clusering vlue of ll closed riples nd he 8

10 ggreged vlue of open riples. The firs fcor c, 1,,, is used o esime he vlue of closed riples. Due o he bsence of T, he vlue of open riples is esimed ih he second fcor o 1,,, hich is he men of he srenghs of T1 nd T (or he srengh of T1/) (Opshl nd Pnzrs, 009). See Appendix B for he corresponding locl clusering mesures using rihmeic nd qudric mens... Ne Globl Clusering Mesure Globl clusering mesures esime he clusering coefficiens of overll socil neorks. Exising globl clusering mesures esime he clusering coefficien of socil neork ih eiher (i) n verged clusering coefficien over ll ego neorks (Brr e l., 004; Nemn, 00), hich ends o infle he clusering coefficien of neork (Opshl nd Pnzrs, 009) or (ii) n ggreged clusering vlue of ll is riples, divided by denominor normlizing he clusering coefficien o be in he 0-1 rnge (Engø-Monsen nd Cnrigh, 011; Opshl nd Pnzrs, 009). In he folloing e pply he proposed riple clusering frmeork o he second mehod. C = c,,,,,, o, () Where, C denoes he clusering coefficien of socil neork. The difference beeen his globl clusering mesure (Equion ) nd is locl version (Equion ) is he riples h re ken ino ccoun. The locl version kes ino ccoun ll riples in he ego neork, hile he globl version kes ino ccoun ll riples in he neork. See Appendix C for he corresponding globl clusering mesures using rihmeic nd qudric mens. 4. Differences mong clusering mesures The heme of his pper is ho o ssess clusering properies for eighed neorks. In he presence of ie srengh vrince, he suggesed, ne clusering mesures score he clusering properies differenly, depending on he ype of men implemened in he clusering formule. We compre he hree neorks in Figure o illusre his. Overll, he hree neorks hve he sme ggreged ie srengh, nd vry only in ie srengh vrince. This vrince is qunified s 0, 0.4, nd 0.6 for he neorks in Figures, b, nd c, respecively. The differen clusering mesures re summrized in Tble. Consisenly, Opshl nd Pnzrs (009) gives he greer vlue, Engø-Monsen nd Cnrigh (011) gives he smller vlue, nd he ne mesure comes ou in he 9

11 middle for he geomeric men versions. Using he qudric men reverses he order. Tble 4 presens he clusering coefficiens of vrious undireced nd eighed socil neorks esimed by he ne globl mesures using he hree differen ypes of men (Equions, 8, nd 9) nd he represenive mesures of he hree exising groups of clusering mesure (geomeric men). Excep for he clusering coefficiens for he firs nd second neorks esimed by Engø-Monsen nd Cnrigh's eighed mesure, he clusering coefficiens esimed ih he uneighed clusering mesures re loer hn he corresponding coefficiens esimed ih he eighed clusering mesures. Inser Tble 4 pproximely here Of he mesures using geomeric men, Opshl nd Pnzrs (009) nd Engø-Monsen nd Cnrigh (011) give he lrges nd smlles clusering coefficiens, respecively, hile he corresponding coefficiens esimed ih he ne mesure re in beeen. 5. Clusering coefficien disribuions of Bernoulli rndom neorks In his secion, e ouline he prmeers of he proposed mesure in Bernoulli rndom grph. Figure 4 conins four clusering coefficien disribuions of 1000 Bernoulli rndom neorks ih he sme size of 00 cors nd rndom ie probbiliy of All ies re ssigned rndom eighs h vry uniformly from 0 o 1. The clusering coefficiens of hese disribuions re esimed ih he rdiionl uneighed clusering mesure nd he ne eighed clusering mesure using geomeric, rihmeic nd qudric mens. The neork size nd/or ie probbiliy increse o (00, 0.0), (500,0.10) nd (500, 0.0) in Figures 4b, 4c nd 4d, respecively. Inser Figure 4 pproximely here In Figure 4, he disribuion of he uneighed clusering coefficiens hs is men equl o he ie probbiliy (0.1000), s expeced. In Figures 4b-d, he disribuion of he eighed clusering coefficiens esimed ih he ne mesure using rihmeic men is overlpping ih h of he uneighed clusering coefficiens. Their men vlue is , , in Figures 1b, c, d respecively. The disribuions bsed on he geomeric men nd he qudric men shif o he lef nd he righ-hnd sides of he o former. The men vlue of he geomeric men-bsed 10

