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Modeling Cities

July 26, 2013

One of the most interesting things about mathematical modeling is that it can be applied not just to gas molecules in a container, but to most systems. This includes biological systems, such as the predator-prey interaction modeled by the Lotka–Volterra equation; and it also includes the behavior of human populations. Its influence goes beyond the "hard" sciences into the social science disciplines of economics, which is perhaps just semi-soft, sociology, psychology and political science.

One current example in the political science area comes from the 2012 US presidential election, in which incumbent president, Barack Obama defeated his Republican opponent, Mitt Romney. It's reported that Obama's re-election team relied heavily on computer modeling as a means to allocate resources.[1-2]

In the weeks prior to the November, 2012, election, Obama analysts used polling data to run 66,000 simulated elections every day.[1-2] This is what I often do, myself, in Monte Carlo simulations, and I suspect that the actual number was 65,535, which is the limiting value of an unsigned integer in most computer languages.

Such simulations were required because of the quirky nature of the US election process. The winner is determined not by the national popular vote, but by the popular vote in each state, weighted by a factor roughly equivalent to the state's population (its number of Representatives in the House of Representatives), plus a constant (two Senators from each state). The full story can be found by looking at the composition of the US Electoral College.

Urban planning is one area in which mathematical modeling has become important, since the world's populations have been steadily leaving the countryside to participate in industries located in larger cities. The figure below, created from data collected by the United Nations, shows that in 2005 more than half of the world's population lived in urban areas, as defined by their respective countries.[3]

World urban vs rural population, 1950-2050The lure of the big city.

Percentage of people living worldwide in urban and rural areas between 1950 and 2050 (projected).

Unless there are advances in farming technique, and hydroponic factories within cities, we may be headed for a food crisis.[4]

(United Nations Data.)[3]

Six years ago I wrote an article summarizing a study, published in the Proceedings of the National Academy of Sciences,[5] on quality of life issues relating to increased urbanization (Speed Dating, May 30, 2007). This study found that some metrics of urban life, such as wealth and R&D jobs, increase exponentially with population. However, most infrastructure metrics, such as fuel consumption and road surface area, have an exponential decline, which might reflect economies of scale.

Here's a list of some of the exponents from that study.[5] All items, except the last two from Germany, are from US statistics. Hopefully, the AIDS exponent has decreased in the last six years.
New patents1.27
Private R&D employment1.34
Super-creative employment1.15
R&D establishments1.19
Total wages1.12
Total bank deposits1.08
New AIDS cases1.23
Serious crimes1.16
Length of electrical cables0.87
Road surface area0.83
Two recent papers on arXiv have used mathematics to model the population dynamics of cities.[6-7] One of these looks at the difference between urban mobility and the known trends for long distance travel. Just as in the example that many qualities of city life are exponentially related to population, it was found that that trip length is exponentially distributed.[6]

Distribution of trip distance for four major citiesDistribution of trip distance for four major cities.

Data are in blue, and the model is shown in red.

(Fig. 3 of ref. 6, modified for clarity, via arXiv.)[6]

The reason for this is quite understandable. People will go to another place only if there's "something there," be it a unique store or another family member. Population density around cities tends to taper as an exponential of distance from the city center, and trip length and the population density have the same exponential decay rate.[6]

The second paper looks at the time evolution of urban sprawl using a 111 year dataset of 45 million people distributed among 8,000 population centers in Spain.[7] In a process much like Ostwald ripening, larger cities grow at the expense of smaller cities. In this case, there's an inertial factor of about fifteen years (no one really likes to move), and a characteristic distance of 70 kilometers. Proximity is the main factor linking the fates of city populations, correlating at 60%.[7]

Figure captionEvolution of a thousand cities after a hundred years..

These virtual cities, placed randomly in a 500 kilometer square, had the same initial population. The size of the circle indicates the population after 100 years.

(Fig. 3a of ref. 7, via arXiv.)[7]


  1. Michael Scherer, "Inside the Secret World of the Data Crunchers Who Helped Obama Win," Time, November 7, 2012.
  2. Sam Biddle, "Obama's Campaign Ran Millions of Virtual Elections to Predict the Future," Gizmodo, November 7, 2012.
  3. Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat, World Population Prospects: The 2010 Revision and World Urbanization Prospects: The 2011 Revision.
  4. Special Report: The Age of Plenty, IEEE Spectrum, 2013.
  5. Lués M. A. Bettencourt, José Lobo, Dirk Helbing, Christian Kühnert, and Geoffrey B. West, "Growth, innovation, scaling, and the pace of life in cities," Proc. Natl. Acad. Sci. vol. 104, no. 17 (April 24, 2007), pp. 7301-7306.
  6. Xiao Liang, Jichang Zhao, Li Dong and Ke Xu, "Unraveling the origin of exponential law in intra-urban human mobility," arXiv Preprint Server, May 28, 2013.
  7. A. Hernando, R. Hernando and A. Plastino, "Space-time correlations in urban sprawl," arXiv Preprint Server, June 16, 2013.

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Linked Keywords: Mathematical model; mathematical modeling; gas; molecule; system; biology; biological; predator-prey; Lotka–Volterra equation; world population; human population; hard science; social science; economics; sociology; psychology; political science; 2012 US presidential election; incumbent; president; Barack Obama; Republican; Mitt Romney; resource allocation; simulation; simulated; Monte Carlo method; Monte Carlo simulation; 65,535; unsigned integer; computer language; national popular vote; state; weight function; weighted; Representative; House of Representatives; Senator; US Electoral College; urban planning; countryside; industry; city; cities; United Nations; country; urban area; rural area; agriculture; farming; hydroponics; hydroponic; famine; food crisis; Proceedings of the National Academy of Sciences of the United States of America; quality of life; urbanization; metric; wealth; research and development; R&D; exponential function; infrastructure; fuel; road; surface area; economies of scale; Germany; statistics; HIV/AIDS; patent; inventor; creativity; wage; bank deposit; crime; electrical cable; arXiv; mathematics; population dynamics; mobility; retail store; family member; Population density; time evolution; urban sprawl; year; Spain; Ostwald ripening; inertia; inertial; kilometer.

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