The first matrix represents the cross-citations between nineteen business and economic journals. The second matrix consists of intersectoral job changes in Belgium. The third matrix indicates the amount of interaction between retail establishments, as indicated by customer movements from one establishment to another

Tobler, W.

International Institute for Applied Systems Analysis (IIASA)

Analysis of Three Interaction MatricesTobler, W.IIASA Working PaperWP-75-1091975 Tobler, W. (1975) Analysis of Three Interaction Matrices. IIASA Working Paper. WP-75-109 Copyright © 1975 by the author(s). http://pure.iiasa.ac.at/319/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at ANALYSIS OF THREE INTERACTION MATRICESW.R. ToblerAugust 1975 WP-75-l09Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor-mation purposes. The views expressedare those of the author, and do notNecessarily reflect those of IIASA.Analysis of Three Interaction MatricesW.R. ToblerThe first matrix represents the cross-citations betweennineteen business and economic journals. The positions ofthese journals relative to each other, as obtained from amultidimensional scaling algorithm, are as illustrated inFigure one. These are extracted from a paper by Hamelmanand Smith, who also provided the cross-citation table.The procedure suggested in IIASA RR-75-19, Spatial Inter-action P?tterns, was applied to this 19 by 19 array, andthe result is shown in Figure Two. The asymmetry of thearray is 29.5 percent. The influence field appears to befrom the substantive economics journals to the statisticaland methodological journals. An interpretation would bethat discussions occurring in the subject matter fieldsare used as a basis, or a justification, for methodologicalinvestigation.The second matrix consists of intersectoral job changesin Belgium. As suggested in IIASA Research Report-75-19,dissimilarities between sectors was calculated from thefollowing formulad" = _12p,p,/M"1J " 1 J 1Jwhere P. and p, are the sizes of the sectors (number of1 J2ｬＧｬｵｬｴｬｇｩｭ･ｾＱｳｴｾｩＮＱＬｊＬＱ 3C.JJ..Lllg t): 19 Husj ｮ Ｈ ｾ ｳ ｳ .Jnd L:onomic ｊ ｾ Ｉ ｬ ｬ ｲ ｮ Ｎ Ｎ ｊ Ｎ ｬ ｳ.},eM!"''')' ofｾ Ｎ ｬ ｜ ｦ ..1!, Ｚ ｲ Ｎ Ｎ ｾ ...\tJOJ"fL.llＬ ｉ Ｂ ｉ Ｌ ､ Ｚ ｬ ｬ Ｑ ｴ ｲ Ｎ ｊ ｬ ｬ ｾ ｣Sc ｬｾｬ｜ｦＮｬ Cl.. ". tll'I1lﾰ Ｑ ｽ ｾ ｲ Ｎ ｬ ｬ ｾ ｏ Ｂ ＧＮ ｾ Ｌ ｣ Ｂ ｬ Ｈ ｨHIlT'lDf.,C.1entSC l.nl;;e 'Jounlll.l ofAlIlcrl'",," -Slot tat I catA••ocl..ltl<Jl\.ｉＺ＼ｏｬｬｬｲＮｬｉｾｴｲｬ･｡.JOtlfnal ofｾ Ｇ ｬ ｮ ･ ..JO\Jllllll offinanc.1III11(v8rd&u,lnellReviewJO\Jrn,ll of&r1utlna •!n"'\IUr1.l• nl1 L.bor• Relatlon,luteYｾ ｣ ｲ ｴ ･ Ｎ ｮ｛ ＼ ｏ ｬ ｬ ｾ ｬ ｣ Ｎ •ReYlewJCllHnd otPoBUc.!-tconor.l1C"n ...dh'"Jourr.,ll oftCO('l<:ntc..,...d PoliticalSclrnt;.11 aｾＧｊｲｴｈｬｹ• J".JrnJ,l orr.coI\Qr:llCIin·lt· ... or[..:;ono.. l .. I and.. SCilllI1UC'Southifrn.1-::oncw"lcJO\l(l,,,lQ\lIrtnlyＮ ･ ｶ ｬ ｾ Ｂ of• ECOII(),';\Ic..and Bus Inc ••NHlon.lllTu •JoucnaLFIGURE I Spatial Positioning of Journals- 3 ->""":e....... ""'"FIGURE II: Vector field of Journal Interactions- 4 -employees) and M.. is the number of people who changed1Jemployment from sector i to sector j. A trilaterationalgorithm estimates x,y