12 disribuion is 0.195, , nd in Figures 1b, c, d respecively, heres he corresponding one of he qudric men-bsed disribuion is 0.00, , nd These orderings nd effecs remin nd he disribuions become more disinc ih incresed neork size nd/or ie probbiliy, hich reduce he disribuions sndrd deviions. Moreover, he Figures 4,b,c,d sho h he mesure using he geomeric men gives he smlles clusering coefficiens heres he mesure using he qudric men give he lrges ones. In oher ords, hey respecively ccenue he roles of ek ies nd srong ies in he esimion of clusering coefficiens. The simulion resuls re consisen ih he resuls of he pplicion of he mesures o empiricl neorks s presened in Tble 4. We hve lso done series expnsion of he clusering mesures using Wolfrm Alph 6, hich re repored in Appendix D. The resuls he sme consn nd liner erm bu difference in he qudric erms re consisen ih he resuls of he empiricl es nd he Bernoulli simulion. 6. Discussion Clusering is mjor propery of socil neorks, hich is used s indicor of neork benefis nd consrins (Brss e l., 004; Bur, 199, 005; Colemn, 1988). We rgue h he exising clusering mesures fil o reflec he imporn, bu heoreiclly disinc, roles h he hird ie plys in ego neorks. In order o explicily incorpore specific hird-ie roles in riples, e define riple clusering s funcion of riple connecedness, he verge srengh of he ies o he lers (T1/) nd he relive srengh of he ie connecing he lers (T) o T1/. Bsed on our proposed definiion, e developed ne mehod for esiming he relive clusering vlue of riples, s ell s ne locl nd globl clusering mesures for eighed, undireced neorks. We shoed ho geomeric, rihmeic, nd qudric mens cn be used o reflec differen heoreicl ssumpions bou mrginl re of subsiuion mong ie srenghs. For exmple, geomeric men my be pproprie hen ssessing opporuniies nd srucurl consrins (Bur, 199, Grnoveer, 197) heres qudric men my be pproprie hen ssessing embeddedness (Colemn, 1988; Grnoveer, 1985; Uzzi, 1997). We provided nlysis, empiricl ess, nd oulined he prmeers of he proposed mesures in Bernoulli rndom grph o sho hen nd ho mesures in heir chrcerizion of neorks. Our exminion of he proposed mesures found significn nd desired differences relive o exising mesures. We clim hree conribuions o socil neork sudies. Firs, by explicily considering he relive srengh of he hird ie in riple, he ne mesures beer cpure is heoreicl effecs h is, 6 Wolfrm Alph: hp://.olfrmlph.com/ 11

13 he hird ie ffecs he focl cor by deermining he relive posiion of he focl cor in relion o he o prners in he riple (Bur, 199; Colemn, 1988; Grnoveer, 1978). Second, e hve shon ho he use of differen ypes of men in he esimion of riple clusering coefficiens cn be used o ddress differen socil neork perspecives. For insnce, n informion perspecive emphsizes he vlue of ek ies (Bur, 199; Grnoveer, 1978; Hnsen, 1999) hile relionl perspecive emphsizes he vlue of srong ies (Colemn, 1988; Uzzi, 1997; Xio nd Tsui, 007). Third, by nlysis, empiricl ess, nd simulion e found suppor for he ne mesures relibly nd vlidly discrimining he desired properies he neork level, hus providing socil neork reserchers ih ne insrumens h cn be ilored o reflec he heoreicl perspecives of heir sudies. Implicions Our mesures nd findings llo incorporion of imporn heoreicl ssumpions bou he role of he hird ie in fuure neork sudies. Furhermore hey imply h reserchers should py greer enion o heoreicl ssumpions bou he effecs of ie srengh vrince ihin rids s ell s he effec of such vrince on ll cors in he rid, even hen he ego is focused. A riple is he pproprie uni of nlysis for sudying clusering in neork. Sudying ho riple clusering ffecs he focl cors sregic neorking behviour (e.g., creing, minining, desroying ies or riples) could improve our undersnding of he evoluion of riples, rids nd socil neorks. Fuure reserch should lso ssess he empiricl implicions of he ne clusering mesures, nd empiriclly evlue nd esblish guidelines for he usge of hese mesures on neorks reflecing differen heoreicl properies. The proposed mehod for esiming riple clusering vlues nd clusering mesures is limied o eighed, undireced neorks. Ties in socil neorks re no only eighed, bu lso ofen direced. The purpose of developing clusering mesures is o beer cpure ll he dimensions of socil neorks (see Tble 1). We leve he chllenging, bu imporn, exension of he proposed mesures nd mehods o eighed, direced neorks for furher reserch. Conclusions There is incresed enion o lrge neorks cross ide vriey of res in boh cdemi nd prcice. Socil neork nlysis is vibrn field ih on-going imporn heoreicl developmens nd empiricl esing. As ih ny field, he developmen of insrumens is condiion for he dvncemen of knoledge. The recen upsurge in he vilbiliy of d nd compuing resources necessies furher developmen of ools nd insrumens o ke dvnge of he exiing opporuniies presened. We believe he mesures proposed nd esed in his pper is 1

