Comparability of Urban Street Networks: Consideration of the Size Effect in the Evaluation of Network Characteristics and a Proposal for Determining an Appropriate Size of the Catchment Area for Pedestrian Networks

Compa abili y o U ban S ee Ne wo ks: Conside a ion o he Size E ec in he E alua ion o Ne wo k Cha ac e is ics and a P oposal o De e mining an App op ia e Size o he Ca chmen A ea o Pedes ian Ne wo ks Hsiao-Hui Chen*, Udo Die ich Resou ce E iciency in A chi ec u e and Planning, Ha enCi y Uni e si y, Hambu g 20457, Ge many Co esponding Au ho Email: hsiao-hui.chen@hcu-hambu g.de h ps://doi.o g/10.18280/ijsdp.160603 ABSTRACT Recei ed: 23 May 2021 Accep ed: 19 Oc obe 2021 Fo he analysis o u ban ne wo ks indica o s, like a e age node deg ee, connec i i y and be weenness cen ali y, a e widely used. Thei alues a e calcula ed o a ca chmen a ea o he ne wo k. This esea ch in es iga es i and o wha ex en he size o he ca chmen a ea in luences he indica o ’s alues. A me hodology o de e mining he size o ca chmen a ea is p oposed and ecommenda ions o i s app op ia e size o pedes ian ne wo k analysis a e o e ed. Fo he ma hema ical deduc ion o he ela ionship be ween he size o he ca chmen a ea and he indica o alues an idealized egula ne wo k is used as a e e ence model. We ound ha he size e ec on indica o alues is p ominen . I dec eases exponen ially as he size o he ca chmen a ea inc eases. The size e ec is no able un il a side leng h o he ca chmen a ea ha equals 50 imes he a e age s ee leng h, while he a ia ion is s ill accep able i i is 30 o 40 imes he a e age s ee leng h. The ypical a e age s ee leng h in ci ies is abou 100 me e s, a ca chmen a ea la ge han 4000 x 4000m2 is no necessa y. This e e ence model can also be used o e alua e eal ne wo ks and o compa e hem wi h an idealized case. Keywo ds: a e age s ee leng h, ca chmen a ea, edge e ec , ne wo k analysis, node deg ee, size e ec 1. INTRODUCTION 1.1 The challenge When e alua ing he eal-wo ld spa ial ne wo ks, he cen al poin and he ca chmen a ea a ound i need o be de ined be o e ca ying ou ne wo k analysis, so ha he indica o s o di e en ne wo ks can be compa ed wi h each o he . Howe e , he eal-wo ld spa ial ne wo ks a e no s ic ly disc ee sys ems and he bounda ies o hei ca chmen a eas a e o en a bi a ily de ined. This a bi a ily de ined size o he ca chmen a ea o en exe s signi ican dis o ion on he alues o ne wo k indica o s [1-6] o he ollowing easons. I he size o he ca chmen a ea is oo la ge, i is mo e likely ha i con ains mul iple sub-ne wo ks o sub-s uc u es wi h di e en cha ac e is ics. In o he wo ds, he mixed cha ac e is ics o he en i e ne wo k a e o en a combina ion o sub-s uc u es, such as g ids, ees, hubs and spokes, o lines s uc u e. And each ype o ne wo k s uc u e may di e in complexi y [7, 8]. Since he alue o he indica o is he a e age alue o he en i e ne wo k, he complexi ies o he sub-s uc u es wi hin he ne wo k may a ec he calcula ion o he indica o o he e alua ion o he cha ac e is ics and a ibu es. On he o he hand, i he ca chmen a ea is oo small, he ne wo k model excludes he cha ac e is ics beyond he a bi a ily de ined bounda y o he model. Since he analy ic algo i hms a e ela ional, he wo ld ou side he ca chmen a ea o analysis s ill a ec s he wo ld inside i and he mo emen pa e ns wi hin [1]. Conce ns o e he size e ec on he eliabili y o signi icance o he ne wo k analysis esul s ha e been exp essed and sha ed by many esea che s [1, 9-12]. They es he size e ec on he pe o mance o spa ial ne wo k models by a ying he h eshold adius o he ca chmen a ea. Fo example, Yoshimu a e al. [13] ha e in es iga ed be weenness cen ali y wi h adius o he ca chmen a ea a ying om 300 o 5000 me e s wi h 100 