GRAVITY MODELS : USE AND THEORY
1. WHAT IS GRAVITY THEORY?
Everyone is familiar with gravity, the force of attraction between two masses.
F = GM1M2/d2 Gravitational force between two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them. The application of the theory to inanimate masses is common knowledge: gravity is the force which maintains the planets in stately procession around the sun, the force which keeps our feet firmly planted on solid ground, the force which supposedly brought the apple down on Isaac Newton's head in 1686. However, to a number of economists and sociologists the concept of gravity has applicability far beyond its ordinary physical usage. To these scientists the physical masses can be replaced by population masses, gravity becoming an attractive force generated when people concentrate in areas, the gravitational equation becoming a social law explaining interaction between masses of people.
H.C. Carey in 1858 was the first to propose that gravity could be applied to human interaction.1 But the concept was not rigorously developed until the 1940's and the work of John Q. Stewart2 and George K. Zipf.3 Stewart coins the phrase "social physics" to describe a whole constellation of new social laws. His reasoning is that laws ruling the macroscopic realm can only be discovered by examining macroscopic units. Gravity is apparent among the motions of the planets, it is not apparent among the motions of single atoms. Likewise, if there are laws governing and giving order to human societies, then these laws can only be discovered by observation of humans in masses. In its earliest form social physics involved the finding of social analogs to physical laws. Thus gravitational force GM1M2/d2 becomes demographic force GP1P2/d2 by substituting populations for masses. Stuart C. Dodd extends this reasoning with his interactance hypothesis,4 which purports to encompass both physical law and social law as subsets of a general principle of how things interact.
Before going into detail concerning the structure and use of gravity models, it is useful to point our three major differences in the way in which social physics and more traditional economic theory approach the formulation of theory.
First, traditional economic theory is built upon microeconomic concepts. Demand theory begins with a preference function of an individual, labor supply theory begins with the income vs. Leisure choice of the individual, marginal theory is concerned with the impact of a single additional unit, and so on. Gravity theories, as previously stated, use a macroscopic approach, viewing microscopic study as ill-suited for the discovery of general social laws. As a result of these opposing viewpoints, the general areas of investigation also differ. Traditional economic theory concentrates on optimization behavior of individuals and static equilibrium. Gravity theories look for overall relationships and lend themselves to dynamic equilibrium analysis. Gravity models can be used in optimization problems, but that has not been the main thrust of theoretical effort.
Second, traditional theory is concerned with motives and explanations of behavior. Individuals are assumed to maximize utility, firms to maximize profits. Social results can be understood in terms of the goals sought by individuals. Gravity theory disputes the need to understand motives. "Continued insistence on examining people's purposes and motives only blocks the way to a science of society as a whole."5 Again, one can not grasp the macroscopic by understanding the goals of microscopic units. Either social laws are independent of economic motives, or else the motives form a neutral background against which social lwas operate. Although Zipf advances his principle of least effort as a motive, it is a motive unconsciously sought by society as a whole, not actively pursued by individuals.
Third, traditional economics views the whole as being basically the sum of its parts. Individual demand aggregates to market demand, individual firms aggregate to an industry. One can study the various sectors of the economy separately and them combine them with a minimum of additional considerations. Gravity theory says that the above is a gross oversimplification which ignores the most powerful forces acting upon society as a whole. Whether Stewart's demographic force and demographic cohesion, Zipf's forces of unification and diversification, or Isard's forces of agglomeration and deglomeration,6 forces which develop as a result of the operation of society as a whole are believed to play a crucial role in the functioning of society. Variations in the density of human activity are crucial in determining the functional form of a society. One city of 1,000,000 people is not equivalent to 10 cities of 100,000.
As these differences indicate, gravity theory is far more than a handy analytic tool for economic research. The field of social physics represents a radically different philosophical approach to economic and social questions of incredible complexity. Gravity theory has already revolutionized the field of traffic flow projection. If some thorny operational problems can be overcome, it promises to do te same for regional economics, economic growth theory, location theory, and many other frontier areas of economic research.
2. GENERAL FORM OF A GRAVITY MODEL
Numerous gravity models with slightly different forms and characteristics have been developed and tested. Differences between the equations used by Reilly,7 Stewart, and Zipf are a result of their independent development of gravity concepts. Differences between other models (for example, Federal Highway Administration gravity model)8 are a result of modifications made to match specific user needs. All gravity models have this in common: that the level of some social activity between population points is directly proportional to some measure of the mass of each point, and inversely proportional to some measure of the distance between the two points.
