The City Formula or: It’s all about geometry!


Mar 17, 2021

City growth follows universal scaling laws. But where do these laws come from? Two Hub scientists, Stefan Thurner and Carlos Molinero, now offer a simple and elegant explanation: They derive urban scaling laws from the 3D geometry of a city.

 

Their article, titled “How the geometry of cities determines urban scaling laws,” was published today in the Journal of The Royal Society Interface.

 

 

Scaling laws are everywhere

 

A characteristic feature of complex systems is that when they double in size, many of their parts do not. Typically, some aspects will grow by only about 80 percent, others by about 120 percent. The astonishing uniformity of these two growth rates is known as “scaling laws.” Scaling laws are observed everywhere in the world, from biology to physical systems—and in cities. Yet, while a multitude of examples show their presence, reasons for their emergence are still a matter of debate.

 

A new publication in the Journal of The Royal Society Interface now provides a simple explanation for urban scaling laws: Carlos Molinero, a researcher at the Hub until last September, and CSH President Stefan Thurner, derive them from the geometry of a city.

 

Scaling laws in cities

 

One example of an urban scaling law is the number of gas stations: If a city with 20 gas stations doubles its population say, from 100,000 to 200,000, the number of gas stations does not increase to 40, but only to 36. This growth rate of about 0.80 per doubling – it will always be between 0.75 and 0.85 – applies to much of the infrastructure of a city.

 

For example, the energy consumption per person or the land coverage of a town rises by only 80 percent with each doubling. Since this growth is slower than what is expected from doubling, it is called sub-linear growth.

 

On the other hand, cities show more-than-doubling rates in more socially driven aspects. Studies found that people in larger cities earn consistently more money for the same work, make more phone calls, and even walk faster than people in smaller towns. This (super-linear) growth rate is around 120 percent for every doubling.

 

Remarkably, these two growth rates, 0.8 and 1.2., are showing up over and over again in literally dozens of city-related contexts and applications. But why is that?

 

It’s all in the geometry

 

Carlos, who worked extensively on this during his time at the Hub, and Stefan are convinced that these scaling laws can be explained by the spatial geometry of cities. “Cities are always built in a way that infrastructure and people meet,” says Carlos, who is an urban science expert and an architect. “We therefore think that scaling laws must emerge from the interplay between where people live, and the spaces they use to move through a city.”

 

And Stefan points out what makes the idea so special: “The innovative idea here is how the spatial dimensions of a city relate to each other.”

 

Fractal Geometry

 

To come to this conclusion, the researchers first mapped thee-dimensionally where people live. They used open data from every house in more than 4,700 cities in Europe. “We know most of the buildings in 3D, so we can estimate how many floors a building has and how many people live in it,” says Stefan. The scientists assigned a dot to every person living in a building. Together, these dots form sort of a “human cloud” of a city.

 

Clouds are fractals. Fractals are self-similar, meaning that if you zoom in, their parts look very similar to the whole. Based on the human cloud, the researchers were able to determine the fractal dimension of a city’s population: They got a number that describes the human cloud of each particular city. Similarly, they calculated the fractal dimension of cities’ road networks.

 

“Although these two numbers vary widely from city to city, we discovered that the ratio between the two is a constant,” Thurner says. The researchers identified this constant as the “sublinear scaling exponent.”

 

Aside from the mathematical elegance of this explanation, the finding has potential practical value, as the two complexity researchers also point out in their paper. “At first sight the appearance of such a constant looks like magic,” says Stefan. “But it makes perfect sense if one takes a closer look: It’s this scaling exponent that determines how the properties of a city change with its size, and that is relevant because many cities around the world are growing rapidly.”

 

The formula for sustainable urban planning?

 

The number of people living in cities worldwide is expected to roughly double in the next 50 to 80 years. “Scaling laws show us what this doubling means in terms of wages, crime, inventiveness or resources needed per person—all this is important information for urban planners,” Stefan points out.

 

To additionally know the scaling exponent of a city could further help urban planners to keep the gigantic resource demands of urban growth at bay. “We can now think specifically about how to get this number as small as possible, for example through clever architectural solutions and radically different approaches to mobility and infrastructure construction,” Stefan is convinced. After all, the smaller the scaling exponent, the higher the resource efficiency of a city.

 

 

Carlos Molinero, Stefan Thurner, How the geometry of cities determines urban scaling laws, Journal of The Royal Society Interface, March 17, 2021

 

 

Our Press Release in English.

 

Our Press Release in German.


Press

PREPARE FOR THE NEXT SUPPLY CHAIN DISRUPTION. [feat. Stefan Thurner]


Scientific Inquirer, May 17, 2022

Unkategorisiert

May 24, 2022

The economic impact of a halt of Russian gas to Austria

Publication

K. Devriendt, S. Martin-Gutierrez, R. Lambiotte

Variance and Covariance of Distributions on Graphs

SIAM Review 64 (2) (2022) 343–359

Publication

C. Diem, A. Borsos, T. Reisch, J. Kertész, S. Thurner

Quantifying firm-level economic systemic risk from nation-wide supply networks

Scientific Reports 12 (7719) (2022)

Event

CSH Talk by Frank Schweitzer: "Sociophysics: What can we know? What must we do? What may we hope?"