me e s s ep, and he esul s show ha he indica o alue is sensi i e o he size o he ca chmen a ea. To be mo e speci ic, he indica o alue o he same node o link in he ne wo k may change wi h he size o he ca chmen a ea. And his dis o ion is pa icula ly p onounced o he nodes and links a he bo de o he ca chmen a ea [1, 14]. Fo example, Gil [1] has ound ha nodes o links nea he cen e o he ca chmen a ea end o ha e highe deg ee o be weenness cen ali y compa ed wi h hose close o he bo de . Acco dingly, his in luence has been called he “edge e ec ” o “bounda y e ec ” [1, 9, 10, 14]. In his esea ch, we ocus on he e ec o size o he ca chmen a ea on he indica o alue. This e ec is he ea e e e ed o as he “size e ec ”. 1.2 Types o esea ch whe e size e ec should be conside ed Conside a ion o he size e ec may be pa icula ly c ucial in esea ch p ojec s associa ed wi h he ollowing pu poses. 1.2.1 Resea ch aiming a compa ing and classi ying mul iple ne wo ks dis ibu ed in di e en loca ions Fo any compa a i e s udy, whe e mo e han one ne wo k is In e na ional Jou nal o Sus ainable De elopmen and Planning Vol. 16, No. 6, Oc obe , 2021, pp. 1019-1026 Jou nal homepage: h p://iie a.o g/jou nals/ijsdp 1019 included, he size o he ca chmen a ea will ei he ha e o be iden ical o i mus be examined as one o he ac o s ha may explain he a ia ion o he measu emen alues among he mul iple ne wo ks. 1.2.2 Resea ch equi ing he dis inc ion among di e en ypes o mobili y ne wo ks based on he mode o anspo a ion, such as he ne wo ks o pedes ian, cyclis s o mo o ized ehicles Fo example, “a small oad segmen in a esiden ial a ea migh be impo an o pedes ians, bu i is almos negligible o mo o ized anspo a ion. In his sense, he oad segmen should ha e a leas wo kinds o impo ance: one o pedes ians and he o he o mo o ized anspo a ion” [15]. In o he wo ds, he alue o he indica o a ies depending on whe he he ca chmen a ea is a he human o ehicle scale. Wi hou conside ing he size o he s udy a ea, he deg ee o be weenness cen ali y canno ep esen mul iple aspec s o he ci y ha is pe cei ed by people in he eal wo ld [16]. To a oid hese p oblems, Yoshimu a e al. [13] p opose ha , in addi ion o he global be weenness cen ali y o he en i e ci y, a se o local be weenness cen ali y alues a a smalle neighbo hood scale shall also be calcula ed. 1.2.3 Resea ch whe e an a ea needs o be di ided in o sec ions o smalle size wi h egula shape The ci y a ea is o en di ided in o smalle uni s due o he s uc u al complexi y and size associa ed wi h he scale p oblem [17-19]. Fo example, he analysis o connec i i y o he oad and s ee ne wo k s uc u e is o en conduc ed wi h he applica ion o opological measu es and ypically equi es ha a la ge geog aphical space be di ided in o smalle pa s wi h egula shape [8]. In a ecen s udy, in o de o de e mine he size o he ca chmen a ea on he alue o he selec ed measu es, Soczówka e al. [8] ha e ca ied ou a compa a i e analysis by di iding he analyzed a ea in o di e en sizes wi h egula shape and compa ing he alues o connec i i y measu emen s o he oad and s ee ne wo k s uc u e. In such a case, he size o a single basic ca chmen a ea is e y impo an because i can signi ican ly a ec he compu a ional esul s o hese measu es. 1.2.4 Resea ch measu ing he spa ial accessibili y A bi a y adminis a i e bounda ies (such as census ac s o block g oups) a e o en used in he s udies o spa ial accessibili y. The a bi a ily de ined bo de may lead o a me hodological limi a ion [20, 21] because, i s o all, accessibili y in ol es mo emen and a gi en bounda y does no ac ually p e en people o ehicles a eling ac oss he bo de om eaching he acili ies, se ices, ameni ies o any poin s o in e es [22]. Secondly, a gi en bounda y may exclude beha io ou side he ca chmen a ea [12, 20-22, 23- 31]. Thi dly, he esou ce o se ices beyond a de ined a bi a y bounda y may in luence he beha io wi hin he ca chmen a ea [27]. Many esea ch p ojec s s udying he spa ial accessibili y ha e poin ed ou he isk ha he accessibili y o acili ies may be biased [22, 25, 27, 32] o unde - epo ed [12, 23, 28, 29, 33] because “a eas close o he bounda y may be classi ied as ha ing poo geog aphic access e en hough hey may in ac be p oxima e o esou ces ac oss he bounda y” [34]. 