
A mathematical model which was devised to represent a wide range of flow patterns in human geography (migration, telephone traffic, passenger movements, commodity flow, etc.) and subsequently used and further developed as a planning tool.
The original model, proposed by exponents of social physics, was based on a crude analogy with Newton\'s gravitational equation (see metaphor):
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This can be interpreted as follows: the gravitational force (Gij) between two masses (Mi and Mj) is proportional to a gravitational constant (g) and to the product of their masses (MiMj) and inversely proportional to the square of the distance between them (dij2).
The analogy for migration was therefore given as:
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where the migrant flow (F) from i to j was modelled as being proportional to the product of their populations. In such an application the constant g was empirically determined from the data set by simple arithmetic methods. At a later stage the model was fitted by regression methods in logarithmic form:
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In this form both g and the exponent for distance b were empirically determined by calibration with the data set.
Planning applications of these models soon revealed that they gave poor fits to real data sets, so ad hoc adjustments were made to the form of the model. Some focused upon the relationship with distance by fitting, for example, the exponential model:
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where e is the root of Napierian logarithms. Other adjustments were made to the P terms to ensure that the flows predicted by the model either from destinations, or to origins, or both, equalled the actual flow. Where only one of these was attempted the model was described as origin constrained or destination constrained, but where both were adjusted the model was termed doubly constrained and took the form:
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In this form the new symbols Ai and Bj were calibrating constants which had to be empirically determined by an iterative procedure. But such forms were a long way from the original analogy from physics and a stronger rationale was required. There were several attempts to do this in terms of likelihood maximizing, utility maximizing and entropy maximizing. The last, due to Wilson (1974), has been widely accepted; it has the added advantage that it demonstrates the close links between gravity models on the one hand and competing models based on intervening opportunity and the transportation problem in linear programming.
Despite its problems the gravity model is widely used in transport planning: the great variety of possible mathematical formulations means that an approximate fit to empirical data can nearly always be achieved. On the other hand, those who believe that its theoretical basis is too weak fear that it will be a bad predictive tool and advocate other methods (for example transportation models), and others note that it will lead to a perpetuation or intensification of existing patterns even where these are undesirable in terms of environmental issues or distributional equity (Sayer, 1971). (See also distance decay; friction of distance.)Â (AMH)
References Sayer, A. 1971: Gravity Models and spatial autocorrelation, or atrophy in urban and regional modelling. Area 9: 1839.Â Wilson, A.G. 1974: Urban and regional models in geography and planning. Chichester: John Wiley.
Suggested Reading Fotheringham, A.S. 1991: Migration and spatial structure: the development of the competing destination model. In J.C. Stilwell and P. Compton, eds, Migration models. London: Belhaven, 5772.Â Senior, M.L. 1979: From gravity modelling to entropy maximising: a pedagogic guide. Progress in Human Geography 3: 179211.Â Tocalis, T.R. 1978: Changing theoretical foundations for the gravity concept of human interaction. In B.J.L. Berry, ed., The nature of change in geographic ideas. De Kalb, IL: Northern Illinois University Press, 66124. 
