Territorial Subdivision

G. Edward Stephan

Social Forces, Vol. 63, No. 1. (Sep., 1984), pp. 145-159.

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http://www.jstor.org Sun Feb 24 20:05:50 2008 Territorial Subdivision*

G . E D WA R D s T E P H A N, Western Washington University

Abstract From the assumption that social structures evolve under the constraint of mini- mizing the total time expended on their operation, 1 derive equations describing the number, distribution, area, and population of territorial subdivisions. I show how this derivation is connected with earlier ones, how an error stemming from several earlier derivations need no longer arise, and review research done to date. Many of the derived equations are new and have not yet been tested empirically. I suggest that, though the derivation is here developed for human territorial subdi- visions, it should apply to territorial subdivision in other species as well.

August Comte recognized three historically sequential bases of social or- ganization: kinship, territory, and the division of labor. His intellectual successor, mile Durkheim, in his classic The Division of Labor in Society, noted that the bulk of the population is no longer divided according to relations of consan- guinity, real or fictive, but according to the division of territory. . . . All peoples who have passed beyond the clan stage are organized in territorial districts (coun- ties, communes, etc.) which . . . connected themselves with other districts of a similar nature . . . which, in their turn, are often enveloped by others still more extensive (shire, province, department) whose union formed the society (185-6). Durkheim claimed that both consanguinous and territorial societies were based on what he called "mechanical solidarity," the joining together of similar units into larger wholes. These he distinguished from societies based on the division of labor, the whole being comprised of a complex of differentiated parts based on "organic solidarity." The principal thesis of this early work of Durkheim's was that organic solidarity would replace mechanical solidarity as population density increased or the technology of transportation and communication improved. The Durkheim thesis has been the mainstay of studies of social change ever since, in spite of the fact that most societies remain essentially

'Address correspondence to the author, Department of Sociology, Western Washington Uni- versity, Bellingham, WA 98225. 0 1984 The University of North Carolina Press 145 146 / Social Forces Volume 63:1, September 1984 territorial and in spite of the fact that "Durkheim never went back, in later studies, to any utilization of the distinction between the two types of soli- darity, nor to the division of labor as a form of cohesion. . ." (Nisbet, 86). The study of territorial division seems to have been left almost entirely to geographers, as if it had little relevance to the aims of sociology. This is unfortunate in several respects. First, there is nothing in the field of geography per se which leads to a concern with an analysis of social structure, even territorially based social structure. Second, if mod- ern societies are (still) for the most part territorial in character, then the field of sociology ought to be concerned with territorial analysis. Third, even if Durkheim's expectations for the future were being realized, it would seem sensible for sociologists to have a good understanding of the kind of territorial society which the division of labor is supposed to re- place. Finally, perhaps at a more mundane level, those who wish to study the structure of society should be aware that the most readily available data for any society-available in almost any atlas-are figures describing the areas and populations of its component subdivisions. The purpose of this paper is to present a theory of territorial subdi- vision. The theory itself is based on a very general theoretical assumption, that social structures evolve in such a way as to minimize the societal time which must be expended in their operation. After the theory is presented I show the relation between it and previous theoretical derivations and then review the research done in this area to date. Finally, I offer some suggestions for further research.

Theoretical Derivation

Imagine a population of size P distributed over a region of area A. The region is to be subdivided so that each member of the population can interact with a subdivisional center. Centers are to be established so that their number N will minimize T, the total time expended in provision of the centers and in interaction with them. Provision of the centers, their operation and maintenance, will re- quire an expenditure of so many man-hours; if h is the average expendi- ture per center, then hN is the total time expended in provision of the centers. Each member of the population will be located at a certain dis- tance from a center, so interaction between the center and members of its population will require an expenditure of so many man-hours traversing the distances involved; if s is the average distance to a center and u is the average velocity, then slu is the average time expenditure per member of the population and (slu)P is the total time expended in interaction. Under these assumptions the total time expenditure will be Territorial Subdivision / 147