The most general statement of the gravity model is that of Dodd's interactance hypythesis. Most other models either implicitly or explicitly assume some of the parameters to be unity. Dodd hypothesizes that interactance (amount of interacting) is a function of time, distance, population size, indices of level (per capita activity), and a constant.
Ie = K(IAPaA)(IBPbB)Tc/Dd
where: Ie = interactance
K = a constant
PA,PB = population means
T = time period
D = distance
a,b,c,d = exponents
K corresponds to Newton's gravitational constant. It is a mathematical device to bring into equality two sides of an equation which are in constant proportion to one another. By expressing relationships as a ratio, the constant term can be eliminated from the equation. This is what Reilly does with his law of retail gravitation. P1/d12 = P2/d22
PA and PB are generally assumed to have exponents of unity. Exponents other than one would mean that interactance does not vary proportionately with population mass. This may be the case. Stewart finds greater activity in the center of cities than his social physics equations predict. He adjusts for this by proposing a new type of force called demographic cohesion.9 Selection of te proper exponent greater than one would have the same kind of effect.
IA and IB in social physics terminology are molecular masses. They measure the quality of the population in respect to the variable of interest. They are affected by natural resource endowments, relative levels of technology, and cultural differences. For example, if the variable of interest is volume of telephone conversations, then the indices of level would likely be affected by per capita telephone ownership and by cultural biases in favor of written over verbal communication.
T is generally expressed as unity (one month, one year, one day, etc.), and generally assumed to have an exponent of one.
D was originally thought to have an exponent of either 1 or 2, and it is argued by Stewart and Warntz on theoretical grounds that these are the only permissible values. Some empirical studies cast doubts on this conclusion. Also, the FHWA has success in their traffic flow model which replaces a single distance term with multiple impedence factors.
The structure of the interactance relation is a straight line on a double-log graph.
log Ie = log K + log IA + a log PA + log IB + b log PB + c log T - d log D
Some empirical studies suggest that a closer fit to observed data can be obtained by using something other than a log-linear function.10 However, the evidence for this is not compelling, and all major proponents of gravity theory claim a log-linear function.
3. IS SOCIAL PHYSICS FOR REAL?
Gravity theory has a number of enthusiastic supporters, but most social scientists have taken a "wait and see" attitude. Skepticism stems partly from the numerous operational difficulties of applying gravity models, and partly from doubt as to either the reality of the forces it describes or to the theory's ability to measure those forces. What arguments can be made to support the concept of social physics? What evidence can be presented to support the quantitative relationships claimed?
On a qualitative level, there is little disagreement that the major parameters of the gravity equation do operate in the directions predicted. As the population of a city increases, there is increased opportunity for many kinds of human interaction, both within the city and between other cities or rural areas. As distance increases, the costs of many kinds of interactions also increase, which acts to reduce the number of interactions. In fact, if one assumes homogeneous populations and zero costs of interaction, simple probability predicts the P1P2 numerator of the gravity equation. If costs of interaction are assumed to be proportional to distance, one can predicts D (straight line distance) or D2 (area) for the denominator. The objection to this is that the assumptions are extremely restrictive and the ceteris paribus provision includes a long list of potentially relevant variables for any imaginable real-life situation. As Roy Sampson puts it, "In local marketing, however, the distance factor may be such an insignificant portion of other cost and demand elements that it becomes submerged and of little effect."11
Some gravity theorists, notably Stewart, are content to state social "laws" without any attempt to provide a theoretical justification. Others attempt to do so. Zipf constructs his P/D and P1P2/D factors from his principle of least effort. This principle basically states that individuals organize themselves into societies so as to accomplish desired goals with the minimum amount of anticipated work, as perceived by the individuals. Distance is correlated to work both by the work of moving over distances and also by the work of gaining information over distances. However, as Isard points out,12 the logic connecting Zipf's qualitative principle and quantitative relationships is not at all clear.
Samuel A. Stouffer hypothesizes a gravity relationship based on the concept of intervening opportunities.13 This explains that the number of interactions falls off with distance because the number of opportunities for interaction one must pass up increases with distance. Thus opportunity is explicitly identified as the variable of cause. However, the hypothesis of intervening opportunities is directed specifically to the question of human migration and thus is too limited to justify gravity theory in general.
