## The Equations Underlying Cities

Cities are complex systems of which functioning depends upon many social, economic, and environmental factors. **Bettencourt** (p. 1438; see the cover; see the Perspective by **Batty**) developed a theory to explain the quantitative relationships observed between various aspects of cities and population size or land area.

## Abstract

Despite the increasing importance of cities in human societies, our ability to understand them scientifically and manage them in practice has remained limited. The greatest difficulties to any scientific approach to cities have resulted from their many interdependent facets, as social, economic, infrastructural, and spatial complex systems that exist in similar but changing forms over a huge range of scales. Here, I show how all cities may evolve according to a small set of basic principles that operate locally. A theoretical framework was developed to predict the average social, spatial, and infrastructural properties of cities as a set of scaling relations that apply to all urban systems. Confirmation of these predictions was observed for thousands of cities worldwide, from many urban systems at different levels of development. Measures of urban efficiency, capturing the balance between socioeconomic outputs and infrastructural costs, were shown to be independent of city size and might be a useful means to evaluate urban planning strategies.

Cities exist, in recognizable but changing forms, over an enormous range of scales (*1*), from small towns with just a few people to the gigantic metropolis of Tokyo, with more than 35 million inhabitants. Many parallels have been suggested between cities and other complex systems, from river networks (*2*) and biological organisms (*3*–*6*) to insect colonies (*1*, *7*) and ecosystems (*8*). The central flaw of all these arguments is their emphasis on analogies of form rather than function, which limit their ability to help us understand and plan cities.

Recently, our increasing ability to collect and share data on many aspects of urban life has begun to supply us with better clues to the properties of cities, in terms of general statistical patterns of land use, urban infrastructure, and rates of socioeconomic activity (*6*, *9*–*13*). These empirical observations have been summarized across several disciplines, from geography to economics, in terms of how different urban quantities (such as the area of roads or wages paid) depend on city size, usually measured by its population, *N*.

The evidence from many empirical studies over the past 40 years points to there being no special size to cities, so that most urban properties, *Y*, vary continuously with population size and are well described mathematically on average by power-law scaling relations of the form *Y*_{0} and β are constants in *N*. The surprise, perhaps, is that cities of different sizes do have very different properties. Specifically, one generally observes that rates of social quantities (such as wages or new inventions) increase per capita with city size (*11*, *12*) (superlinear scaling,

These empirical results also suggest that, despite their apparent complexity, cities may actually be quite simple: Their average global properties may be set by just a few key parameters (*12*, *13*). However, the origin of these observed scaling relations and an explanation for the interdependences between spatial, infrastructural, and social facets of the city have remained a mystery.

Here, I develop a unified and quantitative framework to understand, at a theoretical level, how cities operate and how these interdependencies arise. Consider first the simplest model of a city with circumscribing land area *A* and population *N*. I write the interactions between people *i*, *j* in terms of a social network, *a*_{0} (a cross section in the language of physics), and have strength *g _{k}*, where

*k*describes social link types (

*14*). The parameters,

*g*, can be either positive (attractive, expressing a social benefit, e.g., mutually beneficial economic relations) or negative (repulsive, expressing a social cost, e.g., crime). All these processes share the same average underlying dynamics of social encounters in space and time, against the background of the city and its infrastructure networks.

_{k}The average number of local interactions per person is given by the product of the volume spanned by their movement, *a*_{0}ℓ, times the population density *14*). The total average social output of a city can be obtained by multiplying the total number of interactions by the average outcome per interaction, *Y*, has physical units set by *g _{k}*, but it is useful to think of all quantities ultimately expressed in terms of energy per unit time (power).

Another crucial property of cities is that they are mixing populations. That is, even if people in the city explore different locations at different times, anyone can in principle be reached by anyone else. This concept, familiar from population biology (*15*), is the basis of definitions of functional cities as metropolitan statistical areas (MSAs), e.g., by the U.S. census bureau. In practice, this means that the cost per person of a mixing population is proportional to the transverse dimension (diameter), *L*, of the city *y*≃*T*/*N*, which implies _{}= 2**/**3 and *a*, increases with more productive interactions, e.g., due to economic growth, and decreasing transportation costs, as is observed in worldwide patterns of urban sprawl over time (*16*). Thus, I obtain *A*, varying sublinearly with *N* (α = 2**/**3 **<** 1), and socioeconomic outputs, *Y*, varying superlinearly (β = 4**/**3 **>** 1). However, this overestimates_{}β because as cities grow, space becomes occupied and transportation of people, goods, and information is channeled into networks. The space created by these networks gives the correct measure of the social interactions that can occur in cities.

