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implicating) Larry Brown and Duane Marble for useful leads and Stewart ..... Roy JR (1990) Spatial interaction modelling: some interpretations and challenges.
J Geograph Syst (2004) 6:43–54 DOI: 10.1007/s10109-004-0125-4

Isard’s contributions to spatial interaction modeling M. E. O’Kelly Department of Geography, The Ohio State University, Columbus, OH 43210, USA (e-mail: [email protected])

Abstract. This short review, surveys Isard’s role in promoting what has become known as spatial interaction modeling. Some contextual information on the milieu from which his work emerged is given, together with a selected number of works that are judged to have been influenced (directly and indirectly) by his work. It is suggested that this burgeoning field owes a lot to the foundations laid in the gravity model chapter of ‘‘Methods’’. The review is supplemented by a rather extensive bibliography of additional works that are indicative of the breadth of the impact of this field. 1 Introduction The gravity model is a ‘‘big’’ metaphor within geography, a point emphasized by Barnes and Curry (1992). The spatial interaction model descends directly from the gravitational ideas in early writings of social scientists such as Zipf and Ravenstein. There is no question that this metaphor has been used as a rallying point and has created a community with common interests in a wide range of structural phenomena: international and interregional trade, migration, information flow, traffic, residential mobility, express package and other time sensitive communications and so on.1 The work of Walter Isard added momentum to, and facilitated the creation of this community. This happened, in part, because of his writing in the area of what is now commonly thought of as the study of spatial interaction. A review of ‘‘Location and Space Economy’’ and later the ‘‘Methods’’ volumes makes it clear that the notions of spatial interaction moved from a minor passing mention (in the former) to a major chapter in the later book. It is also I am grateful to the editor, Serge Rey, for his encouragement to write this piece. I thank (without implicating) Larry Brown and Duane Marble for useful leads and Stewart Fotheringham for carefully reading an early draft. Omissions and opinions are my responsibility. 1

For reviews of the transition in context see among others the work of Wilson (1971), Senior (1979) and Haynes and Fotheringham (1984), Fotheringham, Brunsdon and Charlton (2000).

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apparent that the usage of the term spatial interaction as a synonym for the suite of gravitational models was foreshadowed in Isard’s writing. These works grew out of a milieu in which the standard approach in geography and transportation economics was characterized by disjointed and partial models. His work intersected with that of other influential scholars, among them William Warntz who was a frequent visiting lecturer in Regional Science at the University of Pennsylvania in the early 1960s. (see Janelle 1997). 2 What were the antecedents? At the time of ‘‘Methods,’’ Losch, Christaller, Weber and von Thunen (in location models) plus Zipf, Carrothers, Stewart, Ravenstein and others (in movement models) held sway in basic economic geography (see for example the excellent intermediate text by Haggett, et al. 1977; see also recent review by Greenwood and Hunt 2003). Books in urban and regional economics and indeed transport economics focused in the mid 20th century on descriptive and regulatory matters and had not yet reached a high degree of sophistication. To the extent that there was an empirical analysis, coefficients and parameter estimates were characterized by a high degree of ad hoc curve fitting. The notion of a unified derivation of wellspecified models, and their calibration from a rigorous maximum likelihood basis came much later. But these increasingly rigorous specifications grew as a result of the cumulative build-up of ideas, in large part emerging on top of the early work of Isard and others. In thinking about the mathematical analysis of regional models, there is no doubt that Isard contributed to what Scott (2000) has called the great half-century (1950– 2000) of progress in spatial analysis beginning with the regional science movement of the 1960s. The impact and influence of the work that Isard produced in this area is notable: perhaps not directly measured by the number of citations, [although these are considerable – see selected bibliographic examples in the references], but through the influence and training provided to a cohort of numerate and mathematically sophisticated modelers who came after the original work from Methods …, Location and Space Economy, … It is important to note the dates of his main works, and to clearly see in that work the signs of a simultaneous break through in several important areas of spatial analysis. These came from, among others, Serck-Hansen (1970) who made early advances in connecting location theory to the theory of the firm (leading today to the accomplishments of Fujita and Thisse and their colleagues). Takayama and Judge (1971) developed a careful analysis of space-price equilibrium, leading today to new ideas from Nagurney and her extensive contributions to network economics (see for example Moore and Nagurney 1989). Webber’s (1972) early work on the impact of uncertainty on location leads directly to current models and thinking with probabilistic components (e.g. agent-based models). So the great half century of Scott has indeed many notable accomplishments and space here will not be devoted to a wider review of those. Instead, it is proposed to examine the major theoretical and methodological insights in spatial interaction modeling that have emerged post-Isard, with special emphasis on those areas that

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accomplished a high degree of integration and synthesis of previous ideas. After all, one of the hallmarks of Isard’s writing has been to tie diverse components together.

