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Scaling laws between population and facility densities

Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved June 24, 2009

↵^{1}J.U. and S.W.S. contributed equally to this work. (received for review February 20, 2009)
Abstract
When a new facility like a grocery store, a school, or a fire station is planned, its location should ideally be determined by the necessities of people who live nearby. Empirically, it has been found that there exists a positive correlation between facility and population densities. In the present work, we investigate the ideal relation between the population and the facility densities within the framework of an economic mechanism governing microdynamics. In previous studies based on the global optimization of facility positions in minimizing the overall travel distance between people and facilities, it was shown that the density of facility D and that of population ρ should follow a simple power law D ∼ ρ^{2/3}. In our empirical analysis, on the other hand, the powerlaw exponent α in D ∼ ρ^{α} is not a fixed value but spreads in a broad range depending on facility types. To explain this discrepancy in α, we propose a model based on economic mechanisms that mimic the competitive balance between the profit of the facilities and the social opportunity cost for populations. Through our simple, microscopically driven model, we show that commercial facilities driven by the profit of the facilities have α = 1, whereas public facilities driven by the social opportunity cost have α = 2/3. We simulate this model to find the optimal positions of facilities on a real U.S. map and show that the results are consistent with the empirical data.
Starbucks Coffee Company, at the height of its success, opened 4.5 new stores every day, and the convenience store chain Seven Eleven opened 6. Interesting questions for these companies to consider when opening a new store might include where should we locate it, and what factors do we need to consider as we select a location? Floating populations, flowby traffic, and nearby commercial areas are often important factors in determining optimal positions with respect to economics and business growth. Clearly, locating stores is a complicated but very important issue because the provision of better services to more people with fewer facilities is a challenge not only for various kinds of commercial service industries but also for government social welfare agencies. Although various fields such as business economics, systems engineering, computer science, geography, and even biology have approached this issue from an optimal positioning standpoint (1–8), in most cases adhoc local solutions without a theoretical basis have been applied in practice.
An unevenly distributed population makes this positioning problem even more difficult. Because the world's populations are aggregated in urban districts centered around old settlement points, the distribution patterns of populations are very uneven. The population density is nearly lognormal distribution, not only on a global scale but also on subnational ones (1, 9). Economic geography has been investigating the problem of optimal positioning of facilities given uneven population densities (2–4), and has found that the relation between the population density ρ and the facility density D follows the simple scaling D ∼ ρ^{α} in a broad range of ρ, and that the exponent α depends on types of facilities (3–5). Intuitively, within a given country, the more populated an area is the more facilities exist, whereas there are fewer facilities in places where the population density is lower. In that sense, the exponent α = 1, which represents a linearly proportional relation between population and the number of facilities, looks natural, but the empirical data do not support this argument. For example, in the case of public service facilities, the exponents have been observed to be 2/3. Although the exponent α = 2/3 is derived analytically by minimizing the total travel distance between people and facilities from the global optimization scheme (5, 10, 11), the theory has failed to explain α = 1 in the same approach, and only a phenomenological reformulation of the theory has been tried (5).
Generally, commercial facilities like small stores have to attract large numbers of people to make a profit. If they cannot do this, such stores will need to close and move to a place with larger population. On the other hand, public facilities such as fire stations and public schools need to be built in positions where they are as close as possible to their clients, because the consumed travel time or spatial distance yields the social opportunity cost by depriving visitors of time for producing other products or services. For example, public schools that exist in big cities require students who live in nearby small towns or villages to commute for a long period of time. To reduce the social opportunity cost caused by such a longdistance commute, new schools should be built in places where the population is sparse but not negligible. Therefore, one can expect that the public facilities should be more evenly distributed spatially than commercial ones. This inference is consistent with the global optimization of facility positioning to minimize the total travel time in the previous studies (5, 10, 11). However, the global optimization is not applicable to the real economic system because it costs a great deal to regulate the locations of all individual facilities. Therefore, a simple but more realistic model is needed to explain the positions of facilities in general. In this work, we propose a simple model that incorporates relevant economic factors suitable both for public and commercial facilities to explain the two different exponents, 2/3 and 1, in a unified argument.