I propose a more realistic model by generalizing these ideas in terms of four simple assumptions:

1) Mixing population. The city develops so that citizens can explore it fully given the resources at their disposal. I formalize this principle as an entry condition (*17*), by requiring that the minimum resources accessible to each urbanite, *Y*_{min}/*N* ~ *GN*/*A*, match the cost of reaching anywhere in the city. Because travel paths need not be linear, I generalize their geometry via a fractal dimension, *H*, so that distance travelled ∝ *A ^{H}*

^{/}

*(*

^{D}*14*). Matching interaction density to costs, I obtain a generalized area scaling relation,

*a*as before and

*D*dimensions].

*N*scales like a physical volume (

*14*,

*18*).

2) Incremental network growth. This assumption requires that infrastructure networks develop gradually to connect people as they join, leading to decentralized networks (*6*, *19*). Specifically, the scaling of Fig. 1A is obtained when the average distance between individuals *d *= *n*^{–1/2} = (*A*/*N*)^{1/2} equals the average length of infrastructure network per capita so that the total network area, *6*, *12*, *19*) and tracks the average built area of more than 3600 large cities worldwide (*16*), measured through remote sensing.

3) Human effort is bounded, which requires that *G* is, on average, independent of *N*, i.e., *dG/dN =* 0 (Fig. 1B, inset). The increasing mental and physical effort that growing cities can demand from their inhabitants has been a pervasive concern to social scientists (*20*). Thus, this assumption is necessary to lift an important objection to any conceptualization of cities as scale-invariant systems. Bounded effort is also observed in urban cell phone communication networks (*21*) and is in general a function of human constraints and urban services and structure.

4. Socioeconomic outputs are proportional to local social interactions, so that*22*, *23*), but has been difficult to quantify. The prediction that social interactions scale with *21*). Together these assumptions predict scaling exponents for a wide variety of urban indicators, from patterns of human behavior and properties of infrastructure to the price of land (*6*, *9*–*12*, *16*, *21*, *24*, *25*), summarized in Table 1 (*14*).

Thus far, I obtained estimates for scaling exponents without the need for a detailed model of infrastructure. Next, I show how network models of infrastructure can help to illuminate urban planning issues. Consider the infrastructure in a city described by a network with *h* hierarchical levels (Fig. 2A). The network branching, *b*, measures the average ratio of the number of units of infrastructure at successive levels,*i* = *h*, equals the number of people, so that_{.} These networks are not hierarchical trees (*26*) (Fig. 2A). The length of a network segment (such as a road) at level *i* is *l*_{i}, crossing a land area *a _{i}*, and its transverse dimension is

*s*, an area in 3D networks and a length in 2D. To obtain the above scaling relations, I assume that the transverse dimension of the smallest network units,

_{i}*s*

_{*}, is independent of

*N*. This leads to the scaling of network width,

*6*,

*18*), total network length is area filling,

*l*, decrease with

*N*. The total network length

*L*and network area

_{n}*A*follow from the sum of the geometric series over levels

_{n}where I took *D* > 1.

I can now compute the cost of maintaining the city connected as the energy necessary for moving people, goods, and information across its infrastructure networks. These movements form a set of currents, transporting various quantities across the city and can be quantified by means of the language of circuits. The scaling of *s _{i }*together with total current,

*J*, conservation across levels

*i*, sets the scaling for

*i*, where ρ

*is the density of carriers in the network and*

_{i }*v*their average velocity. This quantity is interesting because it controls the dissipation mechanisms in any network. I obtain

_{i }*i*, so that highways are faster and/or more densely packed than smaller roads (

*27*,

*28*). Making the additional assumption that individual needs,

*N*(

*12*) leads to

*J*

_{i }=

*J*=

*J*

_{0}

*N*, with

There are many forms of energy dissipation in networks, including those that occur at large velocity or density. Here, I make the standard assumption that the resistance per unit length per transverse network area, *r*, is constant (*2*, *5*), leading to the resistance per network segment, *N*_{i} parallel resistors this gives the total resistance per level, *W*, follows from summing

which scales superlinearly, with exponent 1+ δ =_{}1+1**/**6 in *D* = 2, *H* = 1. Thus, energy dissipation scales with population like social interactions, as observed in German urban power grids (*12*), so that the ratio *Y/W*, a measure of urban efficiency, is independent of city size.

Finally, I show that these results can be derived by maximizing net urban output, *L*, as the difference between social interaction outcomes, *Y*, and infrastructure energy dissipation, *W*, under settlement and network constraints,

where *2*, *4*, *5*). The novelty in Eq. 4 is the prediction of an optimal *G* for different cities fluctuate around this value, as observed in Fig. 1B (inset).