3 What has come after? One of the most useful avenues is to consider the major regional science contributors to the areas of spatial interaction. We can ask what areas outside North American Regional Science were influential at the same time (see also Fotheringham and O’Kelly 1989). Here are some personal observations of these breakthrough advances and contributions, with just the briefest allusions to their lineage back to Isard. It is in fact possible that these authors do not owe or acknowledge a direct intellectual debt to Isard, but I argue here that the ideas were ‘‘in the air’’ so to speak, and that one of the catalysts for their blossoming was the extensive organizational leadership of Isard. Alonso: The general model of movement is an enormous intellectual accomplishment providing the rich set of flexible models that can accommodate many other model variants as special cases (see Ledent 1981). Instead of fixing the mass of the origin/destination as either a completely exogenous weight or as a required accounting identity to which the model has to conform, the Alonso insight was to see that the mass of the destination (for example) could serve both as a target constraint, but yet have some slack or flexibility in its role as a determinant of inflows. While this poses added complexities for estimation, this is indeed a ‘‘general’’ theory of movement. The chain of influence of this model stretches down to the present with the further work of Fotheringham and Dignan (1984), Anselin (1982), Porell and Hua (1981) and Miller and O’Kelly (1991) among others who have devoted attention to this model. Anas: One of the most valuable pieces of synthesis was Alex Anas (1983) development of a unified framework and derivation for the discrete choice, maximum likelihood and maximum entropy formulations of the gravity and interaction models. This placed all the variants on a common foundation and showed that what had perhaps previously been seen as competing bases for the model were in fact simply flavors of how one chose to set up the problem, and the selected variant of the optimization goal. Sen and Smith: Derivations from the point of view of Poisson counting process and with less reliance than say the Wilson or Fotheringham approaches on the behavioral determinants of interaction. This work is echoed in Flowerdew and Aitkin (1982) and has provided the foundations also for recent statistical estimation frameworks from standard linear models (Tiefelsdorf and Boots 1995). Tobler and Dorigo: The introduction of vector flows and fields is an important expansion of the original discrete nodal framework (see later for example Clark and Koloutsou-Vakakis (1992). Tobler’s work has occupied a central place in the theory of spatial interaction because of the cutting edge integration of continuous mathematics. In some ways one can see this

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connection through to the Isard sphere in the work of Beckmann and Puu (1985) and Sonis (1981). Fotheringham: The introduction of a proposed ‘‘competing destinations’’ effect and the derivation of appropriate rationale for this (from a behavioral, or spatial choice perspective) as well as detailed econometric examinations of the process is perhaps one of the most stimulating accomplishments of the late 20th in spatial interaction modelling. Previous spatial structure effects (traffic shadow, intervening opportunities) had been raised but prior to the early 1980s these effects were not systematically integrated into models. Indeed during the 1970s a debate on spatial structure (see among others Curry 1972, Cliff and Ord 1974 and Curry and Sheppard 1975) and misspecification in gravity model parameters grew increasingly confused and threatened to spiral out of control. Fotheringham’s rigorous analysis and concerted attack on some of the root causes of these spatial patterns basically set the record straight. Leonardi: During his brief career Leonardi (1983) made enormous advances in the way that we view location and interaction as simultaneously determined and did so by linking the objective functions of the type found in spatial interaction models to those in the location-allocation literature. The relaxation of the ‘‘all or nothing’’ assignment of demand to facilities, where the facility attraction could have differential impacts on the flows was an important insight, one that developed to a high degree of sophistication in Leonardi’s work. Allen and Sanglier: These modelers devised simulations of system dynamics and integrated spatial interaction with the hierarchical concepts from central place theory to produce realistic simulations urban systems changes. It is in this work that one sees the possibilities of adapting numerical simulation techniques from the physical sciences to urban systems development. At the heart of these simulations are models that allocate flows between centers (see Allen and Sanglier 1979, 1981). Weidlich and Haag: Devised a set of master equations to describe the dynamics of systems including migrations. Just as stochastic birth and death processes leads to a natural model of population dynamics, the introduction of spatial dynamics to such processes allowed Weidlich (1994) to provide a stimulating basis for a variety of dynamic interaction and migration models. (See also Weidlich and Haag 1986, in Griffith and Haining 1986). 4 What was the major catalyst for change? As a result of his extraordinarily long and productive career, Isard has had a chance to take a second (and perhaps even a third) look at materials in the light of a more mature field and better developed analytical tools. Thus, his more recent methods text refers to advances that have taken place in the 80s (as summarized for example in Fotheringham and O’Kelly 1989) and in a sense renews the cycle of interest and reaffirms the critical importance of spatial interaction to regional science. This cumulative build up of insight, analytical accomplishment, and training of future researchers is itself the kind of institutionalization of ideas that leads to the consolidation of the field