1.3 P oposed solu ions o mi iga ing he dis o ion In an a emp o mi iga e, con as o diminish he dis o ions s emming om he size e ec , he p oposed ea u es and p inciples unde lying u he classi ica ion can be b oadly ca ego ized in o he ollowing wo app oaches. 1.3.1 Adding a bu e zone wi h he assigned h eshold adius The i s app oach is o c ea e bu e zones a ound he ca chmen a ea in o de o compu e he indica o alue [1, 4, 35-37]. Fo example, he Py hon lib a y OSMnx [38, 39] au oma ically c ea es a bu e o hal a kilome e a ound he eques ed a ea so ha each node has a co ec s ee coun . The bu e zones a e hen immed om he cons uc ed ne wo k model. The indica o alues o nodes and links wi hin he bu e zone a e excluded om he analysis because hey a e no eliable. 1.3.2 Using a homogeneous ea u e as he c i e ion o he di ision Soczówka e al. [8] p opose o di ide he ci y a ea in o smalle ca chmen a eas wi h homogeneous ea u es be o e conduc ing he analysis. They sugges se e al possible c i e ia o he homogeneous ea u es including, i s ly, adminis a i e c i e ia, such as adminis a i e dis ic bounda ies; secondly, s uc u al c i e ia, such as he spa ial dis ibu ion o densi y o inhabi an s in households; and, hi dly, echnical and unc ional c i e ia, such as he public anspo managemen sys em. In sho , any di isions and classi ica ions o geog aphical space should de ine he bounda y o he ca chmen a ea. In esponse o hese conside a ions, he cu en esea ch in ends o con ibu e o his decision p ocess and o quan i y he e ec on he alue o he indica o s by 1) in es iga ing whe he and o wha ex end indica o s, such as he node deg ee, change wi h he size o he ca chmen a ea; 2) p oposing a me hodology o decide he size o he ca chmen a ea based on he a e age s ee leng h; and 3) o e ing he ecommenda ion o he app op ia e size o he ca chmen a ea o he in es iga ion o pedes ian ne wo ks. A heo e ical model ep esen ing an idealized egula ne wo k (see Figu e 1) has been c ea ed as a e e ence model o he clea ma hema ical deduc ion o he ela ionship be ween he size o he ca chmen a ea and he changes in he indica o alues. This allows us o answe he ollowing esea ch ques ions in a mo e con olled en i onmen , whe e he di e ences in he alues o he indica o will only be caused by he size e ec (This is no igno ing he ac ha , in he case o eal ne wo k, he size e ec may s ill exis ega dless o he size o he ca chmen a ea. Because a eal ne wo k is no egula o e e ni y and, he e o e, i s cha ac e is ics change). • How do we decide he app op ia e size o he in es iga ion o pedes ian ne wo ks? • A e he e uppe and lowe limi s o he size e ec on he alue o an indica o ? • How big should he ne wo k ca chmen a ea be in o de o be able o compa e i wi h he e e ence model? 