coordinates from the separations,as listed in Table one. After thus locating the sectorsrelative to each other, the 21 by 21 interaction table(provided by Dr. Vanoverbeke) was used to estimate thegradient of the potential between sectors, as described inRR-75-19. The result is shown in Figure Four. The asym-metry of the original array is only 8.6 percent so thatdifferential effects do not appear strongly. The largevector at sector twelve (Energy) is misleading since thevolume of exchange here is very small. Otherwise the figureis difficult to interpret. There seems to be some movementaway from textiles and tobacco, and movement towards services.The third array, Table II, provided by Prof. John Nystuen,indicates the amount of interaction between retail establish-ments, as indicated by customer movements from one establish-ment to another.obtained from d ..1Jinteraction fieldTable III gives the trilateration solution/ J.?- (P.P. M.. ) . Figure V indicates the1 J 1Jobtained from the array of Table II. Thecluster of businesses in the center of Figure V includesSupermarket, Drug Store, Department store, Bank, Bakery,Restaurant, and variety store. The next ring includes Grocery,Utility Company, Hardware and Paint, Miscellaneous Retail,Appliances, Clothing and Shoes, Furniture, and Theater.The outer ring includes all of the remaining establishments.U1Y.lX.lP.lSECTORI-,1 Landbouw (Agriculture) 11,930 180 -418 !2 Extractie (Extractive) 48,810 700 -610 iII3 Voeding (Food) 82,170 -70 288 ,I4 Tabak (Tobacco) 7,010 -635 224 I5 Scheikunde (Chemicals) 54,640 63 4556 Hout en Kurk (Wood and Cork) 45,870 326 3327 Papier (Paper) 22,180 -171 -268 Boek en foto (Book and Photography) 28,440 -648 -779 Huiden en Leder (Hide and Leather) 8,250 -108 -21210 Textiel (Textile) 113,020 -535 691I 11 Kleding (Clothing) 78,780 -1,000 -377I 12 Energie (Energy) 11,540 -643 , -921II 13 Niet Metaal (Non-metal) 56,690 302 -137I 14 Staal nijverheid (Steel) 103,460 686 -23615 Metaalverwerkende nijverheid (Metal working) 268,700 170 5916 Diverse be-enverwerkende nijverheid (Industry) 30,070 127 36717 Bouw (Construction) 224,590 448 -3018 Transport (Transportation) 61,190 690 19419 Handel enfinanciele instellingen (Commerce) 116,060 -19 4320 Horeca (Hotels, eatingestablishments) 69,440 .,..909 -52321 Overheiden n.e.v. diensten (Service) 103,300 -150 159TABLE I: Belgium Handarbeiders- 6 -to•5."1(,•.b.3.4-.•8ｾＬ• ,<I IS'fｾＸ -1 .. Ii.. '/ -'3.. ｉｾ..FIGURE III: BELGIUM HAUDARBEIDERSSpatial Positioning of Sectors· Ｎｾｾ ..FIGURE IV:- 7 -Belgium HandarbeidersVector field of interaction. ..NUMBER OF STOPS AT OTHER RETAIL ESTABLISHMENTSIN EVERY ONE HUNDRED TRIPS TO EACH BUSINESS TYPEEntrie\ indicate the number of stops in 100 trips, to the business types listed by JaW mode to busine\l types listed b)' coluftVli.ActualNumberof Trips 8VlineSi TypeNumber ofSingle Purpo\eTr;p. ;n 100Trips per8usineu Type00• I 17flo I 19' flo I ..IA.. I ?R?RIIc=====tl I151 Department Store I 7 50 I 22 28 11 • • • 9 • • • • 13 • •• ••••• 1952 Ciothirlg and Shoes 2 4 94 25 is 17 .. .. " .." .. 