Normalized closeness centrality of urban networks: impact of the location of the catchment area and evaluation based on an idealized network

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RESEARCH Chenand Die ich Applied Ne wo k Science (2023) 8:60 h ps://doi.o g/10.1007/s41109-023-00585-0 Applied Ne wo k Science No malized closeness cen ali y o u ban ne wo ks: impac o  heloca ion o  heca chmen a ea ande alua ion based onanidealized ne wo k Hsiao‑Hui Chen1* and Udo Die ich2 Abs ac The decision o whe e o loca e he ca chmen a ea o an u ban ne wo k exe s sig‑ ni ican in luence on he indica o alues and in his esea ch his in luence is e e ed o as he placemen e ec . Placemen e ec has signi ican impac on he s ud‑ ies a he neighbo hood scale ocusing on he s uc u al p ope ies o he ne wo k models, he ne wo k analysis esul s and cen ali y measu es, he in e ed mo emen pa e ns and he accessibili y o des ina ion. Placemen e ec becomes e en mo e sig‑ ni ican when mul iple ca chmen a eas a e sampled o be compa ed o classi ied. This esea ch examines placemen e ec on one o he mos a ec ed indica o s, closeness cen ali y, and p oposes using an idealized ne wo k as a e e ence o be compa ed wi h he eal ne wo k in o de o ind a solu ion o mi iga e he placemen e ec s. By compa ing he no malized closeness cen ali y in he eal ne wo k wi h ha in he ide‑ alized ne wo k, we can (1) e alua e he placemen e ec on he closeness cen ali y and (2) ind he h eshold dis ance in o de o mi iga e he placemen e ec . The esul s show ha he closeness cen ali y o he same node a ies ema kably depend‑ ing on i s posi ion and how cen al i is in he chosen ca chmen a ea. Speci ically, in he selec ed a eas in his esea ch, i he cen e poin o a ca chmen a ea is mo ed by mo e han 100 m away om he o iginal cen e poin , he closeness cen al‑ i y o he same node s a s o be signi ican ly in luenced by he placemen e ec . The h eshold dis ance o 100 m o e s a ecommenda ion ha a di ec compa ison o he closeness cen ali y be ween di e en nodes in he same ca chmen a ea should be d awn only i hese nodes a e less han 100 m away om each o he . In o he wo ds, when compa ing wo nodes loca ed u he han he h eshold dis ance om each o he , i is ad isable o c ea e wo sepa a e ca chmen a eas, whe e hese nodes se e as he cen e poin s. I should be no ed ha he h eshold dis ance o 100 m de i ed speci ically om he cu en esea ch should no be gene alized o o he cases. The h eshold dis ance o di e en case s udies emains open o u he in es iga ion in he u u e as i may a y among ci ies o a eas. Keywo ds: S ee ne wo k, Placemen e ec , Bounda y e ec , Edge e ec , No malized closeness cen ali y *Co espondence: hsiao‑hui.chen@ u‑ b aunschweig.de 1 Technische Uni e si ä B aunschweig, SpACE Lab a ISU – Ins i u e o Sus ainable U banism, Pockelss . 3, 38106 B unswick, Ge many 2 Ha enCi y Uni e si ä Hambu g, A chi ek u Und REAP, Henning‑Vosche au‑Pla z 1, Raum 4.107, 20457 Hambu g, Ge many Page 2 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 In oduc ion U ban ne wo k sys em is a complex spa ial sys em whose membe s connec and in e - ac wi h each o he . Fo any eal-wo ld spa ial ne wo k analysis, i is essen ial o de ine an a i icial bo de o he ne wo k model (Pa k 2009). Spa ially con ined ne wo ks o , in o he wo ds, local sub-ne wo ks o he en i e global ne wo k sys em ha e been e med ca chmen a eas (Chen and Die ich 2021), con ex ual a eas o bounded sys ems (Pa k 2009), subne wo ks o egional ne wo ks (Rheinwal e al. 2012). D awing an a bi a y bounda y ine i ably cu s he links connec ing he ca chmen a ea unde in es iga ion wi h he es o he ne wo k ou side he bounda y (Rheinwal e al. 2012). Howe e , he e en s, s uc u es, beha io and dynamics o he en i e global ne wo k sys em s ill a ec s he local sub-ne wo k inside he ca chmen a ea (G