By dimensional analysis the average distance s must be proportional to the square-root of the average subdivisional area; with q as the constant of proportionality s = qa1/2.Since the average subdivisional area is a = AIN, s = q(AIN)'I2 so

The value of N which will minimize T is that value which makes the first derivative of Equation 2 equal to zero, providing also that the second derivative is greater than zero. The first derivative is

The second derivative

must be greater than zero because all its factors are positive, so setting Equation 3 equal to zero and solving for N determines the value of N which minimizes T

where k = (q/2hv)z3. The value of k in Equation 5 is to some extent arbitrary, being de- pendent on the units of measurement of h and v (the other components are dimensionless). In the short run, it may also be presumed to be con- stant. Given k, then, Equation 5 states a mathematical or functional rela- tion which, for given values of P and A, specifies that value of N which minimizes T. A corresponding linear regression equation can be obtained through logarithmic transformation log N' = f + 213 (log P) + 113 (log A) (5') where f = log k and N' is the value of N expected from time-minimization theory. Testing the theory would amount to testing whether observed par- tial regression coefficients depart significantly from those indicated. Such a test would, however, run into irresolvable difficulties if the two independent variables were correlated with one another. The stand- ard errors of the partial coefficients are directly proportional to the amount of such correlation. High correlation (multicollinearity) means highly un- stable partial coefficients, wide confidence intervals, less trustworthy pre- dictions of the dependent variable, and a lessened probability of rejecting an erroneous hypothesis. 148 / Social Forces Volume 63:1, September 1984

It would be possible to avoid the problem of multicollinearity if regions could be selected in such a way that either A or P were constant. In either case, Equation 5 would reduce to a simple bivariate relation. If A were constant, N = kPY3 log N' = f + 2/3 (log P) If P were constant N = k~l/3 log N' = f + 113 (log A) Unfortunately, such regional divisions are almost non-existent. U.S.con- gressional districts are decennially reconstructed so as to make P approxi- mately constant, and the dkpartements of France were originally supposed to contain equal areas (the goal was never achieved). Most real-world re- gions vary considerably in both area and population, and multicollinearity is always a possibility. The problem of multicollinearity can be avoided by combining the two independent variables of Equation 5 into a single variable, the regional population density D = PlA. Four testable regression equations result. Two of them, below referred to as A-type relations, result from dividing Equation 5 by A NIA = k(Pl~)~~ C = k~~~ log C' = f + 2/3 (log D) where C is the number of centers per unit area, the center density C = Nl A. Inverting Equation 6 AIN = c(P1A)-2/3 a = cD-2'3 log a' = g - 2/3 (log D) where c = Ilk, g = log c, and a = AIN is the average or expected subdivi- sional area within the region. Two P-type equations can be obtained by dividing Equation 5 by P NIP = k(P1A)-'" R = kD -'13 log R' = f - 113 (log D) where R is the number of centers per capita, designated the center rate R = NIP. Inverting Equation 8 PIN = C(P/A)'/~ p = c~'/3 log p' = g + 113 (log D) Territorial Subdivision 1 149 where p = PIN is the average or expected subdivisional population within the region.

Relations Among Equations

Empirically obtained regression coefficients, corresponding to those of Equations 6-9, will be mathematically related to one another. Letting the letters C, a, etc., stand for log C, log a, etc., the following identities can be demonstrated (see note) for the variances: Var(C) = Var(a) Var(R) = Var(p) the regression coefficients:

and the correlation coefficients: rDa = - rDC rDp = - rDR Presuming that significance tests will be based on computation of the standard f-statistic,

it follows from the above identities that a test of Equation 6 is mathemati- cally equivalent to a test of Equation 7. Similarly, a test of Equation 8 is mathematically equivalent to a test of Equation 9. In spite of the mathematical connections between regression coeffi- cients, there does not appear to be a simple correspondence between the correlation coefficients for Equations 6-7 (the A-type relations) and those for Equations 8-9 (the P-type relations). It can be shown (see note) that the correlation coefficients for Equations 6 and 8 are related by and that those for Equations 7 and 9 are related by

where, for example, sc is the standard deviation of C. The A-type and P- type equations might, therefore, appear to constitute independent tests of the theory. This is not, however, the case. While the indicated correlation 150 / Social Forces Volume 63:1, September 1984 coefficients themselves show no simple correspondence, the following identities do hold (see note)