Empirical Data
To explore the scaling relation between population and the facility densities with uptodate data, we have gathered and analyzed extensive empirical data on the positions of various facilities in 2 socially and culturally different countries: the United States (US) and South Korea (SK). For the U.S. data, we used the databases of the U.S. Census Bureau (www.census.gov/) and Fedstats (www.fedstats.gov/), and, especially for educational data such as that site of the National Center for Education Statistics (http://nces.ed.gov/). We gathered the Korean data from the Korean Statistical Information System (http://kosis.nso.go.kr/) and the National Geographic Information Institue (www.ngii.go.kr/). Ideally, if we knew the exact position of each facility, we could introduce the set of points closer to a specific facility than to any other facilities, which is the socalled Voronoi cell. However, due to the resolution limitation of the facility positions, we measure the scaling relation by using the number of people and facilities of each county instead as a coarsegrained scheme (see the Materials and Methods for details). From an analysis of the data, it was found in both the US and SK that although the commercial facilities tend to have an exponent α ≈ 1, the public facilities commonly have an exponent α ≈ 2/3. For commercial facilities such as ambulatory hospitals, the exponent α was found to be 1.13 (top), for public facilities such as schools, the exponent was 0.69 (bottom), as shown in Fig. 1. The other results for various facilities in US and SK are summarized in Table 1.
Interestingly, as a border of exponent 0.8 in Table 1, each facility is clearly classified into 2 categories, commercial and public. In the US data, ‘ambulatory hospital,’ ‘beauty care,’ ‘laundry,’ ‘automotive repair,’ ‘private school,’ ‘restaurant,’ ‘accommodation,’ ‘bank,’ and ‘gas station’ are categorized as commercial facilities more or less, whereas ‘death care,’ ‘fire station,’ ‘police station,’ and ‘public school’ are categorized as public facilities, which provide social welfare services. Even though the 2 countries have very different physical, economic, sociocultural, and political environments, (such as different topography, urban formations, population densities, standards of living, and cultures), similar facilities, like the ambulatory hospital in the US and the primary clinic in SK, have similar exponents.
It should be noted, however, that due to the differences in the educational systems in the US and SK, the resultrelated school system is somewhat different. Because the US has a K12 educational system both in the public and private sectors, the table shows the exponent 0.95 for private schools and 0.69 for public schools. In SK, even though there is a distinction between public and private schools, this distinction is deceptive, as most students are allocated to public and private schools by the ministry of education without any preference between private and public school. Therefore, the exponent of primary and secondary schools in SK is the same, 0.77, close to the public facilities and proving no distinction between private and public schools in SK. However, ‘university/college,’ which is a more profitdriven type of school in SK shows α ∼ 0.93, close to the commercial facility.
As a result, the social facilities, such as police and fire stations and government offices, have respectively low exponents close to 2/3. One of major differences between public and commercial facilities is probably the distance between these facilities and their clients. This is especially the case with respect to social facilities. For example, fire stations need to be located close to people's residences, so that fire fighters can serve quickly if needed, and the same applies to police stations. For a school student, the commute distance is also important. As an extreme case, public health centers in SK show the exponent 0.09. Public health centers are distributed almost evenly independent of population density because their role is to support the medical service to persons who live in a very backward part of the country with no hospital. However, a commercial facility does not care much about how long their clients need to travel, only about how many clients come.
Microdynamics Model for Economic Activity
In our model, we consider the microdynamics for each facility of 2 different types, namely commercial and public. Let us consider that there exist several identical facilities on a given area A where populations are allocated unevenly. We assume that people visit their nearest facility following the first law of geography: “Everything is related to everything else, but near things are more related than distant things” (12). For convenience, we introduce the Voronoi cell V_{i} as the set of points closer to the ith facility than to any other facilities within the area A (11). Therefore, the number of visitors to the ith facility is the number of people living in the ith Voronoi cell V_{i}. We define the area of V_{i} as s_{i} and the population living in V_{i} as n_{i}.
First, for the commercial facilities, the profit of the ith facility would be proportional to n_{i}, the number of potential customers, as we discussed above. We assume that every facility has a similar maintenance expense ignoring the spatial inequality; therefore, a owner of the facility having lower n_{i} is better off moving his/her facility to the location near the facility that gets the higher profit. It is natural procedure following the efficient market hypothesis, which is well known in economics. For example, when facilities are evenly distributed, the facilities in a highly populated area should make higher profits because n_{i} is proportional to the population density. Consequently, we expect that the owners of facilities in a place with a low population density will want to move their facilities to a location with a higher population density. After relocating these facilities, more facilities will exist in the highly populated places than the sparsely populated ones. It is notable that the relocation of facilities in this way should result in almost the same profit for every facility, i.e., n_{i} ≈ N_{p}/N_{f} for all facilities, where N_{p} is the total population and N_{f} is the number of facilities. If the positions of facilities are optimized by consecutive relocations according to n_{i}, the number of visitor n_{i} becomes almost the same with each other, and facilities no longer have to change their location. By using the Voronoi cell and its area s_{i}, we obtain the expressions of the population density at certain position r, ρ(r) = n(r)/s(r) in continuous form and the facility density D(r) = 1/s(r), where n(r) = n_{i} and s(r) = s_{i} if r ∈ V_{i}. At the steady state, n(r) ≈ N_{p}/N_{f} leads to From Eq. 1, we reach the conclusion that the exponent α = 1 for the commercial facilities is consistent with the empirical data.