eenbe g e al. 2020). The ine i able a bi a y delinea ion o he bounda y may induce dis o ion o he esul s o he measu es, which can subsequen ly induce bias ha a ec s he in e - ences based on hese measu es (Paul 2014). Such dis o ion o he esul s is ound o be mo e p onounced when he nodes o links a e close o he bo de o he ca chmen a ea (Okabe and Sugiha a 2012). These bounda y de e mina ion p oblems, which ha e been e med he edge e ec (C uci i e al. 2006; Gil 2017; Ripley 2004) o he bounda y e ec (Pa k 2009; Okabe and Sugiha a 2012), ha e signi ican impac on he s udies ocusing on he s uc u al p op- e ies o he ne wo k models, he ne wo k analysis esul s and cen ali y measu es, he in e ed mo emen pa e ns and he accessibili y o des ina ion. Fi s o all, he choice o he bounda ies decides he in e nal s uc u e o he local ne wo k model ha is spa ially con ined wi hin he ca chmen a ea. This decision di ec ly in luences he membe s o he sub-s uc u e and opology included in he ne wo k model and, he e o e, a ec s ou unde s anding o he spa ial s uc u e and unc ional p ope ies o he ne wo k sys em (Laumann e al. 1989). Secondly, he delinea ion o he model bounda y can cause a ce ain bias in ne wo k analysis esul s (Ra i 2004; Jou siniemi 2010) because he analy ic algo i hms o ne - wo k analysis a e ela ional (Okabe and Sugiha a 2012) and“ne wo k da a by de ini ion includes dependencies among obse a ions” (Laumann e al. 1983). Simila ly, syn ac- ic alues a e meaning ul only wi h e e ence o a sys em bounda y ha a esea che chooses o his o he analysis (Pa k 2009). Excluding any elemen s o membe s o he en i e global ne wo k sys em will a ec he cha ac e is ics, pe o mance and beha io o he measu emen esul o he local ne wo k models. In pa icula , pa h-based measu es, such as closeness cen ali y and be weenness cen ali y, a e e y sensi i e o he bound- a y e ec . Dis o ion o he esul s can be induced when links a e cu o by an a bi a y bounda y and, he e o e, a e no included in he calcula ion. The nodes and links close o he bo de a e less cen al and pe iphe al only because o he p esence o he bound- a y. Usually, he bounda y e ec on he pa h-based measu es is p e alen in all nodes and links in he en i e ne wo k model (Rheinwal e al. 2012) and is pa icula ly p o- nounced o hose a he bo de o he ca chmen a ea. Nodes o links nea he cen e o he ca chmen a ea end o ha e highe closeness cen ali y (Gil 2017) and be weenness cen ali y (Chen and Die ich 2021) compa ed wi h hose close o he bo de . Thi dly, bounda y e ec s can also induce a bias on he in e ence based on such dis o ed measu e esul s. Fo example, K a a (1994) has ca ied ou es s o he Page 3 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 co ela ion be ween di e en de ini ions o bounda ies and pedes ian mo emen . Pa k (2009) has es ed he p edic abili y o human mo emen pa e ns unde a ious bound- a y condi ions and ound ha his p edic abili y eaches i s maximum a a ce ain adius om he bounda y, which is also an indica ion o he p esence o he size-independen bounda y e ec s on he in e nal s uc u e. Finally, bounda y de ini ion also a ec s accessibili y analysis. P e ious s udies (Sha - key and Ho el 2008) ela ed o public heal h issues and o nu i ion and ood accessibil- i y in u al a eas ha e been c i icized o no conside ing esou ces ou side o he s udy a ea, e en hough esou ces ac oss he bounda y may also a ec beha io wi hin he a ea unde in es iga ion (Sadle e al. 2011; an Me e e al. 2010). In esponse o his me hodological de iciency, an Me e e al. (2010) and Sadle e al. (2011) ha e in es- iga ed he bounda y e ec on eaching he e ail shops om he loca ions wi hin he s udy a ea, which is ypically wi hin an a bi a y adminis a i e bounda y. The esul s show ha he bounda y e ec has led o conside able bias in mis-iden i ica ion o ood dese communi