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and solidifies received wisdom. While we can certainly anticipate future paradigm shifts and rethinking, it is extraordinarily hard to imagine a regional science and spatial analysis that would not teach the foundation of spatial interaction model from a rigorous base. That base itself has evolved and increasingly relies on theories from spatial information processing capabilities of human decision makers (Fotheringham et al. 2000, Chapter 9 summarize this argument persuasively). The main initial breakthrough in spatial interaction models came with the publication of Wilson’s (1970) ‘‘Entropy ..’’ and later his ‘‘Urban and Regional Models in Geography and Planning.’’ (Wilson 1974). It was the consistent derivation of gravity and related models from the strong foundations of statistical mechanics (and later information theory) that empowered a generation of urban and regional modelers to devise a new set of tools for spatial interaction. Other breakthroughs (such as Alonso, Anas, Fotheringham, …) and others could not have come from an ‘‘ad hoc’’ based gravity model with its mysterious ‘‘k’’ factors, nor indeed was the foundation in the transportation modeling literature particularly favorable to generalization given its emphasis on histograms of empirical travel time distributions. It was Wilson’s insight to treat the travel impedance constraint as a limitation on the most probable interactions, and from that to derive the probabilistic foundations for the family of spatial interaction models. Depending on the amount of ‘‘constraint’’ inherent in the production and attraction of trips, the model could be made to take on one of several formats, all more or less equally easy to calibrate and refine. This was exactly the kind of open tool kit that the generation of post-Lowry modelers needed. Given this vote of approval for the role of Wilson and his work, are there further kudos to go around? I would argue that the role of Isard’s ‘‘Methods,’’ book, was to lay the groundwork for the Wilson refinements, and that only through the recognition that the gravitational model needed to be set in the context of a more rigorous formalism was it possible to make the tools that further advances in models necessitated. It is clear from the many citations listed in the references that the idea of linking trade flows and regional economic models, inherent in Isard’s own early writing, also received a boost through the provision of improved model building tools. It also served to galvanize a generation of researchers, who, like myself, felt a need for a more rigorous basis for these models, and that they offered far more than an ad hoc set of empirical formulae. In sum though, Isard’s influence in this area of research has been indirect rather than direct. It was a great surprise to me in writing this essay to look at the sparse direct acknowledgement of Isard’s work in citation in key texts. A series of books in the series on studies in regional science and urban economics (Fujita 1978; Beckmann and Puu 1985 for example) make passing reference to Isard. Wilson (1974) in ‘‘Urban and Regional Models’’ cites Isard in a handful of places but not for specific research findings. These works do however draw on and emphasize the early creative inspiration in input-output models provided by Isard. A great deal of relatively futile writing was done on the subject of analogies and the role of analogies in reasoning, and cases being made for various foundational derivations for the gravity equations, from entropy maximiza-

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tion (Wilson 1970, Webber 1976) to utility theory (Beckmann 1973, Niedercorn and Bechdoldt 1969) and so on. This trend continues to this day, with the recent publication of Isard’s (1999) paper: ‘‘Regional science: Parallels from physics and chemistry.’’ I am not convinced that this form of analogy yields anything other than mathematical curiosities. From the abstract of that paper we read Isard’s current assessment: Against the background of the fruitful application of the gravitational force from physics in the development of the gravity model extensively employed in regional science, possible parallels from the analysis of the coplay of the four basic forces of particle physics are examined. In particular, the possible parallels from the analysis of the bondings of the strong force and the diverse ones of the electromagnetic force, and specifically how these parallels can replace the relatively outmoded set of concepts of scale, localization and urbanization economies for regional shopping center analysis and related urban structure theory are studied.