Now, we expand the above argument to the public facility. For the positioning of public facilities, however, the government should consider not the profit but a social opportunity cost caused by the distance between visitors and facilities. The summation of travel distance (cost) c_{i} from each visitor to the ith facility is written as where P(V_{i}) is the set of a population within V_{i} and r_{ik} represents the distance from visitor k to the ith facility. Here, 〈r_{i}〉 is defined as the average distance to the ith facility, 〈r_{i}〉 ≡ (1/n_{i}) ∑_{k∈P(Vi)}r_{ik}. To reduce the cost, the government should move a facility in the jth Vonoroi cell with the lowest c_{j} to a location near the V_{i} which has the highest c_{i}. Even though the commercial facilities compete with other facilities for larger n_{i}, the public facilities endeavor to minimize c_{i}, of which optimization, however, results in the same equalization of n_{i} and c_{i}.
If optimization of facility positions is achieved by consecutive relocations of facilities to positions near the facility of the highest cost, the cost of every facility becomes the same in the steady state, i.e., c_{i} = c. With a plausible assumption that 〈r_{i}〉 ≈ g \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\sqrt{{s}_{i}}$$ \end{document} with a geometrical constant g (∼O(1)), the summation of travel distance for V_{i} can be written as c_{i} = n_{i}g \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\sqrt{{s}_{i}}$$ \end{document} . By using ρ(r) = n(r)/s(r) and D(r) = 1/s(r), we get the following expression from c = n_{i}g \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\sqrt{{s}_{i}}$$ \end{document} , To obtain c as the form of ρ, we rewrite c as where we have used ∫_{Vi}1/s(r)d^{2}r = 1 and ∫_{A}1/s(r)d^{2}r = N_{f} with V_{i} and A denoting the region of integration. From Eqs. 3 and 4, we obtain which gives α = 2/3. Therefore, our model precisely predicts α = 1 for the commercial facility, and α = 2/3 for the public facility. Especially for the case of the public facility, our result is equivalent to the scaling behavior found in the global optimization (5, 10, 11).
Not every facility has the exact exponents α = 1 or 2/3; they have a broad range of exponents as shown in Table 1. This range exists because in reality, each facility can have both commercial and public characteristics. In this respect, in order to explain various values of the exponent α, we suggest the generalized objective function v_{i} for the ith facility, which can be written as v_{i} = n_{i}〈r_{i}〉^{β} to cover above mentioned various types of facilities. From the definition of the Voronoi cell, we write v_{i} as Then, finally we obtain the following relation from the optimization condition which gives \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\alpha =\frac{2}{\beta +2}$$ \end{document} . Consequently, when β = 0, corresponding to the objective function v_{i} = n_{i} for the commercial facility, one gets α = 1, whereas when β = 1, corresponding to the public facility, α = 2/3 is obtained. By tuning the exponent β, it is possible to get the various exponent. For example, a public health center in SK shows α ≈ 0.1 in Table 1. Because the most important factor is distance in this case, the exponent β goes to infinite and α converges to zero.
Simulation Result
We perform extensive computer simulations of the above model with a real US population distribution (see Materials and Methods). Consistent with both the analytic prediction and our empirical analysis for the US and SK, commercial facilities are found to have α ≈ 1, whereas public ones have α ≈ 2/3. In Fig. 2 A and C, the density of ambulatory hospitals in the US and the Voronoi cell diagram from a computer simulation for commercial facilities are displayed, and in Fig. 2 B and D, the density of public schools in the US and the simulation result for public facilities are shown. The difference of the spatial unevenness between the commercial and the public facility distributions is clearly observed, which is consistent with different values of α in Fig. 3 for facility densities versus population densities. In Fig. 2 A and C, the density of the facilities is more uneven than that of Fig. 2 B and D, and the size of the Voronoi cell in Fig. 2 D is more regular and uniform than that of Fig. 2 C.
Social Opportunity Cost
From the simulation result, we find that the facility distribution is changed dramatically in accordance with the facility type, and the facility densities are proportional to the power of the population densities with an exponent of not only 1 but also 2/3. If the public facilities behave like commercial ones for their profit, how much more do we have to spend in terms of the social opportunity cost? From our comparison between public and commercial facilities with the same N_{f}, the visitors' total travel distance to commerical facilities is 1.5 times longer than to public facilities, which means that students have to spend 1.5 times more time attending school and that firefighters and policemen have to go 1.5 times farther, on average, to fight a fire or crime. This interesting result can be interpreted as a location version of the price of anarchy for public facility (13).