14 sep long he y ord boh beer insrumens nd greer flexibiliy in iloring he insrumens o heoreicl lenses. 1

15 References: Brr, A., Brhelemy, M., Psor-Sorrs, R., Vespignni, A., 004. The rchiecure of complex eighed neorks. Proceedings of he Nionl Acdemy of Sciences of he Unied Ses of Americ 101, Bocclei, S., Lor, V., Moreno, Y., Chvez, M., Hng, D.-U., 006. Complex neorks: Srucure nd dynmics. Physics Repors 44, Brss, D.J., Glskieicz, J., Greve, H.R., Tsi, W., 004. Tking Sock of Neorks nd Orgnizions: A Mulilevel Perspecive. The Acdemy of Mngemen Journl 47, Bur, R.S., 199. Srucurl Holes. Cmbridge: MA: Hrvrd Universiy Press. Bur, R.S., 001. Srucrurl Holes versus Neork Closure s Socil Cpil, in: Lin, N., Cook, K., Bur, R.S. (Eds.), Socil cpil: heory nd reserch. Aldine Trnscion, Ne Brunsick, N.J., pp Bur, R.S., 005. Brokerge nd closure: n inroducion o socil cpil. Oxford Universiy Press, Oxford. Colemn, J.S., Socil cpil in he creion of humn cpil. Americn Journl of Sociology 94, S95-S10. Cook, K.S., Emerson, R.M., Poer, Equiy nd Commimen in Exchnge Neorks. Americn Sociologicl Revie 4, Emerson, R.M., 196. Poer-dependence relions. Americn Sociologicl Revie 7, Engø-Monsen, K., Cnrigh, G., 011. Weighed Clusering Coefficiens, Telenor Repor R5. Telenor, Oslo. Fleming, L., Mingo, S., Chen, D., 007. Collborive Brokerge, Generive Creiviy, nd Creive Success. Adminisrive Science Qurerly 5, Grnoveer, M., Threshold Models of Collecive Behvior. The Americn Journl of Sociology 8, Grnoveer, M., Economic Acion nd Socil Srucure: The Problem of Embeddedness. The Americn Journl of Sociology 91, Grnoveer, M.S., 197. The srengh of ek ies. Americn Journl of Sociology 78, Grindrod, P., 00. Rnge-dependen rndom grphs nd heir pplicion o modeling lrge smllorld Proeome dses. Physicl Revie E 66, Hnsen, M.T., The Serch-Trnsfer Problem: The Role of Wek Ties in Shring Knoledge cross Orgnizion Subunis. Adminisrive Science Qurerly 44, Holme, P., Min Prk, S., Kim, B.J., Edling, C.R., 007. Koren universiy life in neork perspecive: Dynmics of lrge ffiliion neork. Physic A: Sisicl nd Theoreicl Physics 7, Kln, G., Highm, D.J., 007. A clusering coefficien for eighed neorks, ih pplicion o gene expression d. Ai Communicions 0, Lor, V., Mrchiori, M., 00. Economic smll-orld behvior in eighed neorks. The Europen Physicl Journl B - Condensed Mer nd Complex Sysems, Luce, R., Perry, A., A mehod of mrix nlysis of group srucure. Psychomerik 14, Nemn, M.E.J., 001. Scienific collborion neorks. II. Shores phs, eighed neorks, nd cenrliy. Physicl Revie E 64, Nemn, M.E.J., 00. The srucure nd funcion of complex neorks. Sim Revie 45, Onnel, J.-P., Srmki, J., Keresz, J., Kski, K., 005. Inensiy nd coherence of moifs in eighed complex neorks. Physicl Revie E 71, Opshl, T., 009. Weighed locl clusering coefficien, Neork Science Blog. Opshl, T., Pnzrs, P., 009. Clusering in eighed neorks. Socil Neorks 1, Schilling, M.A., Phelps, C.C., 007. Inerfirm Collborion Neorks: The Impc of Lrge-Scle Neork Srucure on Firm Innovion. Mngemen Science 5,