ies a he bo de o he s udy a ea, e en i he e is a sou ce o ood igh ac oss he bo de . The ac ual dis ance o a eling necessa y o buying ood has also been o e - epo ed. In o de o imp o e he eliabili y and he consis ency o he ne wo k analysis esul s ac oss loca ions, a numbe o p ocedu es and p ac ices ha e been p oposed o mi iga e he bounda y e ec . The i s app oach, he ca chmen o ca chmen (Hillie e al. 1993), adds an addi ional bounda y o c ea e a bu e a ea ou side he ac ual es a ea. The size o he bounda y o he bu e a ea is la ge han ha o he ca chmen a ea. Ne wo k analysis is hen ca ied ou o bo h he ca chmen and he bu e a ea. Howe e , he esul s o he ne wo k measu e o he bu e a ea a e no included in he analysis because hey a e dis o ed by he bounda y e ec (Gil 2017; Penn e al. 1998). Secondly, ins ead o one ixed bounda y de ini ion, he o he mi iga ion me hod, he adius- adius analy- sis (Hillie 1996) o local adius analysis (Gil 2017), applies a ious bounda y condi ions by c ea ing mul iple ci cles a ound he cen e o he ca chmen a ea unde in es iga ion. Al hough he ca chmen o ca chmen me hod and he adius- adius analysis ha e p o ed o be success ul in mi iga ing he bounda y e ec in many empi ical s ud- ies, he op imal size o he bu e and he adius emains open o u he esea ch. Gil (2017) inds ha he esul s o he ne wo k cen ali y analysis a e e y uns able in small s udy a eas (e.g. on a neighbou hood scale) and sugges s ha he s udy a ea should be embedded in a la ge con ex . Howe e , he e emains he ques ion o how la ge is a la ge enough con ex . In o he wo ds, he p oblem o delinea ing he bound- a y becomes he p oblem o deciding he adius o he size o he s udy a ea (Jou - siniemi 2005). In esponse o his open ques ion, Chen and Die ich (2021) conduc ed a se ies o expe imen s o he size- ela ed bounda y e ec s, i.e. he size e ec , on he indica o alues. Based on hese expe imen s, hey ha e sugges ed ha , i s o all, he a e age s ee leng h can be one o he indica o s o de e mining he size o he ca ch- men a ea and, secondly, “ he size e ec on he indica o is no e y signi ican when he size o he ca chmen a ea is la ge han 4000 × 4000 m2. The e o e, any size la ge han 4000 × 4000 m2 would no be necessa y” (Chen and Die ich 2021). Ano he mi iga ing me hod, namely he mo ing bounda y app oach, consis s in shi - ing he cen e o he ci cula bounda ies wi h ixed size and shape o calcula e ne wo k Page 4 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 measu es (Penn e al. 1998; Hillie and Penn 2004; Tu ne 2007; Gil 2017). Howe e , by keeping he same shape and size o he mo ing ne wo k bounda ies, Gil (2017) shows ha indica o s like closeness cen ali y s ill a y in di e en s udy a eas and a e pa - icula ly sensi i e o he shi o ne wo k cen e s. Since he bounda y shape and size a e iden ical, he a ia ion o he indica o alues can only be explained by he loca ion o he ne wo k. In o he wo ds, al hough he size- ela ed bounda y e ec can be elimina ed by adop ing he mo ing bounda y app oach, he placemen - ela ed bounda y p oblem, which is e med placemen e ec in his esea ch, s ill exis . To be mo e speci ic, place- men e ec e e s o he phenomenon ha he a ia ion o he indica o alues depends on whe e he ca chmen a ea is e ie ed. F om his pe spec i e, a e y impo an ques- ion is whe e he cen e o he ca chmen a ea should be. The cu en esea ch in ends o u he in es iga e his placemen e ec in o de o p o ide mo e e ined guidelines o deciding he loca ion o he ne wo k model. P esen ing heplacemen e ec oncloseness cen ali y The esul s o Gil’s expe imen s (2017) show ha one o he indica o s ha could be mos in luenced by he placemen e ec is closeness cen ali y (Cc), which is he ecip o- cal o he sum o he sho es dis ance be ween he chosen node- and all o he nodes in he ca chmen a ea. The Cc o node- can be o