It is evident that even though the proporfion of explained variance may differ between the A-type and P-type equations, the amounf of unex- plained variation is identical for all four equations. The t-statistic, and hence any probability statement determined from it, will be identical whether one is testing Equation 6, 7, 8, or 9. Testing any one of them is equivalent to testing all four. Whether to test Equation 6 or 8, on the one hand, or Equation 7 or 9, on the other, will generally be dependent on the kind of data which is available. If the only information available concerning subdivisional units is their number within specified regions, then Equation 6 or 8 would be appropriate. If subdivisional areas and populations are available but their regional location is not known, as in a tabular list of subdivisional values, then modified forms of Equation 7 or 9 would be appropriate (see below).

Regional Specification

Regions may be specified in a number of ways. They may be official collec- tions of territorial divisions defined at some higher level of organization. For example, regions may be primary political divisions of a nation, con- taining within them the subdivisional units of interest. As a specific exam- ple, if the subdivisional level of interest is that of the county, then the appropriate regional level might be that of the state; regional populations and areas would then be the state populations and areas. Alternatively, regions may be geographic quadrants, defined inde- pendently from any political subdivisions. Quadrants of so many degrees latitude and longitude may be specified, with the added assumption that subdivisional populations and areas be assigned to that quadrant in which the subdivisional center appears. Regions may also be specified with reference to particular subdivi- sional units. They may, for example, consist of all those units which are contiguous with a particular unit, either including or excluding the unit itself. Regional populations and areas would then be the sums of the populations and areas of the units comprising the regions. Territorial Subdivision I 151

Alternative Independent Variables

Values of the independent variable may be determined in a variety of ways. Regional density may be computed directly !?om regional popula- tion and area figures, D = PIA. This would be appropriate for any of the methods of regional specification just indicated. In other cases it may be difficult or impossible to specify the regional location of particular units; this would be the case if the only population and area figures available were in the form of simple statistical tabulations for subdivisional units. In this case one may estimate a given unit's regional density D from the density of the unit itself (d = pla). Unit or local density d thus serves as a substitute or proxy for regional density Dl giving the unit-level equations a = ,-d-z3 log a' = g - 2/3 (log d) (7-4 p = ,-dl13 log p' = g + 113 (log d) (9-4 The implied regional-level Equations 6-a and 8-a have no real meaning since, presumably, the regional location of units is unknown. The substi- tution of d for D will be more or less accurate depending on the degree to which unit densities tend to be similar to those of surrounding units. While this is nearly always the case, there are some obvious exceptions: an island, for example, or a city which happens to be tabulated among a set of counties or states (e.g., Washington, D.C., as a territorial subdivision of the U.S.). In either case, the unit-density will not be a good estimator of the regional density and the unit should probably be excluded from the analysis. It may be possible to measure regional density through other sub- stitute or proxy variables. For example, it has been observed that rural population density is proportional to the square of population potential (Stewart and Wamtz). Assume that an appropriate measure of population potential V has been made (e.g., the potential at a given subdivisional center, or the average potential throughout some specified region). From Stewart and Wamtz we have, with m as the constant of proportionality, the relationship D = mV2; substituting in the appropriate equations above produces C = Kcv43 log C' = f* + 43 (log V) (6-b) a = FV-43 log a' = g+ - 4/3 (log V) (74) R = pv-z3 log R' = f* - 2/3 (log V) (8-b) 152 1 Social Forces Volume 63:1, September 1984 p = C*vZ3 log p' = g* + Y3 (log V) where KC = mZ3k, j" = log KC, c* = 1/KC, and g* = log c*. A test of these equations is still a test of timeminimization theory; however, since their derivation also includes the assumed relation between density and poten- tial, the equations may fail the test due to the failure of either assumption.