16 Snijders, T.A.B., vn de Bun, G.G., Seglich, C.E.G., 010. Inroducion o sochsic cor-bsed models for neork dynmics. Socil Neorks, Uzzi, B., The Sources nd Consequences of Embeddedness for he Economic Performnce of Orgnizions: The Neork Effec. Americn Sociologicl Revie 61, Uzzi, B., Socil srucure nd compeiion in inerfirm neorks: The prdox of embeddedness. Adminisrive Science Qurerly 4, Uzzi, B., Amrl, L.A.N., Reed-Tsochs, F., 007. Smll-orld neorks nd mngemen science reserch: revie. Eur Mnge Rev 4, Uzzi, B., Ryon, L., 00. Relionl Embeddedness nd Lerning: The Cse of Bnk Lon Mngers nd Their Cliens. Mngemen Science 49, Uzzi, B., Spiro, J., 005. Collborion nd creiviy: The smll orld problem. Americn Journl of Sociology 111, Wssermn, S., Fus, K., Socil neork nlysis: mehods nd pplicions. Cmbridge Universiy Press, Cmbridge. Ws, D.J., Srogz, S.H., Collecive dynmics of 'smll-orld' neorks. Nure 9, Xio, Z., Tsui, A.S., 007. When Brokers My No Work: The Culurl Coningency of Socil Cpil in Chinese High-ech Firms. Adminisrive Science Qurerly 5, 1-1. Zhng, B., Horvh, S., 005. A Generl Frmeork for Weighed Gene Co-Expression Neork Anlysis. Sisicl Applicions in Geneics nd Moleculr Biology 4, Aricle

17 Appendix Appendix A: The developmen of he mehod for esiming riple clusering vlues bsed on rihmeic nd qudric mens Arihmeic men: 1,,, Vi,,, = (4) Qudric men: 1,,, Vi, =,, (5) Here,, 1 or is equl o 1 if he ies T1, T, nd T re presen, nd is equl o 0 oherise. The role of he fcor 1, is o mesure riple inerconnecedness. I is equl o 1 if Triple is, 1,, 1, closed nd 0 if i is open. The fcor in Equion (4) or in Equion (5), 1,, cpures he role of he srengh of T1/, heres he fcor in Equion (4) or, 1,, in Equion (5) llos reling he srengh of T o he srengh of T1/. Inuiively, i ould cpure h, ll else equl, ekening of T disproporionlly decreses he embeddedness of he focl cor. Appendix B: The formuls of ne locl clusering mesures using rihmeic nd qudric mens. Arihmeic men: 1,,,,, = c (6) i c 1,,, o, Qudric men: 1,,,,, c = i (7) c 1,,, o, Where, c i denoes he clusering coefficiens of he ego neork of cor i. In he denominor, c nd o snds for closed nd open riples respecively Appendix C: The formuls of ne globl clusering mesures using rihmeic nd qudric mens. Arihmeic men: 16

18 17 = o c C 1 1,,,,,,,,, Qudric men: = o c C 1 1,,,,,,,,, Where, C denoes he clusering coefficien of socil neork. The only difference beeen his globl clusering mesure (Equions 8-9) nd is locl version (Equions 6-7) sems from hich riples re ken ino ccoun. The locl version kes ino ccoun ll riples in he ego neork, hile he globl version kes ino ccoun ll riples in he neork. Appendix D: Series expnsions of he ne clusering mesure Geomeric men: Qudric men: Arihmeic men: (8) (9)

19 Tbles Tble 1: Clusering mesures Binry Weighed Locl Undireced Ws nd Srogz (1998) Grindrod (00) Lor nd Mrchiori (00) Brr e l. (004) Zhng nd Horvh (005) Onnel, Srmki, Keresz, nd Kski (005) Holme, Min Prk, Kim nd Edling (007) Kln nd Highm (007), Opshl (009b) Direced Globl Undireced Luce nd Perry (1949) Opshl nd Pnzrs (009) Engø-Monsen nd Cnrigh (011) Direced Hollnd nd Leinhrd (1970, 1971) Opshl nd Pnzrs (009) 18