mally exp essed as he ollowing whe e Cc( ) e e s o closeness cen ali y o he chosen node- , S( , u) e e s o he leng h o he sho es dis ance be ween he chosen node- and o he nodes, u, and N e e s o he o al numbe o nodes in he chosen ca chmen a ea. Closeness cen ali y measu es how as a node exe s in luence on all o he nodes. Fo example, i he a ge is o sp ead he in o ma ion in he ne wo k, a node wi h la gecloseness cen ali y means ha i is in a posi ion o sp ead in o ma ion quickly. Nodes wi h highe alue o closeness cen al- i y can beimpo an in luence s in he ne wo k. Closeness cen ali y is no an absolu e alue as i may change depending on he loca ion o he selec ed ca chmen a ea in he en i e global ne wo k. A node in he cen e o he ca chmen a ea has he ad an age o ha ing mo e in luence on o he nodes and has highe alue o closeness cen ali y han a node loca ed on he bo de s o he ne wo k (Gil 2017). The e o e, he closeness cen al- i y o a chosen node migh no necessa ily be small in he en i e ci y s ee ne wo k, bu i may be small in he selec ed ca chmen a ea only because i is no close o he cen e o he ca chmen a ea. In o de o demons a e he placemen e ec , he Plaza Luce os in Alican e has been selec ed as he cen e poin o he s udy a ea. The size o he ca chmen a ea is 3000 × 3000 m2 and he uni o closeness cen ali y is 1/km. We ha e chosen he a ea size ha is la ge han he accep able walking dis ance o he pedes ian because his allows mo e space o mo e he chosen node u he away om he cen e in o de o in es iga e he placemen e ec . One o ou a ge s is o os e pedes ians in ci ies and o help o de elop walkabili y, isibili y and accessibili y o poin s o in e es . The e o e, we would like o in es iga e a ne wo k o pedes ian and Open S ee Ne wo k (OSM) (1) C c( )=1/ N u=1 S( ,u ) Page 5 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 can be a sou ce o da a o ex ac exis ing pedes ian ne wo ks in ci ies. The ollowing highway ags o he OSM a e selec ed o o m he ne wo k wi hin he ca chmen a ea: p ima y, seconda y, e ia y, esiden ial, pedes ian, s eps, pa h and unclassi ied. Eigh ca chmen a eas wi h di e en cen e s we e selec ed and p esen ed in Table1. The cen e o each ca chmen a ea is indica ed by a blue cen e poin . The placemen e ec on he chosen node, which is indica ed by he ed node in each ca chmen a ea,1 will be in es iga ed by examining he changes in he indica o alues o he eigh selec ed s udy a eas in Table1. These eigh ca chmen a eas di e in he dis ance be ween he blue cen e poin and he ed chosen node. In ca chmen a ea 1, he blue cen e poin and he ed chosen node a e o e lapping wi h each o he . The cen e s o he ca chmen a eas o se om 50m in ca chmen a ea 2 o 1500m in ca chmen a ea 8.Tha is, s a ing om ca chmen a ea 2 o ca chmen a ea 8, he blue cen e poin g adually mo es 50m, 100m, 200m, 300m, 500m, 1000m and 1500m o he no h om he ed chosen node. The esul s in Table 1 show ha he closeness cen ali y o he ed chosen node changes om 0.000678 1/km in ca chmen a ea 1 o 0.000438 1/km in ca chmen a ea 8. This change shows ha he closeness cen ali y o he ed chosen node is a ec ed by i s dis ance o he blue cen e poin in all eigh ca chmen a eas. Hence, he e is a place- men e ec on he alue o he closeness cen ali y o he ed chosen node. No maliza ion o  hecloseness cen ali y in he eal ne wo k The p ocesses o no maliza ion has been p oposed o connec he numbe o nodes wi h he indica o (Masucci and Moline o 2016). The cu en esea ch also applies he no - maliza ion p ocedu e and c ea es an indica o , no malized closeness cen ali y, CN, so ha he di e en numbe o nodes in di e en ca chmen a eas is balanced h ough he no maliza ion and he ac ha he numbe o nodes changes wi h he loca ions o he ca chmen a ea is now aken in o he conside a ion. In his sec ion, he esul s in Table1 a e used o explain