How mathematics explains cities – 09/21/2023 – Fundamental Science

It is estimated that more than half of the world’s population lives in cities. The projection for 2050 jumps to almost 70%, which leads to the conclusion that life in the city must offer increasingly stimulating challenges for those responsible for planning these spaces. Among the most effective tools for this task is mathematics, and the instances in which it appears are, to say the least, surprising.

Before talking about concrete tools, and the use of mathematics that has just come out of the oven to make life in cities easier, it is worth mentioning some interesting facts. The first has to do with the population of the largest centers in a given country. The 2022 Census estimates that São Paulo has just over 11 million inhabitants, while Rio de Janeiro has 6 million, and Brasília has almost 3 million inhabitants. From here, the population is divided by two for each position we move down in the ranking.

A similar thing happens in Portugal, the United States and Kiribati. In fact, this is a common pattern known as Zipf’s Law, and there are those who suggest theoretical explanations for this fact. One of them has to do with externalities of scale, and translates into the maxim that the more activity there is in a city, the more activity this city is capable of attracting.

Another class of urban curiosities has to do with scale. For example, if a population doubles in size, how does crime increase? Does it also double? And the number of gas stations? In an article published in 2007 in the Proceedings of the National Academy of Sciences, the prestigious PNAS, Portuguese physicist Luís Bettencourt and his collaborators investigated this question and obtained really interesting answers. The researchers propose an exponential model, in which the occurrence of an event (crime) or the demand for a service (gas stations) are determined by the city’s population raised to an exponent. If the exponent is equal to 1, the attribute in question is linear with respect to the population size. An exponent greater than 1 indicates superlinear growth, while an exponent less than 1 indicates sublinear growth.

In the case of serious crimes, the estimated exponent is 1.16, that is: if the population is multiplied by 2, the number of serious crimes more than doubles. When it comes to gas stations, the scenario is different, with an exponent of 0.77. In other words, increasing the population by a factor K implies increasing the number of gas stations by a factor K^3/4. A curiosity about the 3/4 exponent: the same exponent is found in biology, when estimating energy needs in terms of the weight of a mammal, for example — perhaps a city is just another form of organism.

So far, the conversation is mediated by empirical information, and the mathematics emerges when we look carefully at the data. However, it is to be expected that pure, theoretical mathematics will be able to guide the formulation of public policies that improve the functioning of cities. A fundamental example has to do with traffic.

Urban mobility largely involves transporting a mass (of people) from one point to another in space, in the best possible way. That is, a modern math problem known as optimal transport. The question is how to design a mechanism that transports people through a network of roads throughout the city.

Let’s imagine that a city is like a graph, where the roads are the edges and the intersections are the vertices. What we are interested in is finding a transport plan that allows moving masses of agents along the graph in the most economical way possible. In the literature, such a plane is referred to as Wardrop equilibrium, since the English mathematician John Glen Wardrop introduced the concept in the early 1950s.

One of the challenges of research today is to find realistic conditions under which the existence of these equilibria can be demonstrated. This year, an important advance in this direction was made by mathematicians Héctor Chang-Lara and Sérgio Zapeta, both from the Center for Mathematical Investigations, in Guanajuato, Mexico. They were able to incorporate the traffic volume of the entire transportation network into the model and still prove the existence of Wardrop equilibrium.

Another urban problem in which mathematics can be very useful is crime. Mathematical models based on partial differential equations help predict the density of criminal activities in an urban area and also allow designing public policies to combat crime. The basic principle of the model is the old adage that opportunity makes the thief. Assuming that a city is made up of parallel and transversal streets, and that crimes only occur at the intersections of these roads, the model considers the attractiveness of committing a crime on a given corner.

Measuring attractiveness involves estimating the criminal’s earning potential, the chance of being caught, the success rate of crime in the area in the past, etc. Once attractiveness is characterized, the probability of an occurrence can be formulated. Here, two mutually dependent equations arise: one that models attractiveness and another the probability of a crime occurring. By addressing both simultaneously, in theory it would be possible to know the most effective way to police a given region.

Mathematics always offers beautiful tools for understanding the world around us. In the case of urban life it is no different. Whether analyzing data or formulating sophisticated models to attack concrete problems, mathematics is an oasis in any stone jungle.

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Edgard Pimentel is a professor at the Department of Mathematics at the University of Coimbra and a researcher at CMUC.

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