The expression of total distance between visitors and commercial facilities can be obtained from Eq. 4: where \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\stackrel{\mbox{}}{c}$$ \end{document} represents the average cost. Therefore, we obtain the ratio r of the total travel distances for commercial and public facilities, From Eq. 9, one can find easily that r > 1 comes from the uneven population distribution: If the population is distributed uniformly, r = 1 because the densities of public and commercial facility will have the same uniform distribution, and there is no social opportunity cost caused from the selfinterest of a profitoriented commercial facility.
Summary and Discussion
We have found from the empirical analysis of distributions for a variety of different facilities in the US and SK that the facility density D and the population density ρ are positively correlated to each other, described by D ∼ρ^{α}. Commercial and public facilities have been observed to have different distributions: α ≈ 1 for the former and α ≈ 2/3 for the latter. We have proposed a simple model with a focus on the microscopic activity of 2 different types of facilities, driven by profit and social opportunity cost, and successfully reproduced as α = 1 and α = 2/3 for the commercial and the public facilities, respectively.
If the process of removing a facility is prohibited in numerical simulation, the distribution of facilities is found to differ from Fig. 3. In the graph of facility density versus population density, the data scatter in a broader area above the scaling line D ∼ ρ^{α}. The data scatters because the addition process alone makes the points in the lower part of the curve D ∼ ρ^{α} move to the upper part and not vice versa. On the other hand, if removals of facilities are allowed in addition, the upper points move to the lower part, making the scaling law D ∼ ρ^{α} more visible. The difference in results comes from the presence of regulating processes such as removal or relocation, which are also shown in other studies of optimal networks (11, 14, 15).
Materials and Methods
For empirical study, we use compilations of data (see Empirical Data for details) for the US and SK, which contain information of the number of various facilities and population for 3, 147 counties (mean area 2,916 km^{2}) for the US including Alaska and Hawaii and 234 municipal counties (mean area 426 km^{2}) for SK. Most of the data are analyzed at county level, and when the data of county level, are not accessible, we use the coarsegrained data of subnational level, 50 states and the District of Columbia for the US and 9 provinces and 7 metropolises for SK. In our simulation model, we first put N_{p} people according to the US population density distribution and randomly distribute N_{0} facilities, where N_{0} is less than the final number of facilities N_{f}. With the plausible assumption that people will visit their nearest facility, we compute n_{i} and c_{i} for the case of commercial and public facilities, respectively. At each step, we either eliminate a commercial (public) facility of the lowest n_{i}(c_{i}) or create one near the facility of the highest n_{i}(c_{i}), and then all n_{i}(c_{i}) are updated. We also checked that when we create a facility, even in a randomly chosen place, during the process, it does not change the main result but only changes the time to the steady state. The number of facilities is increased and decreased periodically during simulations with the average number of facilities fixed at N_{f}, which is analogous to the expansion and contraction of a market in reality. For Fig. 2, we set N_{p} = 100,000 and N_{f} = 2,000 and simulation usually starts from N_{0} = N_{f}/10 or N_{f}, which does not change the result. While we repeatedly create and eliminate 1 facility following the above procedure, the averages \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\overline{n}$$ \end{document} and \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\overline{c}$$ \end{document} and standard deviations are calculated. As our optimization procedure proceeds, the system is observed to approach the steady state with σ_{n}/ \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\overline{n}$$ \end{document} ∼ 0.1 or σ_{c}/ \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\overline{c}$$ \end{document} ∼ 0.1, where σ_{n}(σ_{c}) is the standard deviation of n(c), and optimal positions of N_{f} facilities within our framework are obtained to give us Voronoi cell plots such as in Fig. 2 C and D. Each Voronoi cell, by definition, contains 1 facility, and thus the density D_{i} of the facility in V_{i} is nothing but the inverse of the area s_{i} of the Voronoi cell containing the ith facility. By using the gridded 1 km population data of US, we compute the population density within the ith Voronoi cell. The scatter plots for D and ρ like Fig. 3 are made, and the exponent α in D ∼ρ^{α} is computed. A Java applet version of our simulation is available at http://statphys.skku.ac.kr/bjkim/Applet/opof.html.
Acknowledgments
We thank S. Combe, M. T. Gastner and P.J. Kim for useful discussions. This work was supported by Korea Science and Engineering Foundation Grant R012007000200840 (to B. J. K.), the Korean Systems Biology Program from the Ministry of Education, Science, and Technology through Korea Science and Engineering Foundation Grant M1030902000003B500200000 (to H. J.). S. I. L. was supported by the Creative Research Initiative Program. We also acknowledge support from Korea Research Council of Fundamental Science and Technology and Samsung Economic Research Institute.
Footnotes
 ^{2}To whom correspondence may be addressed. Email: hjeong{at}kaist.edu or beomjun{at}skku.edu

Author contributions: S.I.L., H.J., and B.J.K. designed research; J.U. and S.W.S. performed research; and J.U., S.W.S., H.J., and B.J.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
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