Jun 02, 2022 | 10:3011:30

Complexity Science Hub Vienna

Press

Folgen für Österreich ohne Erdgas aus Russland [feat. Stefan Thurner]


Ö1 [Austrian Broadcasting Corp. Radio], May 24, 2022

Event

CSH External Faculty Meeting 2022


May 30, 2022May 31, 2022

Complexity Science Hub Vienna

Event

CSH Panel Discussion: "Managing a dramatically changing world—What science can contribute"


May 30, 2022 | 16:3018:00

Complexity Science Hub Vienna

News

May 12, 2022

CSH study dissects an economy into its finest details

Press

The Logic of the Meat Grinder [feat. Peter Turchin]


American Greatness, May 17, 2022

Publication

C. Doblinger, K. Surana, D. Li, N. Hultman, L. Diaz Anadón

How do global manufacturing shifts affect long-term clean energy innovation? A study of wind energy suppliers

Research Policy 51 (7) (2022) 104558

News

May 5, 2022

How to create a functional bicycle infrastructure

Press

PREPARE FOR THE NEXT SUPPLY CHAIN DISRUPTION. [feat. Stefan Thurner]


Scientific Inquirer, May 17, 2022

Unkategorisiert

May 24, 2022

The economic impact of a halt of Russian gas to Austria

Publication

K. Devriendt, S. Martin-Gutierrez, R. Lambiotte

Variance and Covariance of Distributions on Graphs

SIAM Review 64 (2) (2022) 343–359

Publication

C. Diem, A. Borsos, T. Reisch, J. Kertész, S. Thurner

Quantifying firm-level economic systemic risk from nation-wide supply networks

Scientific Reports 12 (7719) (2022)

Event

CSH Talk by Frank Schweitzer: "Sociophysics: What can we know? What must we do? What may we hope?"


Jun 02, 2022 | 10:3011:30

Complexity Science Hub Vienna

Press

Folgen für Österreich ohne Erdgas aus Russland [feat. Stefan Thurner]


Ö1 [Austrian Broadcasting Corp. Radio], May 24, 2022

Event

CSH External Faculty Meeting 2022


May 30, 2022May 31, 2022

Complexity Science Hub Vienna

Event

CSH Panel Discussion: "Managing a dramatically changing world—What science can contribute"


May 30, 2022 | 16:3018:00

Complexity Science Hub Vienna

News

May 12, 2022

CSH study dissects an economy into its finest details

Press

The Logic of the Meat Grinder [feat. Peter Turchin]


American Greatness, May 17, 2022

Publication

C. Doblinger, K. Surana, D. Li, N. Hultman, L. Diaz Anadón

How do global manufacturing shifts affect long-term clean energy innovation? A study of wind energy suppliers

Research Policy 51 (7) (2022) 104558

News

May 5, 2022

How to create a functional bicycle infrastructure

Unkategorisiert

May 24, 2022

The economic impact of a halt of Russian gas to Austria

News

May 12, 2022

CSH study dissects an economy into its finest details

News

May 5, 2022

How to create a functional bicycle infrastructure

News

Apr 20, 2022

Event discusses cryptoasset analytics

News

Apr 12, 2022

CSH publishes first global Corona Dashboard for animals

News

Mar 28, 2022

An analysis of the economic ties between Ukraine, Russia, and the EU

News

Mar 22, 2022

Colombian Diaspora makes an economic difference

News

Mar 19, 2022

CSH contributes to World Happiness Report 2022

News

Mar 15, 2022

Economic shocks shaking Russia and the World

Spotlight

Mar 11, 2022

Peter Klimek nominated for "Communicator of the Year"

News

Mar 4, 2022

Effects of a total loss in Ukrainian food production

News

Feb 10, 2022

Ranking algorithms can reinforce inequalities in social networks

Press

Folgen für Österreich ohne Erdgas aus Russland [feat. Stefan Thurner]


Ö1 [Austrian Broadcasting Corp. Radio], May 24, 2022

Press

PREPARE FOR THE NEXT SUPPLY CHAIN DISRUPTION. [feat. Stefan Thurner]


Scientific Inquirer, May 17, 2022

Press

The Logic of the Meat Grinder [feat. Peter Turchin]


American Greatness, May 17, 2022

Press

Mapping a country's entire economy to predict—and prepare for—the next supply chain disruption [feat. Christian Diem]


Phys.org, May 12, 2022

Press

Wie man optimale Radweg-Netze für Städte plant [feat. Michael Szell]


Nau, May 4, 2022

Press

Election forensics [Opinion] [feat. Peter Klimek]