21" .. .. ｾ66 _\{nrlcly J 5 91 15 17 • • 17 • • • 15 •• • • ｾＫＭｃＱＢＬＬＷＫｾｦＭＮ］ＭｴＭＭ［ＱＭ］ＵＫｾＧ60 Bonk. ｾ B 40 17 10 •• • 10' • 38 •••• •• ••••••ｾ Ｑ Ｍ Ｒ Ｍ Ｍ ｟ Ａ ｾ ｲ ･ Ａ ｟ ［ ｡ ｮ ｣ Ｎ Ｎ 2 17 Ｍｩｾ｣ＭＲＭＵＭ 17 17 1--_ • 17" • 33 '. ._37 M"celloncou, Retoll -..1. II 43' n II' • 14 II' • • 24 • • 14 ·Tｾ _J.:!'"d .....ace ond Point _-.1:: 28 Ｌｾｬ c-!1-I-c-: . 12 ; ••• 16 •• Ｇ Ｎ ｾ II 29 _lJI,l!'yCompanyOffice Ｎ ｾ Ｑ ｟ Ｑ ｾ 66 • 17 !4' 2! I! • • • 24 • 11 1749 Ｎ Ｎ ｑ ｯ ｯ ｾ ｯ ｲ . ....1.f---P---- 39 •• 1--__ • r-- • ••. • •. ｾ_--.!..3__ ｾ ｾＧｩＧＭＧＭＭ u:.:. 38 • e--- I .. • 15'--'2__ J:!: lei II 4.L---._--' JL 33 33 •18 ｬＮＡＺＡＮｯＧｾＮｲＺＮＮＮＡＮＮＮｾｾｾｌｒ｣ｯｉｅｾｲ｡ｴｾＬｬｯｷｹｴｲｯＮ ,-1. 11 __ ;'_Q r-- ｵ Ｎ ｾ 17 22 17. 11 11 _lL ..145 Re blJrufl! / . II 17 " " 13 .... .. 1 ....... 12""" .. 11 11" .... ..----y-;j-- Ta,-!';:;-----· J'i 29 '14 21 • 50 • •• 21' ｟ ｾ108 Th"o"e I:> 61 9······ • 19 •• •• 9 •••-1-6-- ｾ ｾ ｬ ｏ ｐ ---1J. 31 19 19' 31 • 19' j13 ｬｾＧＧＧＲ､ｲＮｬＧ and D,y Clean;ng -U 0 23 •• 15 • 38 15 23 15' _16 Bo Co" _....d 18 22' • II 22 11 • 17 I' . . 1714 A\.I 0 ａＨＮＮＺＺ･ｾｾｯｲｹ 4 64 _ tA60 GCi Sh],ion J.& 18 .. .... ...... .. - .. .. .. ...... ..30 ;;;",0 Repal' ,t( 23 10 • ••• •• \0' •• 1032 Ice Cream Store .tl 41 .... - .... .. 16160 [);\1!lStore U 36 25"""" 9 .. .. .. .. .. .. 10 .. .. .. 10 ..-2-.-- lAo ＢＧ｟ｾｾｉ｡ｵｬ･ Markel> ---M 25 •• 13 • . 17 .. 21 17 L-. I33 Ba''''-l: ｾ 15 9'" •• •• •• •• •• 12" 21 9-1l144 ｾＮ･ｲｭｾｲｫ･ｬ .'lJ>, 34 26········ • 10 ••• •• 11 • • • • •264 . G ｲｾｯＺｩＱ - J:.1- 60 -- -==;- • •• ••• ••••• •• •••••• I •18 f'iL"llure :Lj ,------.lL I 508:t:>'tot"ltIjHH(-) ｩ ｮ ､ ｩ ｣ ｡ ｴ ･ ｾ eight (8) or f. Ner Ａｴｯｰｾ were mode ot the ｢ ｖ Ｇ Ｎ ｯ ｩ ｮ ･ ｾ ｳ type listed in the column in every lOa ｴ ｲ ｩ ｰ ｾ to the ｢ ｵ ｾ ｩ ｮ ･ ｓ Ｑ type ｉ ｩ ｾ ｴ ･ ､ in the row.Blank ｣ ･ ｬ ｬ ｾ indicate the b.. sint.u types werp. never auocioted in the soene trip.- 9 -SOLljT I ｏｾｊ C[)[j'll) J ｾｾａ ｲｬＭ｟ｾ (X, If)e1.!"q.. ＶＮｲｾＮｾ-1.48-n.q1-2h.34-21.1.lr,-(.'3.4..,-1 4 ,,3U25.64ｾ Ｂ Ｓ Ｎ ｱ ｬ Ｇ ｜19.734.7b-"'.51Ｍ ﾷ ｱ Ｎ Ｕ ｾ ｾ-1:5.1:1:"-ＭＨＧｾＮＷＴ-50.1qＭ ﾷ ｂ Ｎ Ｚ Ｍ ｾ Ｔ1(".77£i.31.lｾ ｬ Ｎ Ｘ ｾ17.6P,-3.5'538.(1\2-?.iJ71.q0L1. 7 32Vl.75ii1 • ｦｾ ｾ1:S. l-1 E''-l. tiS1 • ( ,10.'1 ••,(1Ｍ ｾ Ｎ Ｇ Ｏ Ｑ Ｘ.. ｾ ｾ • " "1Ｍ｢Ｎ＿Ｚｾ-10.1t.Jｾ Ｒ Ｎ Ｏ Ｚ Ｇ ｯ Ｗ7?:.bF3;" • 1 ｾI-:; • '-J (--,ＭＮｾＶＮＧ［｜ｬｾＮＬＮｬｨＵｾＱＬＮｃＵ［ｊＭＱＷＮｾＺＧ_?-6.YG-£l1. Q q-15.74-16. 1.1\ＭＨｊＰＮｾＧＮＳt:'.;;u-U. IJ '5-1 • 7 1iJ.t'1.. R.L!FIc./12TABLE III:- 10 -,. ｾＬＮＭ "FIGURE V: Retail EstablishmentsVector Field of Interactions- 11 -This grouping seems to resemble that of a modern shoppingcenter. The array exhibits strong assymetry (48%) and theinfluence field seems directed towards the central cluster,with some exceptions.- 12 -BibliographySmith, C., and P. Hamelman, "Modeling the Structure of Science",TIMS/ORSA National Meeting, April 30, 1975, ｃ ｨ ｩ ｣ ｡ ｧ ｯ ｾFigure 2.Tobler, W., Spatial Interaction Patterns, Research Report75-19, IIASA, Laxenburg, Austria, 1975.VanOverbeke, L., Sectoriele Arbeidsmobiliteit in Belgie,Centrum voor Economische Studien, KatholiekeUniversiteit, Leuven, June 1974, pp. 27 - 30.

Analysis of Three Interaction Matrices

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