Relation to Previous Derivations

The earliest size-density derivation was that of Mycielski and Trzeckia- kowski. In deriving other equations they passed en route through two which could be shown to be equivalent to Equations 6 and 7 in the above derivation, though no attention was called to either one. Virirakis derived Equations 7-a and 9-a. Palmer derived Equation 6. These earlier deriva- tions were based on the assumption of distance- or cost-minimization. Stephan (c) derived Equation 7-a from the assumption of time-minimiza- tion. Stephan and McMullin derived Equations 5 and 6 as algebraic exten- sions of Equation 7-a. In criticism of the Stephan (c) derivation Vining et al. derived a further relationship from Equation 7-a and the identity d = pla: where w = c3. Virirakis derived a similar relationship in the form pt. = constant (where r is the square-root of a). Vining et al. argued that Equa- tion 7-a was artifactual since the dependent variable appeared as the de- nominator of the independent variable. Their own equation failed empiri- cal tests, leading Vining et al. to reject Stephan's earlier derivation from time-minimization theory. "The fallacy of Stephan's statistical work," they wrote, "cannot, however, detract from the strong visual impression given by density maps that closely settled areas tend to be subdivided more than sparsely settled areas. To confirm this relation statistically will require the measurement of density independently of area." One of the chief virtues of the present derivation is that it puts Equation 7-a in its proper place for the first time. The frame of reference for the derivation is regional rather than local; it thus puts the theory in direct correspondence with the sort of visual impressions to which Vining et al. refer. Equation 7 also clearly indicates measurement of density inde- pendently of unit area (as does Equation 7-b). Most importantly, Equation 7-a-the principal focus of all prior empirical and theoretical work-has been reduced to a conditional hypothesis; if d is a good estimate of Dlthen it can be said that Equation 7-a is an appropriate test of the theory; other- Territorial Subdivision / 153 wise Equation 7-a is meaningless. Still further derivations from Equation 7-a itself, such as that by Vining et al., become completely pointless. An additional and related advantage of the present derivation is that it conforms to the requirement, in regression analysis, that the inde- pendent variables be error-free. The variables P, A, and D are presumed to be mathematical variables; their values in the set of derived equations are simply given, once the region is specified. N is the only strictly stochastic variable; it may vary around its expected value ("expected" here taken in either its statistical or time-minimizing sense). As N varies, so will C, a, R, and p. But in order to derive the Vining et al. equation in the present context it is necessary for N to appear as a component in both the depen- dent and the independent variable, thus automatically violating the as- sumption of an error-free independent variable. It may be noted that the present derivation is superior to previous efforts in that it permits the derivation of an entire set of equations simul- taneously. The equations obtained earlier were isolated and their deriva- tions piecemeal. Now those equations, and the new ones not previously derived, can be obtained from the same initial assumption. The present derivation also conforms more closely to the original statement of the time-minimization assumption (Stephan, c): Social structures evolve un- der the constraint of minimizing the total societal time expended in their operation. Previous derivations minimized per capita, rather than total, time.

Review of Research Findings

By far, the bulk of the research conducted to date has been concerned with establishing or testing Equations 7 and 7-a, the so-called size-density law. A weak form of this law, stating simply that small units appear in regions of high density, was reported impressionistically (that is, without quanti- tative data analysis) for counties in the United States and for parishes in England (Haggett), and for United States counties during the entire period of their formation (Stephan, a). Haggett presented a scatterplot indicating a negative size-density relation for counties in the Brazilian state of Santa Catarina; Skinner reported such a relationship for market areas in rural mainland China; and Whitney has shown similar results for the hsien and provinces of China. Statistical analyses, testing the hypothesized negative relation against the null hypothesis, were conducted on data for the primary politi- cal divisions of 98 modern nations (Stephan, b); negative relations were found in 94 cases. Further tests revealed negative relations for aboriginal tribal territories of the Pacific Northwest (Stephan and Wright), Africa 154 / Social Forces Volume 63:1, September 1984