At these analogies, I think I would draw the line. I am of the opinion that the most fruitful avenues for insight to the realm of human spatial interaction is most likely to come from social and behavioral sciences, including psychology (search; behaviors in uncertainty and Luce choice axioms) economics (random utility) and geography (spatial organization). For example Sheppard (1978) makes these foundations clear in his discussion of theoretical underpinnings for the gravity model. Thus, there is no necessity for the analogy to be pushed to the ultimate limit in order for it to provide some useful insights. Simply thinking laterally as inspired by the derivation of most likely states, from the point of view say of information theory, or dealing with the complexities of spatial choice through information processing strategies gives adequate inspiration and in my opinion opens fascinating parallels. I do not think that there is need for a physical science ‘‘imprimatur’’ to validate the essential mechanisms that govern spatial interaction. On that one somewhat critical note then I will stop. The metaphors and analogies have served us well; but the fascination with a possible reification of physical science analogies may take us away from the basic social and behavioral core concepts underlying spatial interaction and its modeling. We have been well-served by Isard’s integrative thinking about spatial interaction, as indicated by the selected areas singled out. From the much larger number of topics identifiable from the citations, it is clear that this has been an extraordinarily fruitful area for regional science and geography. References Bibliographic notes: Items with a * are from works citing Walter Isard in the area of spatial interaction. These were selected using key words ‘‘flow’’ or ‘‘gravity’’ or ‘‘interaction.’’ Other papers are listed without necessarily having a citation in the body of the text to convey some examples of the range of influence of interaction modeling. Almeida LMW, Goncalves MB (2001) A methodology to incorporate behavioral aspects in tripdistribution models with an application to estimate student flow. Environment and Planning A 33(6): 1125–1138 Allen PM, Sanglier M (1979) A dynamic model of growth in a central place system –I, Geographical Analysis 11: 256–272 Allen PM, Sanglier M (1981) A dynamic model of growth in a central place system – II, Geographical Analysis 13: 149–164

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Alonso W (1978) A theory of movement. In: Hansen NM (ed) Human Settlement Systems Ballinger: Cambridge Amrhein CG, Flowerdew R (1992) The effect of data aggregation on a poisson regression-model of canadian migration. Environment and Planning A 24(10): 1381–1391 Anas A (1983) Discrete choice theory, information-theory and the multinomial logit and gravity models. Transportation Research Part B-Methodological 17(1): 13–23 Anselin L (1982) Implicit functional relationships between systemic effects in a general model of movement. Regional Science and Urban Economics 12: 365–380 Bailey TC, Munford AG (1994) Modeling a large, sparse spatial interaction matrix using data relating to a subset of possible flows. European Journal of Operational Research 79(3): 489–500 Baker RGV (1994) An assessment of the space-time differential model for aggregate trip behavior to planned suburban shopping-centers. Geographical Analysis 26(4): 341–363 Baker RGV (2000) Towards a dynamic aggregate shopping model and its application to retail trading hour and market area analysis. Papers in Regional Science 79(4): 413–434 * Barnes TJ, Curry MR (1992) Postmodernism in economic-geography - metaphor and the construction of alterity. Environment and Planning D-Society & Space 10(1): 57–68 * Batten DF (2001) Complex landscapes of spatial interaction. Annals of Regional Science 35(1): 81–111 Batty M, Sikdar PK (1982) Spatial aggregation in gravity models .1 An information- theoretic framework. Environment and Planning A 14(3): 377–405 Batty M, Sikdar PK (1982) Spatial aggregation in gravity models .2 One-dimensional population-density models. Environment and Planning A 14(4): 525–553 Batty M, Sikdar PK (1982) Spatial aggregation in gravity models .3 Two-dimensional trip distribution and location models. Environment and Planning A 14(5): 629–658 Batty M, Sikdar PK (1982) Spatial aggregation in gravity models .4 Generalizations and largescale applications. Environment and Planning A 14(6): 795–822 Bavaud F (1998) Models for spatial weights: a systematic look. Geographical Analysis 30(2): 153– 171 Bavaud F (2002) The quasi-symmetric side of gravity modelling. Environment and Planning A 34(1): 61–79 Beaumont JR (1980) Spatial interaction models and the location-allocation problem. Journal of Regional Science 20(1): 37–50 * Beaumont PM (1990) Supply-and-demand interaction in integrated econometric and inputoutput models. International Regional Science Review 13(1–2): 167–181 Beckmann MJ, Puu T (1985) Spatial economics: density, potential, and flow. North Holland Amsterdam Bennett RJ, Haining RP, Wilson AG (1985) Spatial structure, spatial interaction, and their integration: a review of alternative models. Environment and Planning A 17: 625–645 * Black WR (1991) A note on the use of correlation-coefficients for assessing goodness-of-fit in spatial interaction models. Transportation 18(3): 199–206 Black WR (1995) Spatial interaction modeling using artificial neural networks. Journal of Transport Geography 3: 159–166 * Brown S (1992) The wheel of retail gravitation. Environment and Planning A 24(10): 1409–1429 Brown WM, Anderson WP (2002) Spatial markets and the potential for economic integration between Canadian and US regions. Papers in Regional Science 81(1): 99–120 Bucklin LP (1971) Retail gravity models and consumer choice: a theoretical and empirical critique. Economic Geography 47: 489–497 Cadwallader M (1981) Towards a cognitive gravity model - the case of consumer spatialbehavior. Regional Studies 15(4): 275–284 * Carlino G, Lang R (1989) Interregional flows of funds as a measure of economic- integration in the united-states. Journal of Urban Economics 26(1): 20–29 Cliff A, Ord JK (1974) Evaluating the friction of distance parameter in gravity models. Regional Studies 8: 281–286 Congdon P (2000) A Bayesian approach to prediction using the gravity model, with an application to patient flow modeling. Geographical Analysis 32(3): 205–224 Curry L (1972) A spatial analysis of gravity flows. Regional Studies 6: 131–147

