The Mathematics of Cities

Cities are not just random collections of buildings and people; they follow surprising mathematical regularities. From the distribution of city sizes to the pace of innovation, equations can describe the pulse of urban life.

Zipf’s Law and City Sizes

One of the most famous empirical regularities in urban science is Zipf’s Law. It states that the population of the $n$-th largest city is proportional to $1/n$ of the largest city’s population.

Mathematically, if $P_r$ is the population of the city with rank $r$, then:

$$P_r = \frac{P_1}{r}$$

Where:

• $P_1$ is the population of the largest city.
• $r$ is the rank of the city.

This power-law distribution implies a scale-invariant hierarchy in urban systems.

Urban Scaling Laws

Cities also exhibit allometric scaling. Many urban indicators $Y$ (like GDP, road surface area, or crime) scale with population $P$ according to a power law:

$$Y = Y_0 P^\beta$$

Taking the logarithm of both sides, we get a linear relationship:

$$\ln(Y) = \ln(Y_0) + \beta \ln(P)$$

The exponent $\beta$ reveals the nature of the scaling:

• $\beta = 1$ (Linear): Household water consumption typically scales linearly with population.
• $\beta < 1$ (Sublinear): Infrastructure (roads, cables) often scales sublinearly (e.g., $\beta \approx 0.85$), indicating economies of scale. A city twice as large needs less than twice the infrastructure.
• $\beta > 1$ (Superlinear): Socio-economic outputs (wealth, patents, crime) often scale superlinearly (e.g., $\beta \approx 1.15$), indicating increasing returns to scale. A city twice as large produces more than twice the wealth.

The Gravity Model of Trade

Modeled after Newton’s law of gravitation, the interaction $I_{ij}$ (e.g., trade, migration, commuting) between two cities $i$ and $j$ is proportional to their populations and inversely proportional to the distance between them:

$$I_{ij} = K \frac{P_i P_j}{d_{ij}^\alpha}$$

Where:

• $P_i, P_j$ are the populations of the two cities.
• $d_{ij}$ is the distance between them.
• $K$ is a constant.
• $\alpha$ is a distance decay parameter (often close to 2).

Modeling Traffic Flow

Traffic flow $q$ (vehicles per hour) is related to density $k$ (vehicles per km) and speed $v$ (km/h) by the fundamental equation:

$$q = k \cdot v$$

A simple linear model for speed (Greenshields’ model) assumes speed decreases linearly with density:

$$v = v_f \left( 1 - \frac{k}{k_j} \right)$$

Where:

• $v_f$ is the free-flow speed.
• $k_j$ is the jam density.

Substituting this back into the flow equation gives a parabolic relationship:

$$q = k \cdot v_f \left( 1 - \frac{k}{k_j} \right) = v_f k - \frac{v_f}{k_j} k^2$$

This helps urban engineers determine the optimal density for maximum traffic flow.

Conclusion

Mathematics provides a powerful language to decode the complexity of our cities. By understanding these underlying laws, we can design more efficient, resilient, and equitable urban environments.