1020 2. METHOD 2.1 The in es iga ed indica o The indica o we ha e chosen o examine he size e ec is he a e age node deg ee, k, which is exp essed by 𝑘 = 2 × 𝑀 𝑁 (1) whe e, M e e s o he o al numbe o links and N e e s o he o al numbe o nodes. In addi ion, he a e age node deg ee p o ides he in o ma ion o he ne wo k pa e n and can also be used o e alua e he le el o "g idness.". A ne wo k wi h a lo o nodes connec ing o 4 links, i.e. k  4, means ha he ne wo k is mo e likely o be g id-pa e n. And "mo e-g idded ci ies ha e highe connec i i y (i.e., highe node deg ees, mo e ou -way in e sec ions, ewe dead-ends e c.) and less-winding s ee pa e ns"[39]. This means ha , holding he numbe o nodes cons an , i he a e age node deg ee o a eal ne wo k is smalle han ha o a g id-pa e n ne wo k, he connec i i y o he eal ne wo k is wo se han he connec i i y o a g id- pa e n ne wo k. Be o e examining he size e ec on he node deg ee, we need o p o ide some de ini ions conce ning ou heo e ical model. 2.2 The heo e ical ma hema ical model Fo he in es iga ion o he size e ec , a eal ne wo k migh no be a good basis o be he e e ence o compa ison. The main p oblem is ha he size e ec canno be clea ly sepa a ed om o he e ec s. The e o e, a heo e ical model o he p ognosis o he size e ec was c ea ed in o de o p o ide he me hod o ma hema ical analysis o an idealized egula ne wo k. In his way, he di e ences in he alues o he indica o will only be caused by he size e ec . Ou de ini ion o an idealized ne wo k consis s o he ollowing componen s. 2.2.1 De ini ion o a single quad a as he basic elemen To illus a e he ules o how he ne wo k is c ea ed, we s a wi h a ne wo k ha is jus a quad a (squa e). As shown in (a) in Figu e 2 and Table 1, a quad a has ou links as i s bounda y and i se e as he basic uni o he quad a ic ne wo k wi h ou nodes (N = 4) and ou links (M = 4). We use d o e e o he leng h o a link in he ne wo k, which is also he side leng h o a quad a . 2.2.2 De ini ion o a ne wo k A quad a is expanded and ex ended in he same quad a ic pa e n, as shown in (b), (c) and (d) in Figu e 2 and Table 1. The size o he ne wo k is measu ed by he o al numbe o nodes, N, and he o al numbe o links, M. Elemen (b) in Figu e 2 and Table 1 show ha a ne wo k consis ing o ou quad a s has nine nodes (N = 9) and 12 links (M = 12). 2.2.3 De ini ion o a ca chmen a ea In he case o eal ne wo ks, a ne wo k is di e en om a ca chmen a ea and he selec ed ca chmen a ea will be smalle han he en i e ne wo k. As illus a ed in Figu e 1 (a) and (b), he black solid line indica es he links and also he bounda y o a quad a ic ne wo k and he ed dashed line indica es he bounda y o a ca chmen a ea. A ca chmen a ea is like he sec ion ha cu s o links connec ing nodes om inside o he ca chmen a ea o nodes ou side o he ca chmen a ea (In a eal ne wo k, i is no necessa ily he case ha a ca chmen a ea cu s o links connec ing nodes om inside o he ca chmen a ea o nodes ou side o he ca chmen a ea. Some imes, he bounda y o he ca chmen a ea may coinciden ly lie exac ly on a link as shown in Figu e 1(b).). The link cu o by he ca chmen a ea could be coun ed as a comple e o a hal link. And he alue o he indica o will be a ec ed by how hese links a e coun ed. Al e na i ely, hey migh be excluded om he calcula ion al oge he . Fo he pu pose o his pape , we assume ha hese cu -o links a e neglec ed and hey a e excluded om he calcula ion. This means ha he bounda y o he ca chmen a ea consis s o he links o he ne wo k as shown in Figu e 1(b). The e o e, he side leng h o a ca chmen a ea, D, consis s o one o mul iple links o he ne wo k. And he size o he ca chmen a ea is D x D. (a) (b) Figu e 1. Abs ac ed exp ession o he bounda ies o he idealized ne wo k and he ca chmen a ea The e can be wo kinds o ela ionships be ween he ne wo k and he ca chmen a ea. 2.2.4 When he ca chmen a ea consis s o one quad a and D consis s o single d (i.e. D = d) In he case o he smalles ca chmen a ea (as shown in elemen (a) in Figu e 2 and Table 1), he e is only one quad a and, he e o e, he side leng h, D, o his ca chmen a ea equals o d. The size o he ca chmen a ea is D×D = d2. The ne wo k in his ca chmen a ea has 4 links (M = 4) and 4 nodes (N = 4). The node deg ee, k, is acco dingly 2. 