(Stephan and Tedrow), Bougainville Island (Callen and Stephan), and California (Myers and Stephan). Additional studies showed that the dioce- san boundaries of national churches are related to the density of church members (Suggs) and that the size of urban police car patrol districts is related to the density of crimes (Hall). The stronger form of the size-density law, that indicated by Equa- tion 7-a, was discovered empirically when the 1,764 divisions of 98 mod- ern nations were analyzed as an aggregate set (Stephan, b). Subsequent statistical analyses have tested the hypothesis that size and density are related by a -213 slope. The original study has been replicated for most of the original data (Vining et al.) and with more recent data for a total of 2,028 divisions within 113 nations (Callen). One of the nations which, in the Stephan (b) study, failed to con- form to Equation 7-a, the United Kingdom, was shown to steadily ap- proach conformity as earlier censuses were analyzed back to the first in 1801; reorganization of British counties for the first time in nearly a millen- nium (in 1973) returned that nation to conformity with Equation 7-a (Mas- sey and Stephan). A reanalysis of the British data provided the first test of the regional density formula, Equation 7, with results even more striking than the earlier study (Massey et al.). Equation 7 has since been supported with data for 20 modern nations (Callen). Analysis of the 27 other deviant cases from Stephan (b) suggested that the pattern of size-density decay found in the United Kingdom might not be unique to that nation. Deviation from Equation 7-a appeared to be related to urbanization in the face of fixed subdivisional boundaries (Mc- Mullin). Removal of statistical outliers and inappropriate units brought apparently deviant nations into conformity with Equation 7-a; for those nations where historical data were available, earlier censuses showed greater conformity, as in the case of the United Kingdom (Stephan et al.). A study of the size-density relation in a number of Arab nations (Alko- baisi) has shown that recent population redistribution in many of them is producing an erosion of the size-density relation comparable to that ob- served in industrial nations. The other equations derived above have received little or no empiri- cal attention. Equation 6 was tested for the density of U.S. county seats, by state or territory, from 1790-1970 (Stephan and McMullin). A slightly modified form of Equation 5-a was tested for U.S. county seats, by geo- graphic quadrant, from 1790-1970, with results similar to those for state units (Moe). Two of the population potential relationships, Equations 7-b and 9-b, were supported with 1970 data for a random sample of 100 U.S. counties (Stephan, d). Territorial Subdivision / 155