The chief argument in favor of gravity models, and the origin of their development, is empirical evidence. All gravity theorists, led by Zipf and Stewart, have searched for and found rectilinear (log-linear) relationships for numerous and varied economic and social activities. Rank size of cities, railroad freight movements, telephone messages, rural land values, newspaper obituaries, attendance at national parks, all have been shown to have rectilinear form. Gravity theory was developed to explain these empirical findings.
When graphed on double-log paper, the predictive power of the gravity model for many very diverse activities is visually striking. There definitely appears to be some general social law of interactance in operation. The burden of an alternative explanation of a large and growing body of data is place upon anyone who would reject gravity concepts. However, closer examination of the data indicates that it is not quite so compelling. Many researchers do not calculate statistical measures of correlation to accompany their graphs. When this is done, as by Cavanaugh,14 the results are significant, but not especially impressive. It is difficult not to feel a little suspicious when measures of explanatory power are given in terms of the coefficient of correlation instead of the more relevant (and smaller) coefficient of determination. The slopes of the double-log straight lines found by Zipf all fall within the bounds of statistical error around the predicted value of one. However, Zipf notes a tendency for many of the slopes to approach 0.8. This he can not explain. As mentioned before, Stewart's predicted values for interaction within cities are so low that they cause him to hypothesize a new type of force: demographic cohesion. Numerous researchers find exponent values different than those proposed by Stewart and Zipf.
In short, the empirical evidence is strong, but does not of itself prove the case for gravity theory. There are too many unexplained irregularities in the data. A prudent reading of the evidence is that there is something there, some type of force at work across broad ranges of human activity; but that it is unclear if current gravity models accurately describe the workings of this force.
Although gravity theory has not had as large an impact on theory and research as its developers feel it deserves, it has had some notable successes. The Federal Highway Administration has great success with a sophisticated gravity model incorporating socio-economic factors as needed, and relying on survey feedback to "calibrate" the model to a high level of accuracy. Also, the principles of social physics are not limited to economics, but are being applied in other disciplines. Thus Stephen Jay Gould15 uses Zipf's theory of variation, a corollary of his principle of least effort, to explain the massive extinction of species at the end of the Permian era: the uniting of the earth's land masses into the supercontinent Pangaea reduced the total area of shallow seas: a smaller area supports fewer distinct species. Also, Lewis Thomas16 speculates on the similarity between human and insect societies, and that humans may be driven by societal forces (forces generated when people congregate in masses) in much the same way that bees and termites are. Although many may object to being compared to termites or molecules, and indeed there seems a great gulf between location theory and the extinction of trilobites, remember that Dodd's interactance hypothesis claims to explain how all things interact.
4. ILLUSTRATION OF OPERATIONAL DIFFICULTIES
Once a decision is made to apply gravity model analysis to some aspect of human interaction, there are five basic questions which must be answered.
1. Into how many masses will the area in question be divided, and along what lines?
2. What will be used as a measure of mass?
3. What will be used as a measure of distance?
4. What weights will be assigned to the masses?
5. What form of the general interactance formula will be used; what exponents will be given to the parameters?
These five questions will be discussed in the context of a gravity model analysis of membership patterns at the First Christian Church of Falls Church. The study, performed expressly for this paper, illustrates difficulties inherent in the choice and measurement of variables and of data availability. The hypothesis tested is that residence patterns among members of the First Christian Church of Falls Church can be predicted by a gravity model; that is, total church membership from a particular community is directly proportional to the mass of the community and inversely proportional to the distance from the church.
The first question is how to determine the masses to be used. The first consideration is to define the entire region relevant to the analysis. The proper size for the region varies depending upon the type of interaction being studied. For example, a study of DC Metro subway ridership can effectively be limited to the Washington, DC SMSA. However, a study of commuting patterns of Connecticut residents can not be limited to the state of Connecticut, because the proximity of New York City certainly has a large effect. Basically, the region must be chosen so as to enclose the great bulk of the interactions under study. In the church membership study the relevant region is defined as Arlington County, City of Alexandria, City of Fairfax, and Fairfax County. Only a few church members live outside the Northern Virginia area.