2.2.5 When he ca chmen a ea consis s o mul iple quad a s and D consis s o mul iple d In elemen (b) in Figu e 2 and Table 1, he e a e 4 quad a s and, he e o e, he side leng h, D, o his ca chmen a ea equals o 2d. The size o he ca chmen a ea is D x D = 4d2. The ne wo k in his ca chmen a ea has 12 links. The o al numbe Is his a comple e o hal link, o no ega ded as a link a all? An idealized ne wo k wi h links (black solid line) as i s bounda y. A ca chmen a ea wi h ed dashed line indica ing i s bounda y. 1021 o links, M, in his ne wo k is 12 and he o al numbe o nodes, N, is 9. The node deg ee, k, is acco dingly 2.67. The ela ionship be ween he quad a , he ne wo k and he ca chmen a ea o a ious sizes a e summa ized in Table 1 and i shows ha , wi h he inc easing side leng h, D, o he ca chmen a ea, he node deg ee, k, also inc eases. This means ha he e is a size e ec on he indica o o node deg ee. This end is no ewo hy, so we b eak down he p ocess in o s eps in o de o in es iga e he de ails. The ollowing discussion is di ided in o wo pa s: he addi ional links and he addi ional nodes. Figu e 2. Abs ac ed exp ession o he p ocess o inc easing he size o he ca chmen a ea Table 1. Rela ionship be ween he quad a , ne wo k and ca chmen a ea o a ious sizes Elemen s in Figu e 2 (a) (b) (c) (d) Numbe o quad a s 1 4 16 36 Ne wo k To al numbe o links (M) 4 12 40 84 To al numbe o nodes (N) 4 9 25 49 Node deg ee (k) 2 2.67 3.2 3.43 Ca chmen a ea Side leng h o ca chmen a ea (D) d 2d 4d 6d Size o ca chmen a ea (D×D) d2 4d2 16d2 36d2 2.3 P ocess o inc easing he size o he quad a ic ne wo k 2.3.1 Inc easing he numbe o links Figu e 2 p esen s he abs ac ed exp ession o he p ocess o inc easing he size o he ca chmen a ea. Figu e 2 (a) shows one quad a (squa e). The ne wo k wi h one quad a is he smalles ne wo k. Figu e 2 (b) shows a ne wo k consis s o 4 quad a s. To ex end he ne wo k om (a) o (b) in Figu e 2 and Table 1, h ee links a e added o c ea e he yellow quad a s. And hen wo mo e links a e added o c ea e he blue quad a . Fo he ne wo k in (c) in Figu e 2 and Table 1, which is a ne wo k consis s o 16 quad a s, we begin wi h a co ne and, i s ly, c ea e he g een quad a e wi h ou links. Nex , wo mo e links a e added o c ea e ano he wo blue quad a s. Thi dly, h ee links a e added o c ea e he yellow quad a in he co ne . Finally, hese h ee s eps a e epea ed un il he inal pu ple quad a e, which needs only one addi ional link o be c ea ed. As he ne wo k becomes bigge and bigge , he e a e mo e and mo e blue quad a s, which a e o med by wo addi ional links, among he newly c ea ed quad a s. In o he wo d, he numbe o blue quad a s inc eases much as e han o he quad a s. E en ually his ype o quad a becomes mo e and mo e impo an and hus dominan he pa e n o he inc easing side leng h, D. Meanwhile, he yellow co ne quad a , which consis s o 3 links, becomes less dominan . 2.3.2 Inc easing he numbe o nodes Since he blue quad a is mo e dominan han he o he ypes o quad a s on he inc easing side leng h, D, he ocus o he in es iga ion is on his ype o quad a s. Fo e e y blue quad a , i akes wo addi ional links o c ease one addi ional node. E en ually he o al numbe o nodes, N, will be hal o he o al numbe o links, M. The e o e, 𝑁 = 1 2𝑀 (2) o , in o he wo ds, 𝑀 = 2 𝑁 (3) This means ha node deg ee, k, will e en ually app oach he inal limi . 