Suggestions for Further Research

Fourteen equations have been derived from the initial theoretical assump- tion of time-minimization. Equation 7-a, in both its weak and strong forms, has been tested fairly extensively and with supportive results; to a lesser extent, so has Equation 7. Aside from these tests, little has been done. Equations 5-a, 6, 7-b, and 9-b have each been tested once; Equations 5, 5', 5-b, 6-b, 8, 8-b, 9, and 9-a have never been tested directly (though it should be recalled that Equations 6-9 are mathematically equivalent to one another). This suggests that a good ded more research remains to be done. Almost all of the research which has been done to date has been concerned with the areal distribution of governmental centers (provincial capitals, county seats and the like). It would seem useful to broaden the scope of such work to include the study of nongovernmental institutions. Churches, schools, hospitals, and retail outlets, for example, must interact with a dispersed population just as governmental units must. It would seem useful to test the equations derived here on such units, not only to extend the scope of their application, but because the theory-if true- should serve as a measure of maldistribution of such services. If any such units are over- or under-distributed in a given region, then cost is not being minimized. I want to suggest an additional area for further research, prompted by a study currently being conducted of territorial subdivision in France. When a revolutionary government established the French d+artements in 1789, the original purpose was to bring governmental services directly to the people. Each subdivision was set up so that no citizen would be more than a one day ride from the seat of local government. Within a few years, Napoleon used these same structures to enforce dictatorial control: the one-day rule applied to his soldiers as well as to citizens. Whether we think of governments as servicing or controlling, they must interact with a dispersed population. Minimally, they must obtain resources essential to their own survival (e.g., taxes) from that dispersed population. They are, in this sense, predatory (perhaps "parasitic" would be more accurate, perhaps "symbiotic" more kindly). If we think of gov- ernments as one species living off another species (their populations), the suggestion for extending the above derivation to the study of non-human territorial subdivision comes immediately to mind. That there is, at least in qualitative terms, an inverse relation be- tween territory size and resource density is well known in biology (Andre- wartha; MacArthur and Connell), so well known that it "has in fact al- ready been accepted or implied by most of the authors on the subject" (Wynne-Edwards, 151). It may be that precise quantitative relations be- 156 1 Social Forces Volume 63:1, September 1984 tween territory size and food-density (Equation 7 or 7-a) have already been observed by biologists; I do not know. But if the principle of time- minimization and the argument leading to Equation 1can be extended to non-human species, then the rest of the equations ought to follow from there. I am not suggesting a mere analogy. Time is a scarce resource for all living things. If time were infinite, no other cost factors (e.g., distance, energy) would have any meaning. Any genetic or behavioral adaptation which minimizes the time required to accomplish necessary tasks should carry with it a competitive evolutionary advantage. Let P be a population of prey distributed through a region and slv be the average time it takes for predators to reach them. Let N be the population of predators and h be the average time per predator spent in non-predatory "maintenance" ac- tivities (nest-building, pairing, display, grooming, resting). Equation 1will then describe the total time spent by the predatory species. If the principle of time-minimization applies to their behavior, the derived Equations 6- 9-describing predator density, territory size, predatory-prey and prey- predatory ratios-should apply as well. I know of no published findings which bear directly on this sugges- tion, but I have been able to conduct a very simple test of Equation 5-a. Wynne-Edwards (2) displays a map of the north Atlantic showing the geo- graphic distribution, by quadrant, of sea birds and of macroplankton, the base of their food chain. Specifically, what is shown are the average num- ber of sea birds spotted daily and the amount of macroplankton taken in standard hauls during a survey conducted in 1930. He computed a correla- tion between the two sets of numbers (r = 0.85) to demonstrate structure in the food chain. From his data I computed the slope relating predators (N) to prey (P). The value expected from Equation 5-a was 0.67, and the value actually observed was 0.69. One study is hardly conclusive, nor can I suppose many sociolo- gists would be particularly interested in such a result. But it does seem interesting that, in view of the long history of accusations of "biological reductionism," "organic analogies," and the like in our field, a theory de- veloped to account for the distribution of human social institutions might actually prove useful in the study of strictly biological distributions. I won- der, though, if biologists would tend to accuse one another of "sociological reductionism."

Note From the identities log D = log (PIA) = log P - log A log C = log (NIA) = log N - log A log R = log (AIN) = log A - log N Territorial Subdivision 1 157

log R = log (NIP) = log N - log P log p = log (PIN) = log P - log N it follows that the covariances in Equations 6-9 are SDc = SpN - SAN - SPA + SM SDa = SAN - SpN + SPA - SAA SDR = SPN - SAN + SPA - SPP SDp = SAN - SPN - SPA+ SPP where, for example, SPAis the covariance of log P and log A, and SM is the variance of log A. These identities, plus SDD = spp+ SAA - 2spA lead to SDa = -SDc SDR = SDC - SDD SD~= SDD - SDC Since the regression coefficient bq = sds,, bDa = -bDc bDR = bDC - 1 bDp = 1 - bK From the additional identities See = SNN + SAA - 2SAN S, = SAA + SNN - 2SAN SRR = SNN + SPP - 2SPN S, = SPP + SNN - ~SPN it follows that Scc = sw S, = s, Since the correlation coefficient rq = sdy/iyY) rDa = - ~DC TD~= -~DR Finally, since