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Curry L, Griffith DA, Sheppard ES (1975) Those gravity parameters again. Regional Studies 9: 289–296 Drezner T, Drezner Z (2001) A note on applying the gravity rule to the airline hub problem. Journal of Regional Science 41(1): 67–73 Esparza A, Krmenec AJ (1994) Business services in the space economy: a model of spatial interaction. Papers in Regional Science 73: 55–72 Esparza A, Krmenec AJ (1996) The spatial extent of producer service markets: hierarchical models of interaction revisited. Papers in Regional Science 75: 375–395 Fik TJ (1988) Hierarchical interaction: the modeling of a competing central place system. Annals of Regional Science 22: 48–69 Fik TJ, Mulligan GF (1990) Spatial flows and competing central places: towards a general theory of hierarchical interaction. Environment and Planning A 22: 527–549 Fik TJ, Mulligan GF (1998) Functional form and spatial interaction models. Environment and Planning A 30(8): 1497–1507 Fischer MM (1998) Computational neural networks: a new paradigm for spatial analysis. Environment and Planning A 30(10): 1873–1891 Fischer MM, Reismann M (2002) A methodology for neural spatial interaction modeling. Geographical Analysis 34(3): 207–228 * Flowerdew R, Aitkin M (1982) A method of fitting the gravity model based on the poissondistribution. Journal of Regional Science 22(2): 191–202 * Foot DK, Milne WJ (1984) Net migration estimation in an extended, multiregional gravity model. Journal of Regional Science 24(1): 119–133 Fotheringham AS (1983) A new set of spatial-interaction models: the theory of competing destinations. Environment and Planning A 15: 15–36 Fotheringham AS (1983) Some theoretical aspects of destination choice and their relevance to production-constrained gravity models. Environment and Planning A 15(8): 1121–1132 Fotheringham AS, Dignan T (1984) Further contributions to a general theory of movement. Annals, Association of American Geographers 74: 620–33 Fotheringham AS, O’Kelly ME (1989) Spatial interaction models: formulations and applications. Dordrecht Netherlands: Kluwer Academic Fotheringham AS, Webber MJ (1980) Spatial structure and the parameters of spatial interaction models. Geographical Analysis 12: 33–46 Fotheringham AS, Wong D (1991) The modifiable areal unit problem in multivariate statisticalanalysis. Environment and Planning A 23(7): 1025–1044 Fotheringham AS, Brunsdon C, Charlton M (2000) Quantitative Geography: Perspectives on Spatial Data Analysis Sage Publications, London [especially chapter 9 Spatial modeling and the evolution of spatial theory] Fujita M (1990) Additive-interaction models of spatial agglomeration. Journal of Regional Science 30: 51–74 Getis A (1991) Spatial interaction and spatial autocorrelation: a cross- product approach. Environment and Planning A 23: 1269–1277 * Goodchild MF, Anselin L et al (2000) Toward spatially integrated social science. International Regional Science Review 23(2): 139–159 Goodchild MF, Kwan M (1978) Models of hierarchically dominated spatial interaction. Environment and Planning A 10: 1307–1317 Greenwood M, Hunt (2003) International Regional Science Review vol 26: 3–37 Griffith DA, Haining RP (1986) Transformations Through Space and Time. Nijhoff Dordrecht Guldmann JM, Wang FH (1998) Population and employment density functions revisited: a spatial interaction approach. Papers in Regional Science 77(2): 189–211 Guy CM (1987) Recent advances in spatial interaction modelling: an application to the forecasting of shopping travel. Environment and Planning A 19: 173–186 Guy CM (1991) Spatial interaction modeling in retail planning practice – the need for robust statistical-methods. Environment and Planning B-Planning & Design 18(2): 191–203 Haggett P, Cliff AD, Frey A (1977) Locational Models. Wiley New York Hallefjord S, Jornsten K (1986) Gravity models with multiple objectives–theory and applications. Transportation Research B 20B: 19–39