𝑘 = 2 × 𝑀 𝑁= 2 × 2𝑁 𝑁= 4 (4) The esul s a e shown in Figu e 3 and Table 2. Figu e 3. Rela ionship be ween he alue o he node deg ee, k, and he size o he ca chmen a ea, which is exp essed by he numbe o links on each side o he ca chmen a ea 2.4 App op ia e size o he ca chmen a ea o pedes ians Fo he in es iga ions o pedes ian ne wo ks, i makes no sense o explo e a la ge a ea. I we conside ha he pedes ians’ maximum accep able walking ime is abou 15 o 20 minu es, he size o he ca chmen a ea would be be ween 1500x1500m² o 2000x2000m². Also, ollowing he indings ega ding he size e ec , he compa ison be ween di e en “pedes ian” ne wo ks is only co ec and hus possible i all ca chmen a eas ha e he same size. 2.00 2.50 3.00 3.50 4.00 010 20 30 40 50 60 70 80 90 100 node deg ee, k Numbe o links, which is equi alen o D/d, on each side o he ca chme a ea. 1022 Table 2. Rela ionship be ween size o he ca chmen a ea and he alues o indica o s. The size o he ca chmen a ea is indica ed by i s side leng h, D, and D is indica ed by he numbe o links, which is equi alen o D/d Side leng h, D, o he ca chmen a ea Links in he ne wo k Numbe o links on each side o he ca chmen a ea To al numbe o links, M To al addi ional links o base quad a , indica ed by he black links To al addi ional links equi ed o c ea e he new quad a , indica ed by colo ul links 1 4 4 4 2 12 4 8 4 40 12 28 6 84 40 44 8 144 84 60 10 220 144 76 12 312 220 92 14 420 312 108 16 544 420 124 18 684 544 140 20 840 684 156 40 3280 2964 316 50 5100 4704 396 100 20200 19404 796 Side leng h, D, o he ca chmen a ea New quad a Numbe o links on each side o he ca chmen a ea Numbe o new g een quad a s o med by 4 addi ional links Numbe o new yellow quad a s o med by 3 addi ional links Numbe o new blue quad a s o med o 2 addi ional links Numbe o pu ple quad a s o med o 1 addi ional link 1 1 0 0 0 2 0 2 1 0 4 1 3 7 1 6 1 3 15 1 8 1 3 23 1 10 1 3 31 1 12 1 3 39 1 14 1 3 47 1 16 1 3 55 1 18 1 3 63 1 20 1 3 71 1 40 1 3 151 1 50 1 3 191 1 100 1 3 391 1 Side leng h, D, o he ca chmen a ea Indica o s Numbe o links on each side o he ca chmen a ea To al numbe o nodes, N Node deg ee, k Nodes / a ea A ea 1 4 2.00 4.00 1 2 9 2.67 2.25 4 4 25 3.20 1.56 16 6 49 3.43 1.36 36 8 81 3.56 1.27 64 10 121 3.64 1.21 100 12 169 3.69 1.17 144 14 225 3.73 1.15 196 16 289 3.76 1.13 256 18 361 3.79 1.11 324 20 441 3.81 1.10 400 40 1681 3.90 1.05 1600 50 2601 3.92 1.04 2500 100 10201 3.96 1.02 10000 3. RESULTS AND IMPLICATIONS The bene i o in es iga ing he size e ec wi h a heo e ical model is clea because, wi h he con olled condi ion, we can ind ou he scena io when he size e ec is (nea ly) sa u a ed and deli e a co ec p ognosis abou he o al size o he ne wo k. Figu e 3 shows he ela ionship be ween he alue o he node deg ee, k, and he size o he ca chmen a eas, which is exp essed by he numbe o links on each side o he ca chmen a ea. I can be concluded ha he size e ec is e y no able un il he side leng h o he ca chmen a ea, D, equals 10 imes o he leng h o he link, i.e. D = 10d. And i is s ill 1023 no able wi h a de i a ion o 3.8/4 = 5% when D = 20d. The de i a ion is educed o 2.5% when D=40d, which can be ega ded as a h eshold and ecommended as he size ha is big enough o allow compa isons be ween di e en ca chmen a eas and/o di e en ne wo ks. E en ually, i comes close o sa u a ion whe e D = 50d. Fo alues D >= 50d he node deg ee, k, eaches he ideal alue o he ex ended ne wo k and he size does no play a ole a e ha . The p incipal implica ion o his heo e ical model is wo old. Fi s o all, he abo e in es iga ion p o ides e idence o show ha he size e ec is ema kable and canno be neglec ed un il D = 40d. Secondly, Table 3 shows he ela ionship be ween he a e age s ee leng h and he ca chmen a ea size ange. The esul s o e a p inciple guideline o de e mining he size o he ca chmen a ea whe e he eal ne wo k can be compa ed wi h he heo e ical model. In he eal s ee ne wo k, he a e age leng h o a s ee , which is he leng h o he link, d, in he heo e ical model, is mos ly