Once the region has been selected, it must be divided into the masses to be used in the gravity equation. Division is generally made along administrative lines, primarily for reasons of data availability. Care should be taken not to group together areas with widely divergent socio-economic or cultural characteristics, since this will greatly complicate the later assignment of weight to the masses. For the church membership study the following divisions are considered.
a. Real estate areas, as suggested by Northern Virginia Association of Realtors publications.
b. Map grids.
c. Zip code areas.
d. Local government, cities and counties.
Only (c) and (d) prove feasible. Choice (d) is rejected because it yields only three mass points. The region is divided into 27 zip code masses. Outer portions of Fairfax County are excluded because they have no First Christian Church membership.
The second question is what to use as a measure of mass. Population is the measure most widely used, but it is not always the most appropriate measure. For a study of library use in an underdeveloped country, the literate population is a more appropriate measure. For expenditure studies, income is a better measure of mass. In each case the best measure of mass is that which most clearly responds to the attractive force under study. In the church membership study the preferred measure of mass is number of Christian church-attending families, since this defines total potential church membership at a point in time (church membership data is by family). However, this data is not available. The best available proxy is 1970 total population figures.
The third question is what to use as a measure of distance. The most commonly used measure is straight line distance in miles, and for national or broad regional studies encompassing large areas this is a good choice. For more localized studies better measures are distance along major traffic routes, travel time, or a weighted combination of the two. In industrial location studies a measure of transport cost (economic distance) should be used. In each case the best measure of distance is that which most closely represents the forces which impede the interaction being studied. For the church membership study the preferred measure of distance is travel time. Membership data is from 1978, prior to gasoline shortages and the recent leaps in fuel cost, so that time in transit rather than actual mileage is the more significant impedence. Lacking data on travel time, straight line distance is used as an acceptable proxy for the preferred measure.
The fourth question is whether the masses need to be weighted to reflect qualitative differences in the populations, and if so, how this weighting is to be done. Two general approaches are used. The method cited by Isard is to weight the population masses by socio-economic factors felt to be relevant to the issue at hand. For example, in a study of the distribution of high-class fashion stores the masses can reasonably be weighted by average income, since an area of higher income is expected to exert a stronger attractive force on such stores. Likewise, a study of disco attendance might weight by an age factor. Note that these weights have the effect of changing the population masses to total income and total population belonging to the selected age group. Thus the use of socio-economic weights is really nothing more than a redefinition of the most appropriate measure of mass. This does not really correspond to the concept of molecular mass as used by Stewart.
Stewart uses an empirical approach to determine appropriate weights. He begins by assuming basic homogeneity between masses; that is, weights of unity. He then examines his empirical results for patterns of variation between masses. If a group of masses consistently has higher than expected results across a broad range of economic and social activities, then the ratio of actual to predicted values is the weight to be applied to those masses. For example, Stewart uses weights of 2.0 for the Far West, 0.8 for the Deep South, and 1.0 for all other states.17 These weights differ from those used by Isard in that they are not associated with any socio-economic measures, nor do they vary depending on the type of activity being studied. In effect, Stewart's weights are adjustment factors to eliminate irregularities in the observed data.
For the church membership study weights of one are used for all areas. The region is felt to be sufficiently homogeneous that weighting by socio-economic factors is not needed. The empirical results also do not indicate any need for weight adjustments.
The last question is what exponents to apply to the parameters. Strict social physicists argue that the appropriate exponents are provided by the equation of the analogous physical law. Thus masses have an exponent of one, distance an exponent of one or two. The argument for other, including fractional, exponent values is based on empirical studies which have found a better fit to observed data by using exponents ranging from 0.7 to 3.0. Such studies are suggestive but far from conclusive as yet. For the church membership study, lacking any means to rationally chose fractional exponents, a strict social physics approach is adopted. The relevant concept is demographic potential, as a measure of aggregate accessibility. The formula used is:
Mj = (G)(Pj/Dj)
where: Mj = projected church membership from area j
Pj = population of area j
Dj = straight line distance from midpoint of area j to church
G = a constant, defined as Total Church Membership
: Summation from j=1 to 27 of Pj/Dj
The data used are presented in attachment 1. In attachment 2 predicted membership by zip code area is plotted against actual membership. Since the range of values is only 1 to 34, simple graph paper is used rather than double-log paper. The expected 45 degree straight line relationship is added for visual reference. No attempt is made to actually fit a straight line by least squares. The plotted results roughly support the gravity theory, although there are significant deviations for both high and low membership areas. These generally poor results might be explained by any of a large number of weaknesses in the data used.