Business Mirror, May 4, 2022

Press

Estatinas: el vínculo con la osteoporosis depende de la dosis [Spanish] [feat. Caspar Matzhold]


1in4mentalheatlh, May 4, 2022

Press

The pretend proletariat [feat. Peter Turchin]


The Week, Apr 29, 2022

Press

Austrijski znanstvenik: Ovog će ljeta biti više novozaraženih nego lani [Croatian] [feat. Peter Klimek]


Vecernji, Apr 29, 2022

Press

Historians' first verdict on Trump: You're fired! But there's more to it than that [feat. Peter Turchin]


Salon, Apr 23, 2022

Press

Digital humanism aims to balance human needs, emerging tech [feat. Hannah Metzler]


TechTarget, Apr 22, 2022

Publication

C. Doblinger, K. Surana, D. Li, N. Hultman, L. Diaz Anadón

How do global manufacturing shifts affect long-term clean energy innovation? A study of wind energy suppliers

Research Policy 51 (7) (2022) 104558

Publication

C. Diem, A. Borsos, T. Reisch, J. Kertész, S. Thurner

Quantifying firm-level economic systemic risk from nation-wide supply networks

Scientific Reports 12 (7719) (2022)

Publication

K. Devriendt, S. Martin-Gutierrez, R. Lambiotte

Variance and Covariance of Distributions on Graphs

SIAM Review 64 (2) (2022) 343–359

Publication

F. Neffke, L. Nedelkoska, S. Wiederhold

Skill Mismatch and the Costs of Job Displacement

CESifo Working Paper No. 9703, CESifo, Munich (2022)

Publication

R. Crescenzi, A. Dyèvre, F. Neffke

Innovation catalysts: How multinationals reshape the global geography of innovation

Economic Geography (2022) 1–29

Publication

K, Bhattacharya, A. Chakraborty

Aggregation of self-propelled particles with sensitivity to local order

Phys. Rev. E 105 (2022) 044124

Publication

K. Ledebur, M. Kaleta, J. Chen, S.D. Lindner, C. Matzhold, et al.

Meteorological factors and non-pharmaceutical interventions explain local differences in the spread of SARS-CoV-2 in Austria

PLoS Comput Biol 18 (4) (2022) e1009973

Publication

F. Momeni, F. Karimi, P. Mayr, I. Peters, S. Dietze

The many facets of academic mobility and its impact on scholars' career

Journal of Informetrics 16 (2) (2022) 101280

Publication

M. Papoutsoglou, E. S. Rigas, et al.

Online labour market analytics for the green economy: The case of electric vehicles

Technological Forecasting and Social Change 177 (2022) 121517

Publication

C. J. de Montgomery, M. Norredam, et al.

Labour market marginalisation in young refugees and their majority peers in Denmark and Sweden: The role of common mental disorders and secondary school completion

PLOS ONE 17 (2) (2022) e0263450

Publication

M. Raddant, H. Takahashi

Interdependencies of female board member appointments

International Review of Financial Analysis (2022) 102080

Publication

L. Espín-Noboa, et al.

Inequality and inequity in network-based ranking and recommendation algorithms

Scientific Reports 12 (2022) 2012

Event

CSH External Faculty Meeting 2022


May 30, 2022May 31, 2022

Complexity Science Hub Vienna

Event

CSH Panel Discussion: "Managing a dramatically changing world—What science can contribute"


May 30, 2022 | 16:3018:00

Complexity Science Hub Vienna

Event

CSH Workshop: "Stochastic dynamics for complex systems"


Jun 01, 2022Jun 03, 2022

Complexity Science Hub Vienna

Event

CSH Talk by Frank Schweitzer: "Sociophysics: What can we know? What must we do? What may we hope?"


Jun 02, 2022 | 10:3011:30

Complexity Science Hub Vienna

Event

CSH Talk by Marc Wiedermann: "Big Data and Citizen Science in the fight against COVID-19 — The Corona Data Donation Project"


Jun 03, 2022 | 15:0016:00

Event

CSH Workshop: "Decentralized Finance: Hype or Disruption?"


Jun 13, 2022

Complexity Science Hub Vienna

Event

2nd Workshop on Efficient Machine Learning


Jun 13, 2022 | 14:0019:30

Complexity Science Hub Vienna

Event

Aleszu Bajak, CSH Journalist in Residence, introduces his work & project


Jun 15, 2022 | 15:0015:30

Complexity Science Hub Vienna

Event

CSH Workshop: "Move or Perish—Scientific trajectories, inclusion, and inequality, and their consequences for transformative science"


Jun 16, 2022Jun 17, 2022

Event

CSH Workshop: "Cities as Complex Systems"


Jun 20, 2022Jun 21, 2022

Complexity Science Hub Vienna

Event

Complexity-GAINs International Summer School 2022


Jul 04, 2022Jul 15, 2022