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* Harrigan F, McGilvray JW et al (1981) The estimation of inter-regional trade-flows. Journal of Regional Science 21(1): 65–78 Harris B, Wilson AG (1978) Equilibrium values and dynamics of attractiveness terms in production constrained interaction models. Environment and Planning A 10: 371–388 * Haynes KE (1997) Labor markets and regional transportation improvements: the case of highspeed trains - An introduction and review. Annals of Regional Science 31(1): 57–76 Haynes KE, Fotheringham AS (1984) Gravity and spatial interaction models. Beverly Hills CA: Sage Hopkins LD, Los M (1979) Location-allocation algorithms for land use plan design with fixed and substitutable interactions. Journal of Regional Science 19(3): 345–361 * Hua C (1990) A flexible and consistent system for modeling interregional trade-flows. Environment and Planning A 22(4): 439–457 Irwin MD, Hughes HL (1992) Centrality and the structure of urban interaction: measures, concepts, and applications. Social Forces 71: 17–51 Isard W (1956) Location and Space-Economy. MIT Press, Cambridge Isard W (1960) Methods of Regional Analysis. MIT Press, Cambridge * Isard W (1999) Regional science: parallels from physics and chemistry. Papers in Regional Science 78(1): 5–20 Janelle DG (1997) In memoriam: William Warntz, 1922–1988, Annals of the Association of American Geographers 87(4): 723–731 * Karemera D, Oguledo VI et al (2000) A gravity model analysis of international migration to North America. Applied Economics 32(13): 1745–1755 Kawate H, Oyama T (1994) Space interaction-model analyses for the interregional telephone call flows in the metropolitan-areas. Journal of the Operations Research Society of Japan 37(2): 114–132 * Kim TJ, Boyce DE, Hewings GJD (1983) Combined input-output and commodity flow models for interregional development-planning - insights from a korean application. Geographical Analysis 15(4): 330–342 * Kim TJ, Ham H et al (2002) Economic impacts of transportation network changes: implementation of a combined transportation network and input- output model. Papers in Regional Science 81(2): 223–246 Kirby HR (1974) Theoretical requirements for calibrating gravity models. Transportation Research 8: 97–107 Krmenec AJ, Esparza A (1993) Modeling interaction in a system of markets. Geographical Analysis 25: 354–368 Kwan MP (1998) Space-time and integral measures of individual accessibility: a comparative analysis using a point-based framework. Geographical Analysis 30(3): 191–216 Ledent J (1981) On the relationship between alonso theory of movement and Wilson family of spatial-interaction models. Environment and Planning A 13(2): 217–224 Leonardi G (1983) The use of random-utility theory in building location-allocation models. In: Thisse J-F, Zoller H (eds) Locational, Analysis of Public Facilities, North Holland Amsterdam pp 357–383 Liew CK, Liew CJ (1991) A multiregional, multiproduct, household interactive, variable inputoutput model. Annals of Regional Science 25(3): 159–177 Linthorst JM, van Praag B (1981) Interaction-patterns and service-areas of local public services in the Netherlands. Regional Science and Urban Economics 11: 39–56 Lo L (1991) Spatial structure and spatial interaction - a simulation approach. Environment and Planning A 23(9): 1279–1300 Lo L (1991) Substitutability, spatial structure, and spatial interaction. Geographical Analysis 23(2): 132–146 Lo L (1992) Destination interdependence and the competing-destinations model. Environment and Planning A 24(8): 1191–1204 Long WH (1970) Air travel, spatial structure, and gravity models. Annals of Regional Science 4: 97–107 Luoma M, Palomaki M (1983) A new theoretical gravity model and its application to a case with drastically changing mass. Geographical Analysis 15(1): 14–27