To see this, consider that, keeping ε fixed and*Y* and *W* grow with *G*, because *G* follow from the solutions to *G* = *14*). Itfollows that However, there is an upper value of

*G*are less than average, would typically benefit from measures that promote greater mobility or density, in order to achieve more intense and beneficial city-wide social contact. Conversely, cities with

That many cities are becoming more global in their economic relations and political and cultural influence (*29*) does not alter the basic premises of the theory. The internal dynamics and organization of cities (as social networks of people and institutions) produces new socioeconomic functions that allow cities to exchange goods, services, people, and information within and across national borders (*22*, *23*, *30*). Thus, even if some singular places such as Hong Kong, Singapore, or Dubai are primarily part of international economies, the majority of the world’s most global cities, such as Tokyo, New York, Los Angeles, Beijing, Shanghai, Berlin, or Frankfurt, show clear scaling effects in line with their own national urban systems (Fig. 1 and figs. S1 to S3).

All cities have spatial and social pockets of greater and lower mobility, social integration, better or worse services, and so forth (*1*, *17*). It should be emphasized that the theory does not predict density profiles or socioeconomic differences inside the city, but the scaling for the properties of the city as a whole*.* None of these pockets exist in absolute isolation; they are just more or less “connected,” so they must be understood with reference to the rest of the city (*17*).

The interactions between people also provide the basis for institutional relationships via the appropriate groupings of individuals in social or economic organizations and by the consideration of the resulting links between such entities. Institutions and industries that benefit from strong mutual interactions may aggregate in space and time within the city in order to maximize their *Y – W*, a point first made by Marshall (*23*) in the context of industrial districts. Other organizations may benefit primarily from the general effects that result from being in the wider city and collecting a diversity of interactions, an argument often attributed to Jacobs (*22*). These results establish necessary conditions for urban areas to express certain levels of socioeconomic productivity, but it remains a statistical question (*21*, *25*) how well they are realized in specific places.

Most urban systems for which reliable data exist confirm almost exactly the simplest predictions of the theory developed here. Examples are the scaling of area for about 1800 cities in Sweden (*14*, *18*), or for roads in several hundred American (Fig. 1A) and Japanese metropolitan areas (fig. S3). One of the most spectacular agreements is for the scaling of total area of paved surfaces for all cities worldwide above 100,000 people (over 3600 cities) (*14*, *16*). These examples illustrate the result derived above that urban infrastructure volume scales faster with population than land area (and both are sublinear). This effect is visually apparent in large, developed cities, where roads, cables, and pipes become ubiquitous and eventually migrate into the third dimension, above or below ground.

Measurements of electrical cable length and dissipative losses in German urban power grids (*12*) further confirm these expectations and support another key result obtained above: The energy loss in transport processes scales like socioeconomic rates (and both are superlinear). This shows how cities are fundamentally different from other complex systems, such as biological organisms (*4*, *5*) or river networks (*2*), which are thought to have evolved to minimize energy dissipation. Thus, the framework developed here also brings into focus efforts for sustainable urban development, by showing what kind of energy budget must be expended in order to keep cities of varying sizes socially connected.

The predictions of the theory are further supported by data on the size of urban economies from hundreds of cities in several continents, such as those in the United States (Fig. 1B), Japan (fig. S3), China (fig. S2A), or Germany (fig. S2B). In particular, the specific result that scaling exponents remain invariant over time, and are independent of population size and level of development, is confirmed by data for wages in U.S. metropolitan areas spanning 40 years (fig. S3). Direct empirical tests on the predictions made here for individual properties remain more difficult, but are confirmed, for example, by measurements for the scaling of social interactions with city size in the cell phone networks of two European nations (*21*), and for certain other patterns of individual behavior (*12*, *20*, *31*). Nevertheless, for most nations, we cannot yet access all predicted urban quantities simultaneously, especially in developing countries. This provides many future tests and applications for the theory, especially where understanding urbanization is most critical.

The spatial concentration and temporal acceleration of social interactions in cities has some striking qualitative parallels in other systems that are also driven by attractive forces and become denser with scale (*20*, *30*). The most familiar are stars, which burn faster and brighter (superlinearly) with increasing mass. Thus, although the form of cities may resemble the vasculature of river networks or biological organisms, their primary function is as open-ended social reactors. This view of cities as multiple interconnected networks that become denser with increasing scale (*32*) may also help to elucidate the function of other systems with similar properties, from ecosystems to technological information networks, despite their different relationships to physical space.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/340/6139/1438/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S3

Tables S1 to S3

## References and Notes

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The smallest conceivable city has population size
*N*= 2, as I assume that a city is intrinsically a social network and, as such, is predicated on the existence of social interactions. - ↵
**Acknowledgments:**I thank J. Lobo, G. West, and H. Youn for discussions. This research was supported by grants from the Rockefeller Foundation, James S. McDonnell Foundation (grant 220020195), NSF (grant 103522), Bill and Melinda Gates Foundation (grant OPP1076282), John Templeton Foundation (grant 15705), and Bryan J. and June B. Zwan Foundation.