he no maliza ion o closeness cen ali y de i ed om wo di e en indica o s: (1) he closeness cen ali y and (2) he sho es dis ance be ween he chosen node and all o he nodes. A common way o de e mining he no malized closeness cen ali y is mul iplying he closeness cen ali y o he chosen node wi h he numbe o nodes in he ca chmen a ea. The no malized closeness cen ali y can be o mally exp essed as he ollowing. whe e CN( ) e e s o he no malized closeness cen ali y o node- , N e e s o o al numbe o nodes in he ca chmen a ea, Cc( ) e e s o he closeness cen ali y o he chosen node- . (N-1) e e s he numbe o connec ions be ween a chosen node o all o he nodes because he chosen node has no (o ze o) connec ion wi h i sel . In he case o a small u ban a ea, in which N is no e y la ge, (N-1)is used. In he case whe e N is e y la ge, he ‘1’ can be d opped om (N-1). (2) C N ( ) = (N − 1) × C c ( ) 1 This ed node is also he node ha is closes o Plaza Luce os. Page 6 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 Table 1 Ca chmen a eas wi h di e en cen e (blue) nodes and indica o alues o ( ed) node unde in es iga ion in he selec ed ca chmen a eas Dis ance be ween ed chosen node and blue cen e node m Closeness cen ali y o he ed chosen node A e age dis ance om all nodes o chosen node To al numbe o nodes No malized closeness cen ali y C( ) N n=1 S( ,n)/(N−1 ) NCN( ) Uni 1/km m 1/km Ca chmen a ea 1 0 0.000678 917 1606 1.089 Ca chmen a ea 2 50 0.000674 906 1637 1.103 Ca chmen a ea 3 100 0.000668 899 1665 1.112 Ca chmen a ea 4 200 0.000649 891 1728 1.12 Ca chmen a ea 5 300 0.000635 874 1802 1.145 Ca chmen a ea 6 500 0.000611 854 1916 1.170 Page 7 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 To illus a e his, we can use ca chmen a ea 1 in Table1 as an example. The o al numbe o nodes, N, is 1606 nodes and he closeness cen ali y o he ed chosen node- , C( ), is 0.000678 1/km. Based on he o mula (2), he no malized closeness cen ali y o he ed chosen node- , CN( ), accoun s o (1606−1) × 0.006784 = 1.089 1/km. Idealized ne wo k Knowing he alue o he no malized closeness cen ali y o he chosen node- , CN( ), in he eal ne wo k, we need o ha e a e e ence o di e en ne wo ks o be compa ed wi h in o de o e alua e he alue o CN( ). In his esea ch we p opose o use he ideal- ized ne wo k o be he e e ence. Fo ou pu poses, he idealized ne wo k is de ined o a The leng h o any link be ween wo nodes can be calcula ed wi h a simple Py hago as’ Theo em Table 1 (con inued) Dis ance be ween ed chosen node and blue cen e node m Closeness cen ali y o he ed chosen node A e age dis ance om all nodes o chosen node To al numbe o nodes No malized closeness cen ali y C( ) N n=1 S ( ,n ) /(N−1 ) NCN( ) Uni 1/km m 1/km Ca chmen a ea 7 1000 0.000528 918 2062 1.088 Ca chmen a ea 8 1500 0.000438 1190 1919 0.840 Side leng h o he ca chmen a ea equals o 3000m 1500m Cen e One o he possible sho es pa hs o he cen e Fig. 1 The idealized ne wo k Page 8 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 be ma hema ically ideal, and i s nodes a e e enly dis ibu ed in a quad a ic g id. The idealized ne wo k has he s a -like pa e n, as shown in Fig.1. In his sec ion he cen e node o he idealized ne wo k is he chosen node- , which is used o explain how he closeness cen ali y o his chosen node- changes wi h (1) di - e en numbe o nodes in he ca chmen a ea, N; and (2) he co esponding sum o he sho es dis ance be ween he chosen node- and all o he nodes, N u=1 S( ,u ) . Figu e1 p esen s he idealized ne wo k. The e is a di ec connec ion be ween he cen e node and all o he nodes. The nodes and links in such a ne wo k o m a s a -like pa e n. The sho es dis ance om any node o he cen e node is indica ed by he yel- low a ow. I should be no ed ha such a ne wo k is only a o able o one node, which is he cen e node in his case. Fo all