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Luoma M, Palomaki M (1993) The threshold gravity model and transport geography. Journal of Transport Geography 1: 240–247 Masser I, Brown PJ (1975) Hierarchical aggregation procedures for interaction data. Environment and Planning A 7: 509–523 Masser I, Scheurwater J (1980) Functional regionalisation of spatial interaction data: an evaluation of some suggested strategies. Environment and Planning A 12: 1357–1382 Matthes N (1994) Allocation of mobile communication flows - from microeconomic demand theory to a gravity model. Annals of Regional Science 28(4): 395–409 * Miller HJ (1999) Potential contributions of spatial analysis to geographic information systems for transportation (GIS-T). Geographical Analysis 31(4): 373–399 Miller HJ, O’Kelly ME (1991) Properties and estimation of a production-constrained alonso model. Environment and Planning A 23(1): 127–138 Murray AT (2000) Spatial characteristics and comparisons of interaction and median clustering models. Geographical Analysis 32(1): 1–18 * Moore C, Nagurney A (1989) A general equilibrium-model of interregional monetary flows. Environment and Planning A 21(3): 397–404 Neidercorn JA, Bechdolt BV (1969) An economic derivation of the ‘gravity law’ of spatial interaction. Journal of Regional Science pp 273–282 Nijkamp P, Poot J (1987) Dynamics of generalized spatial interaction models. Regional Science and Urban Economics 17(3): 367–390 Noronha VT, Goodchild MF (1992) Modelling interregional interaction: implications for defining functional regions. Annals of the Association of American Geographers 82: 86–102 O’Kelly ME (1999) Trade-area models and choice-based samples: methods. Environment and Planning A 31(4): 613–627 O’Kelly ME, Bryan D (2002) Interfacility interaction in models of hub and spoke networks. Journal of Regional Science 42(1): 145–164 O’Kelly ME, Song W, Shen Q (1995) New estimates of gravitational attraction by linearprogramming. Geographical Analysis 27(4): 271–285 * Oregan KM, Quigley JM (1996) Spatial effects upon employment outcomes: the case of New Jersey teenagers. New England Economic Review pp 41–58 Ottensmann JR (1995) Using a gravity model to predict circulation in a public library system. Library & Information Science Research 17(4): 387–402 Ottensmann JR (1997) Partially constrained gravity models for predicting spatial interactions with elastic demand. Environment and Planning A 29(6): 975–988 * Plane DA, Mulligan GF (1997) Measuring spatial focusing in a migration system. Demography 34(2): 251–262 Plane DA (1982) Redistricting reformulated: maximum interaction/minimum separation objective. Socio-Economic Planning Sciences 16(6): 241–244 Plane DA (1984) Migration space - doubly constrained gravity model mapping of relative interstate separation. Annals of the Association of American Geographers 74(2): 244–256 Pooler J (1992) Spatial uncertainty and spatial dominance in interaction modelling: a theoretical perspective on spatial competition. Environment and Planning A 24: 995–1008 Pooler J (1993) Structural spatial interaction. Professional Geographer 45(3): 297–305 Pooler J (1994) An extended family of spatial interaction models. Progress in Human Geography 18(1): 17–39 Pooler J (1994) A family of relaxed spatial interaction models. Professional Geographer 46(2): 210–217 * Popkov YS, Shvetsov VI et al (1998) Settlement formation models with entropy operator. Annals of Regional Science 32(2): 267–294 * Porell FW, Hua C (1981) An econometric procedure for estimation of a generalized systemic gravity model under incomplete information about the system. Regional Science and Urban Economics 11(4): 585–606 Porojan A (2001) Trade flows and spatial effects: the gravity model revisited. Open Economies Review 12(3): 265–280