20 Tble : Triple clusering definiions nd esimion mehods nd srenghs of clusering mesures Group 1 Group Group Group 4 Group-represenive mesures Ne Mesure* Globl Clusering Mesures (Geomeric Men) Clssic uneighed mesure C = i,,, C = Engø-Monsen nd Cnrigh (011) s eighed clusering mesure,,, C = Opshl nd Pnzrs (009) s eighed clusering mesure,,,,, C = c,,,,,, o, Triple clusering vlue esimion mehod Triple clusering definiion Srenghs V i, =,, i, i,, Triple clusering is funcion of: - Triple connecedness V = V i, =,,, V = Triple clusering is funcion of: - Triple connecedness - The men of he srengh of T1, T nd T Triple clusering is funcion of: - Triple connecedness - The men of he srenghs of T1 nd T, denoed he srengh of T1/ in he ex. i,,,, Triple clusering is funcion of: - Triple connecedness - The men of he srenghs of T1 nd T, denoed he srengh of T1/ in he ex. - The relive srengh of T compred o he srengh of T1/ - The srenghs of Group - Cpuring he heoreic role of he relive srengh of T o he srengh of T1/. - Abiliy o - The srengh of Group 1 - The srenghs of Group differenie closed - Abiliy o differenie riples - Abiliy o differenie riples in riples from open consiued by differen rids rid here he focl cors re riples ih differen clusering differen in he exen o hich degrees. hey re embedded in he rid. Noes: - i is he focl cor. j nd h re he firs nd second prners of he focl cor i. denoes Triple in he neork. -, 1,,, or, denoes he presence of he ie connecing cors i nd j, connecing cor i nd h, or connecing j nd h, respecively. I is equl o 1 if he ie is presen nd equl o 0 if i is bsen.,,, or, denoe he eighs of ies connecing cors i nd j, connecing cor i nd h, or connecing j nd h, respecively. I is greer hn 0 if he ie is presen nd equl o 0 if i is bsen. - Clusering coefficiens rnge from 0 o 1. - * In he denominor, c sndrds for closed riples hile o snds for open riples 19

21 Tble : Clusering coefficiens esimed by differen clusering mesures for he hree exmple neorks in Figure : Types of men Arihmeic Men Geomeric Men Mesures Neorks nd Vrince* A B C Engø-Monsen nd Cnrigh (011) Opshl nd Pnzrs (009) Ne Mesure Engø-Monsen nd Cnrigh (011) Opshl nd Pnzrs (009) Ne Mesure Engø-Monsen nd Cnrigh (011) Qudric Men Opshl nd Pnzrs (009) Ne Mesure Noes: * Vrince denoes he ie srengh vrince ihin he neork. - In Tble 1, he denominor of he ne clusering mesure is differen from h of he exising mesures. When he ne mesure hs he sme denominor s he exising ones, he difference beeen he coefficiens esimed by he ne mesure nd hose esimed by he exising mesures is he sme in pern. 0

22 Tble 4: Clusering coefficiens of vrious undireced socil neorks Neorks Uneighed mesure Exising mesures Engø-Monsen nd Opshl nd Cnrigh (011)'s Pnzrs (009)'s eighed mesure eighed mesure Ne Clusering Mesure Geomeric men Arihmeic men Qudric men Coefficien Coefficien % Coefficien % Coefficien % Coefficien % Coefficien % 1 Freemn EIES (ime 1) Freemn EIES (ime ) Freemn EIES (Messges) Online communiy Consuling (dvice) Consuling (vlue) Reserch em (dvice) Reserch em (reness) Scienific collborion (pper) Averge Noes: - % is he difference beeen he corresponding eighed clusering coefficien nd he clssic uneighed clusering coefficien (Column ), mesured in percenge. - Neorks 1, nd re Freemn EIES cquinnce neorks h include cors. Neork 4 is he fcebook-like socil neork used in Opshl (009). Neork 5, 6, 7 nd 8 re inr-orgnizionl neorks used in Cross nd Prker (1981). Neork 9 is he co-uhorship neork (pper) used in Nemn (001). The d of Neorks 1,, re obined from Wssermn nd Fus (1994) heres he d of he oher neorks is obined from Opshl (009b). The deiled descripions of hese neorks cn be seen in Opshl (009b) or Opshl nd Pnzrs (009). Neork 9 is originlly o-mode neork nd rnsformed ino one-mode eighed neork by couning he number of common nodes. The remining neorks re originlly direced neorks nd rnsformed ino he corresponding undireced ones by king he rihmeic men of he eighs of relionship s o direced ies s he eigh of he relion or he undreced ie. The clusering coefficiens esimed ih he uneighed mesure nd Opshl nd Pnzrs (009) s eighed mesure re clculed ih Tne sofre developed by Opshl (009) heres he coefficiens esimed ih he remining mesures re clculed ih sofre developed by he second uhor. The coefficiens in he columns,, 4 re clculed ih he geomeric men mehod. - The ne clusering mesure hs he denominor differen from h of he exising mesures. When he ne mesure hs he sme denominor s he exising ones, he differences beeen he coefficiens esimed by he ne mesure nd hose esimed by he oher mesures remin he sme in pern. 1