o he nodes, such a ne wo k is no ideal because he sum o he sho es dis ances be ween all nodes and any o he non-cen e node is la ge han he sum o dis ances be ween he cen e node and all o he nodes. This ype o ne wo k is o en ound in eal u ban ne wo ks, such as Place Cha les es de Gaulle in Pa is o Connaugh Place in New Delhi. I is designed o gi e he cen al place a high and excep ional impo ance. Idealized ne wo k as he e e ence o compa ison This sec ion explains (1) he calcula ion o he no malized closeness cen ali y in he ide- alized ne wo k and (2) he compa ison o he no malized closeness cen ali y in he ide- alized and eal ne wo ks. Calcula ion o no malized closeness cen ali y o idealized ne wo k In o de o calcula e he no malized closeness cen ali y in he idealized ne wo k, he i s s ep is o acqui e he sum o he sho es dis ances om he chosen node o all o he nodes. Al hough he side leng h o he ca chmen a ea (which is indica ed by he black ec angula in Fig.2) is 3000m, because o he symme y i is su icien o ocus on jus one quad an o he ca chmen a ea, which has he side leng h o 1500m and is indi- ca ed by he ed ec angle in Fig.2. Figu e3 p esen s he ela ionship be ween he inc easing numbe o links on each side o he 1500m × 1500m quad an 2 and he a e age dis ance be ween he cen e node and all o he nodes in he idealized ne wo k. Table2 p esen s he a e age dis ance be ween he cen e node and all o he nodes in Fig.2, wi h an inc easing numbe o nodes on each side o he quad an . As he numbe o nodes on each side o he ca chmen a ea Side leng h o he ca chmen a ea equals o 3000m 1500m One quad an o ca chmen a ea Bounda y o he selec ed ca chmen a ea Fig. 2 Rela ionship be ween he ca chmen a ea and one quad an o he ca chmen a ea 2 In he idealized ne wo k, as he numbe o links on each side inc eases, he node densi y o he ne wo k also inc eases. Page 9 o 14 Chenand Die ich Applied Ne wo k Science (2023) 8:60 inc eases, he node densi y o he ne wo k also inc eases. The esul s show ha , wi h an inc easing numbe o nodes on he side o he quad an , he a e age dis ance be ween he cen e node and all o he nodes, i.e. N u=1 S( ,u)/(N−1 ) dec eases and eaches sa - u a ion a 1148m. Applying o mula (1) and (2), one can calcula e he no malized close- ness cen ali y o he cen e node in he idealized ne wo k, namely Compa ing heno malized closeness cen ali y in heidealized and he eal ne wo k Ou nex ask is o compa e he no malized closeness cen ali y in he idealized and eal ne wo k and o examine he ela ionship be ween he numbe o nodes and he close- ness cen ali y. The closeness cen ali y o he chosen node in he eigh ca chmen a eas o he eal ne wo k in Table1 is plo ed in Fig.4a. The no malized closeness cen ali y o he chosen node in he eigh ca chmen a eas o he eal ne wo k is shown by he ed (3) C N( )=(N−1)/ N  u=1 S( ,u)=1/1148m=0.8711/ km A e age dis ance be ween cen e poin o all nodes (m) Numbe o links on each side o 1500x1500m 2 quad an Fig. 3 Rela ionship be ween numbe o links on each side o quad an and a e age dis ance be ween cen e and all nodes Table 2 Changes o measu emen s in quad an wi h di e en numbe o nodes on each side o he quad an Numbe o nodes on each side o he quad an A e age dis ance be ween he cen e node and all o he nodes N n=1 S ( ,n )/( N−1 ) Sum o sho es possible dis ance be ween each node and he cen al poin a, N n=1 S( ,n ) To al numbe o nodes, N No malized closeness cen ali y, CN m M 1/km 5 1342.176 33,554 25 0.745 15 1212.577 272,830 225 0.825 25 1186.662 741,664 625 0.843 50 1167.228 2,918,069 2500 0.857 100 1157.511 11,575,105 10,000 0.864 150 1154.272 25,971,110 22,500 0.866 200 1152.652 46,106,082 40,000 0.868 300 1151.033 103,592,930 90,000 0.869 500 1149.737 287,434,240 250,000 0.870 750 1149.089 646,362,654 562,500 0.870 1000 1148.765 1,148,765,266 1,000,000 0.871