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* Prentice BE, Wang ZK et al (1998) Derived demand for refrigerated truck transport: a gravity model analysis of Canadian pork exports to the United States. Canadian Journal of Agricultural Economics 46(3): 317–328 * Relethford JH (1986) A gravity model of human-population structure. Human Biology 58(5): 801–815 Rietveld P, Janssen L (1990) Telephone calls and communication barriers - the case of the Netherlands. Annals of Regional Science 24(4): 307–318 * Rogerson P, Mackinnon RD (1982) Inter-regional migration models with source and interaction information. Environment and Planning A 14(4): 445–454 Roy JR (1990) Spatial interaction modelling: some interpretations and challenges. Environment and Planning A 22: 712–716 Roy JR (1999) Areas, nodes and networks: some analytical considerations. Papers in Regional Science 78(2): 135–155 * Scott AJ (2000) Economic geography: the great half-century. Cambridge Journal of Economics 24(4): 483–504 * Serck-Hansen J (1970) Optimal Patterns of Location. North Holland, Amsterdam Sen A, Pruthi RK (1983) Least-squares calibration of the gravity model when intrazonal flows are unknown. Environment and Planning A 15(11): 1545–1550 Senior ML (1979) From gravity modelling to entropy maximizing: a pedagogic guide. Progress in Human Geography 3: 175–210 Shen GQ (1999) Estimating nodal attractions with exogenous spatial interaction and impedance data using the gravity model. Papers in Regional Science 78(2): 213–220 Sheppard ES (1978)Theoretical underpinnings of the gravity hypothesis. Geographical Analysis 10: 386–402 Slater PB (1985) Point-to-point migration functions and gravity model renormalization approaches to aggregation in spatial interaction modeling. Environment and Planning A 17(8): 1025–1044 Smith TE (1987) Poisson gravity models of spatial flows. Journal of Regional Science 27(3): 315– 340 Smith TE (1991) A simple decision-theory of spatial interaction - the alonso model revisited. Environment and Planning A 23(9): 1247–1268 * Sonis M (1981) Flows, hierarchies, potentials. Environment and Planning A 13(4): 413–420 * Srivastava RK, Green RT (1986) Determinants of bilateral trade-flows. Journal of Business 59(4): 623–640 Tabuchi T (1984) The systemic variables and elasticities in alonso general- theory of movement. Regional Science and Urban Economics 14(2): 249–264 Tabuchi T (1986) Existence and stability of city-size distribution in the gravity and logit-models. Environment and Planning A 18(10): 1375–1389 * Takayama T, Judge GG (1971) Spatial and Temporal Price and Allocation Models. North Holland Amsterdam Taylor PJ (1975) Distance decay in spatial interactions. Norwich UK: Geo Abstracts Thorsen I, Gitlesen JP (1998) Empirical evaluation of alternative model specifications to predict commuting flows. Journal of Regional Science 38(2): 273–292 Tiefelsdorf M, Boots B (1995) The specification of constrained interaction models using the SPSS loglinear procedure. Geographical Systems 2: 21–38 Tobler W (1977) Spatial interaction patterns. Journal of Environmental Systems 6: 271–301 * Vanlierop W, Nijkamp P (1980) Spatial choice and interaction models - criteria and aggregation. Urban Studies 17(3): 299–311 * Webber MJ (1972) Impact of uncertainty on location. MIT Press, Cambridge Webber MJ (1976) The meaning of entropy maximizing models. In: Papageorgiou GJ (ed) Mathematical Land Use Theory. Lexington Books, Heath Weidlich W, Haag G (1986) Stochastic migration theory and migratory phase transitions. In: Griffith DA, Haining RP (eds) Transformations Through Space and Time, Nijhoff Dordrecht * Weksler I, Freeman D et al (1986) Estimation of interregional trade-flows - a markov-chain approach. Environment and Planning A 18(1): 123–132

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* Willekens F (1994) Monitoring international migration flows in europe - towards a statisticaldata base combining data from different sources. European Journal of Population 10(1): 1–42 Wilson AG (1970) Entropy in urban and regional modelling. Pion London Wilson AG (1971) A family of spatial interaction models, and associated developments. Environment and Planning 3: 1–32 * Wilson AG (1974) Urban and regional models in geography and planning. Wiley New York