- 1978 membership data, but 1970 population data.
- Straight line distance from geographical midpoint, whereas travel time from population midpoint would be more appropriate.
- No adjustment to population masses based on percentage of Christian church-attending families.
- No adjustment to population masses based on average family size.
The use of 1970 population data in particular might be expected to distort the results, since Fairfax County has grown 17% since 1970, whereas Alexandria shows a 5% decrease and Arlington an 11% decrease in population since 1970.
However, analysis of the data indicates that the single overriding error is in the selection of the number of masses. Zip code areas is too detailed a disaggregation of Northern Virginia when the population of interest is only 270 families. Inherent probability variability disguises the underlying gravity forces. Attachment 3 shows the results of the experiment repeated using only three masses: Arlington County, City of Alexandria, and City and County of Fairfax combined. The predicted values are very close to the actual values, clearly a straight line relationship. These results are obtained despite all the data weaknesses previously discussed. This indicates that the selection of masses is the single most crucial aspect of gravity modeling. The study suggests that, at a sufficiently aggregated level, the gravity model is able to output useful predictions even when faced with serious weaknesses in the available data inputs.
5. STRENGTHS AND WEAKNESSES OF GRAVITY MODELS
In a practical vein, one of the great attractions of gravity models is their simplicity. They are easy to construct and easy to use. The computations are not complex and are easily programmed for computers. Gravity models generally use very few variables. This not only reduces the amount and variety of data which must be obtained, but also reduces statistical complications caused by interrelatedness of multiple variables. Similar models are applicable for many different areas and types of interaction. It is not necessary to make detailed socio-economic adjustments to fine-tine ones model to specific situations. Thus Stewart uses the same weights for Western, Southern, and other states in one study after another. Thus the FHWA traffic model, although equipped to handle socio-economic adjustment factors, rarely needs to use them.18
A direct result of simplicity is low cost. It is generally inexpensive to conduct a gravity study, both in time and money. It is not necessary to conduct expensive and time-consuming surveys to gather mountains of data. Often all the data required by the gravity model are already available from other sources. Low cost makes it feasible to perform more studies, to replicate studies, to experiment with different parameter values and exponents. This flexibility can be very valuable at the start of a study if one is not quite sure what relationships are at work.
Gravity models are particularly well suited to the distribution of total activity among sub-areas. Within roughly homogeneous regions both probability theory and considerable empirical evidence lend credence to the use of gravity models. These models also have advantages in projection of future distribution patterns, since they do not simply apply proportional changes to current values. Gravity models can project positive values for sub-areas with zero current values.
Clearly, though, the great contribution of social physics is in its novel approach to the functioning of society. People in masses are explicitly identified as qualitatively distinct from people as individuals. Gravity models identify and take into account societal forces which are difficult to incorporate into traditional means of analysis. Social physics forces one to see institutions from the macroscopic point of view, to consider the extent to which society itself defines the range and intensity of activity of its members.
Gravity theory is a new way of looking at ourselves. It suggests fresh approaches to difficult problems and new interpretations of old data. Consider Stewart's demographic potential map of the United States.19 New York City is the peak; east of the Sierras every general equi-potential contour line closes around New York City. This dramatically illustrates the impact of large population masses. It unambiguously establishes New York City as the population center of the United States, and reveals as meaningless the Census assertion that Fort Wayne, Indiana is the population midpoint. A dozen other examples could be cited as well; gravity theory has already proved its usefulness.
This is not to say that gravity models do not have weaknesses as well. A serious problem with social physics in general is the lack of a solid theoretical base to build upon. If gravity relationships are the result of some kind of optimization process, it would be very valuable to have an explicit statement of that process. No such statement has yet been made.
Gravity models work best when identifying patterns of relationships for large populations, large areas. Results are less satisfactory when the aggregate is broken down into smaller units, when more detail is sought on more narrowly defined groups. Better predictions are obtained for total interaction than for interaction by sub-group. Peculiarities and irregularities of interaction between Sub-groups net out in the aggregate, but become more important as the masses under study are more finely defined. There appears to be a trade-off between detail and reliability inherent in the operation of gravity models. In addition, some versions of the interactance equation are highly sensitive to small changes in the choice of parameters. Measures of demographic force and energy can vary dramatically given relatively small changes in the number of masses chosen. It is partly for this reason that Stewart used demographic potential in so many of his studies. Potential, measuring the effect of all other masses on one mass, is less subject to this induced variability.