be ween 50m and 100m. Assuming ha d = 100m, he size o he selec ed ca chmen a ea has o be a leas 20d x 20d = 2000x2000m2 in o de o be able o compa e i wi h he heo e ical model. I he side leng h o he ca chmen a ea is 40d, i.e. 4000m, and he size o he ca chmen a ea is a leas 4000x4000m2, he size e ec will be e en less signi ican in he heo e ical model. The e o e, we p o ide he e idence o suppo ha he lowe and uppe limi s o he size o he ca chmen a ea should be 2000x2000m2 and 4000x4000m2. Table 3. Recommenda ions o he lowe and uppe limi s o he side leng h o he ca chmen a ea A e age s ee leng h, d 50m 100m Lowe limi o he side leng h o he ca chmen a ea 20d=1000m 20d=2000m Uppe limi o he side leng h o he ca chmen a ea 40d=2000m 40d=4000m Range o he sizes o ca chmen a eas 1000x1000m2 ~ 2000x2000m2 2000 x 2000m2 ~ 4000 x 4000m2 To sum up, he esul s in his esea ch con ibu e o he a gumen ha a ca chmen a ea wi h an a ea size ha is oo la ge would no be p ac ical due o he ollowing easons. Fi s o all, he cha ac e is ics and pa e ns o he s ee ne wo ks in a eal ci y may a y om qua e o qua e . Fo example, he cha ac e is ics o he ne wo k in he his o ical cen e would be di e en om i s su ounding a eas. The e o e, i he size o he ca chmen a ea is oo big, he alues o he indica o s would e lec no he in o ma ion abou one ype o ne wo k bu a he abou a sum o mul iple ypes o ne wo ks in se e al neighbo ing and connec ed qua e s. Such mixed in o ma ion would be less aluable o in es iga ing he ela ionship be ween he s ee ne wo k s uc u e and he indica o s o o classi ying he s ee ne wo ks. Secondly, 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 chmen a ea. Acco ding o ou analysis, in he heo e ical ne wo k, 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 x 4000m2. The e o e, any size la ge han 4000 x 4000m2 would no be necessa y. Thi dly, he node deg ee o an idealized egula ne wo k changes wi h he size o he ca chmen a ea. Bu he a ia ion becomes nea ly neglec able when he size o he ca chmen a ea is la ge han D = 40d and anishes wi h D >= 50d. This means ha , i he size o he ca chmen a ea o he eal ne wo k is la ge han D = 40d o 50d and he node deg ee is 4, he ne wo k pa e n has a g id-like cha ac e . Howe e , i he node deg ee o a eal ne wo k is smalle han 4, i s connec i i y is wo se han ha o he heo e ical g id-pa e n ne wo k and ice e sa. Fou hly, we sugges ha in all u u e in es iga ions he size o he ca chmen a ea should be de ined be o e ca ying ou u he analysis. Fi hly, when compa ing he indica o s o mul iple ca chmen a eas, all o he ca chmen a eas should ha e he same size as long as D is less han 40d o 50d. Finally, node deg ee has been chosen because i s beha io can be calcula ed o he heo e ical and idealized ne wo ks p oposed in he cu en esea ch in o de o quan i y and o demons a e he size e ec on he indica o alues s ep by s ep. 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Na u al mo emen : O , con igu a ion and a ac ion in u ban pedes ian mo emen . En i onmen and Planning B: Planning and Design, 20: 29-66. h ps://doi.o g/10.1068/b200029 [38] Boeing, G. (2017). OSMnx: New me hods o acqui ing, cons uc ing, analyzing, and isualizing complex s ee ne wo ks. Compu e s, En i onmen and U ban Sys ems, 65: 126-139. h ps://doi.o g/10.1016/j.compen u bsys.2017.05.004 [39] Boeing, G. (2019). U ban spa ial o de : S ee ne wo k o ien a ion, con igu a ion, and en opy. Applied Ne wo k Science, 4(67): 1-19. h ps://doi.o g/10.1007/s41109-019-0189-1 NOMENCLATURE d leng h o a link in he ne wo k and also he side leng h o a quad a , m D side leng h o a ca chmen a ea, m D×D size o ca chmen a ea k dimensionless a e age node deg ee M o al numbe o links N o al numbe o nodes 1026