SDR = SDC - SDD it follows that

SDRISDSR= (SDC- SDD)ISDSR ~DR= (SKISD - SD)/SR ~DR= (~DCSC- SD)ISR Similarly,

rDp = (r~asa+ SD)/S~ To show that (1 - ?DC)SCC = (1 - ?DR)SRR express each ? as a covariancelvariance ratio to obtain see - sZDC/sDD= SpJ - sZDRIsDD 158 / Social Forces Volume 63:1, September 1984

Multiplying by SDD

Each factor or term in this equation is expressed above as a sum of variances or covariances of P, A, or N. Making these substitutions and carrying out the indicated operations will result in cancelling all terms on either side of the equation, thus demonstrating the sug- gested equality. A similar argument applies in the case of "Da" and "Dp" relationships.

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References

The Size-Density Hypothesis in Great Britain: Analysis of a Deviant Case Douglas S. Massey; G. Edward Stephan Demography, Vol. 14, No. 3. (Aug., 1977), pp. 351-361. Stable URL: http://links.jstor.org/sici?sici=0070-3370%28197708%2914%3A3%3C351%3ATSHIGB%3E2.0.CO%3B2-H

Marketing and Social Structure in Rural China: Part I G. William Skinner The Journal of Asian Studies, Vol. 24, No. 1. (Nov., 1964), pp. 3-43. Stable URL: http://links.jstor.org/sici?sici=0021-9118%28196411%2924%3A1%3C3%3AMASSIR%3E2.0.CO%3B2-S

Variation in County Size: A Theory of Segmental Growth G. Edward Stephan American Sociological Review, Vol. 36, No. 3. (Jun., 1971), pp. 451-461. Stable URL: http://links.jstor.org/sici?sici=0003-1224%28197106%2936%3A3%3C451%3AVICSAT%3E2.0.CO%3B2-3

International Tests of the Size-Density Hypothesis G. Edward Stephan American Sociological Review, Vol. 37, No. 3. (Jun., 1972), pp. 365-368. Stable URL: http://links.jstor.org/sici?sici=0003-1224%28197206%2937%3A3%3C365%3AITOTSH%3E2.0.CO%3B2-D http://www.jstor.org

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Territorial Division: The Least-Time Constraint Behind the Formation of Subnational Boundaries G. Edward Stephan Science, New Series, Vol. 196, No. 4289. (Apr. 29, 1977), pp. 523-524. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819770429%293%3A196%3A4289%3C523%3ATDTLCB%3E2.0.CO%3B2-%23

The Historical Distribution of County Seats in the United States: A Review, Critique, and Test of Time-Minimization Theory G. Edward Stephan; Douglas R. McMullin American Sociological Review, Vol. 46, No. 6. (Dec., 1981), pp. 907-917. Stable URL: http://links.jstor.org/sici?sici=0003-1224%28198112%2946%3A6%3C907%3ATHDOCS%3E2.0.CO%3B2-S

Tribal Territories in Africa: A Cross-Cultural Test of the Size-Density Hypothesis G. Edward Stephan; L. M. Tedrow The Pacific Sociological Review, Vol. 17, No. 3. (Jul., 1974), pp. 365-369. Stable URL: http://links.jstor.org/sici?sici=0030-8919%28197407%2917%3A3%3C365%3ATTIAAC%3E2.0.CO%3B2-Y

Statistical and Historical Analyses of Nations which Deviate from the Size- Density Law G. Edward Stephan; Douglas R. McMullin; Karen H. Stephan Demography, Vol. 19, No. 4. (Nov., 1982), pp. 567-576. Stable URL: http://links.jstor.org/sici?sici=0070-3370%28198211%2919%3A4%3C567%3ASAHAON%3E2.0.CO%3B2-W

Political Subdivision and Population Density Daniel R. Vining, Jr.; Chung-Hsin Yang; Shi-Tao Yeh; G. Edward Stephan Science, New Series, Vol. 205, No. 4402. (Jul. 13, 1979), pp. 219-220. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819790713%293%3A205%3A4402%3C219%3APSAPD%3E2.0.CO%3B2-0