The major problem preventing widespread use of gravity models in location theory is an inability to assign monetary value to trade-offs between different gravity measures. For example, New York City is the point of maximum market potential, and Fort Wayne, Indiana is the point of minimum transport cost potential. To chose the optimum location based on these two factors, one must be able to say that an X% advantage in transport cost potential just offsets a Y% disadvantage in market potential. No one has been able to figure out how to compare the subjective advantages of market potential against the objective cost advantage of lower transport costs. Until this is done gravity models can only supply input to the industrial location decision-maker, they cannot make that decision themselves.
A final weakness of social physics concepts is that it is difficult to identify, in any practical sense, the meaning of the social analogs to physical laws. For example, although the concept of population density (people per square mile) is fairly clear, what meaning should be assigned to demographic force (people squared per mile)? Along the same line Isard comments that, "The interpretation to be given to the concept of demographic potential is not yet entirely clear."20 This goes back to understanding the why of gravity theory; gravity theorists have found something important, but they don't really know what it is or what to do with it.
In conclusion, gravity models amply demonstrate the existence of macroscopic societal forces which operate outside the domain of traditional economic theory. Empirical studies demonstrate that these forces are significant. Major unresolved questions are: how significant?, how do these forces operate?, how can they be measured?, how relevant are they to microeconomic optimization studies? Until these questions can be better answered gravity models will remain, in the minds of most economists, little more than tantalizing curiosities, ripe with unfulfilled promise, and short on practical significance.
|Zip code area||1970 population (1,000's)
|Average distance (miles)
(G x P/D)
|1978 actual membership||Actual to projected ratio|
|Total 1978 membership = 270
Total potential (P/D) = 180,419
Constant term (G) = 0.0015
G = Total membership/Total Potential
1. Carey,H.C., Principles of Social Science, J. Lippincott, Philadelphia, 1858-1859.
2. Stewart, John Q., "Demographic Gravitation: Evidence and Applications," Sociometry, Vol. 11 (Fed. And May 1948).
3. Zipf, George K., Human Behavior and the Principle of Least Effort, Addison-Wesley Press, Reading, Massachusetts, 1949.
4. Dodd, Stuart C., "The Interactance Hypothesis: A Gravity Model Fitting Physical Masses and Human Groups," American Sociological Review, Vol 15 (April 1950).
5. Stewart, op. cit.
6. Isard, Walter, Location and Space Economy, MIT Press, Cambridge, Massachusetts, 1956.
7. Reilly, W.J., The Law of Retail Gravitation, Fillsburg Publishers, New York, 1953.
8. U.S. Department of Transportation, Federal Highway Administration, Calibrating and Testing a Gravity Model for Any Size Urban Area, reprinted October 1973.
9. Stewart, John Q. and Warntz, William, "Macrogeography and Social Science,: Geographical Review, Vol 48 (April 1958).
10. Isard, Walter, Methods of Regional Analysis, MIT Press, Cambridge, Massachusetts, 1960,
11. Sampson, Roy J. and Farris, Morton T., Domestic Transportation Practice, Theory, and Policy, Houghton Mifflin Company, Boston, 1979, p. 244.
12. Isard, Location and Space Economy, op. cit., p. 60.
13. Stouffer, Samuel A., "Intervening Opportunities: A Theory Relating Mobility and Distance," American Sociological Review, Vol. 5 (Dec. 1940).
14. Cavanaugh, Joseph A., "Formulation, Analysis and Testing of the Interactance Hypothesis," American Sociological Review, Vol. 15 (April 1950).
15. Gould, Stephen Jay, Ever Since Darwin: Reflections in Natural History, WW Norton and Co., New York, 1977.
16. Thomas, Lewis, The Lives of a Cell: Notes of a Biology Watcher, The Viking Press, New York, 1974.
17. Stewart, op. cit.
18. U.S. Department of Transportation, op. cit., p. III-3.
19. Stewart, op. cit.
20. Isard, Methods of Regional Analysis, op. cit., p. 501.
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