To determine the relative smartness level of a city, there are several indicators that have been defined to arrive as such a number. These indicators, among many others, are:

**Natural Environment**,**Water and waste**,**Transport**,**Energy**,**Economy**,**Education**,**Health**,**ICT**and**Governance**

In the swamp of these unbalanced indicators lies a mathematical law which is common to them all. It is called as Kleiber’s law.

## Background

Kleiber’s Law was found by scientist Max Kleiber, who was a Swiss mathematician doing his undergraduate studies in Zurich in 1910. He moved to California to study agriculture at the University of California Davis. He started his research on the metabolism rates in animals which was important to determine how much food the animals would require and how much meat they would produce.

It was then he stumbled upon a mysterious pattern that would be found in almost all of the animals. It was called as **“negative power quarter scaling”**. If mass vs metabolism was plotted on a logarithmic scale, the result was a perfectly straight line that was common to all animals.

The equation was clear, metabolism scales to mass to the negative quarter power. If you take square root of 1000, it is almost 31, and square root of 31 is almost 5.5 . If a cow that weighs 1000 times as much as a bird, the cow will live 5.5 times as long as a bird, and the heart rate would be 5.5 times slower than that of a bird.

Physicist Geoffery West decided to apply Kleiber’s law to the largest (mimicking) organism on the planet, cities. Did they obey it? Did they slow down as they expanded in size? His team conducted a research and found that cities did follow this law.

Number of street lights, gas stations, length of roads, electricity used by the city, all these indicators followed Kleiber’s law, which was found in biological life. They also found that a city that was 10 times as bigger than another was not 10 times more innovative, but 17 times more innovative. A city 50 times bigger was 130 times more innovative.

West proved that similar to animals, as cities get bigger, they slow down. But, they generate ideas faster than a smaller city. **This is called as super linear scaling.** Now, we are going to see it in action. The code used in this article can be found here.

## Applying Kleiber’s Law

Now that we are familiar with Kleiber’s law, i will now apply it to the cities of today.

I have gathered data (source) for several different cities for 3 indicators, street lights, road length and electricity. The data in it’s raw form looks like this.

```
City Name, Length of Roads (in km), Population (100k)
Davanagere, 1278.7, 4.5
```

The data for electricity and street lights is in the same format.

After some cleaning and aggregating, I then calculate log to the base 10 of length of roads and log to the base 10 of the population. The code is

After aggregation the data looks like this:

```
City Name Length of Roads (in km) Population RoadLengthPerCapita RoadLengthPerCapitaLog PopulationLog LengthOfRoadsLog
2 Davanagere 1278.700 4.50 284.155556 8.150537 0.653213 3.106769
4 Jalandhar 2360.000 8.62 273.781903 8.096883 0.935507 3.372912
0 Belagavi 979.670 4.88 200.752049 7.649271 0.688420 2.991080
8 indore 3477.577 20.00 173.878850 7.441939 1.301030 3.541277
3 Gwalior 1732.000 10.70 161.869159 7.338684 1.029384 3.238548
5 Jhansi 731.575 5.07 144.294872 7.172876 0.705008 2.864259
7 Namchi 41.410 0.40 103.525000 6.693835 -0.397940 1.617105
9 raipur 1254.920 16.42 76.426309 6.255997 1.215373 3.098616
6 Muzaffarpur 550.360 48.00 11.465833 3.519269 1.681241 2.740647
1 Dahod 112.145 21.30 5.265023 2.396440 1.328380 2.049780
```

The data for street lights is almost the same. For electricity, it is a little different.

When log of population to the base 10 (x axis) is plotted against log of length of roads to the base 10, using the following code

We get the following graph:

Let us try to make a comparison. Let’s take the cities *Jhansi* and *Indore*.

Jhansi has a population of 5 * 10^6, and Indore has a population of 20 * 10^6, almost 4 times as big. If we compare log of length of roads, we see that the difference is more than 4 times. Log of length of roads for Indore is 10^3.5, and log of length of roads for Jhansi is 10^2.75. 10^3.5 / 10^2.75 is 5.6 . So, Indore is 5.6 times more creative than Jhansi, even though it is 4 times as bigger.

Similarly, we can make a comparison based on street light indicator.

The data looks like the following:

```
City Name Number of Poles Population PolesPerCapita PolesPerCapitaLog PopulationLog NumberofPolesLog
3 Dharamshala 3548 0.50 0.070960 -3.816850 -0.301030 3.549984
5 Kohima 837 1.15 0.007278 -7.102191 0.060698 2.922725
8 Thanjavur 11124 2.91 0.038227 -4.709272 0.463893 4.046261
0 Atal Nagar 7904 5.60 0.014114 -6.146700 0.748188 3.897847
1 Bareilly 31280 9.04 0.034602 -4.853010 0.956168 4.495267
4 KOTA 12909 10.00 0.012909 -6.275479 1.000000 4.110893
7 Srinagar 63750 11.80 0.054025 -4.210218 1.071882 4.804480
6 Raipur 54068 16.42 0.032928 -4.924535 1.215373 4.732940
2 Bhopal 20000 18.00 0.011111 -6.491853 1.255273 4.301030
```

The plot of log of population (x axis) vs log of number of street lights is as follows:

If we take *Kohima* and *Raipur* for the purpose of comparison, Kohima has a population of 1.15 * 10⁶, and Raipur has a population of 16.42 * 10⁶, which means Raipur is 14 times as big as Kohima, population wise. Log of number of street lights to the base 10 for Kohima is 10³, and for Raipur is 10^(4.75). That means Raipur is 10^(4.75) / 10³ , or 56 times more creative than Kohima.

If we continue to Electricity, log of electricity consumption to the base 10 does not yield a straight line that follows Kleiber’s law. Instead, if we plot log of electricity consumption per capita to the base 2 (x axis) vs log of population to the base 2, we get a line that follows kleiber’s law.

The code to aggregate data is:

The aggregated data looks like:

```
City Name Electricity Population ElectricityPerCapita ElectricityLog ElectricityPerCapitaLog PopulationLog
10 Tiruchirappalli 23.37000 9.17 2.548528 1.368659 1.349664 3.196922
2 Belagavi 40.30000 4.88 8.258197 1.605305 3.045827 2.286881
0 Agartala 319.10000 4.38 72.853881 2.503927 6.186934 2.130931
7 Jaipur 2873.50100 30.70 93.599381 3.458411 6.548427 4.940167
4 Bhopal 7339.37013 18.00 407.742785 3.865659 8.671516 4.169925
9 Thiruvananthapuram 4040.40000 9.58 421.753653 3.606424 8.720257 3.260026
5 Davanagere 2627.36000 5.21 504.291747 3.419520 8.978115 2.381283
1 Amritsar 8059.09000 11.30 713.193805 3.906286 9.478150 3.498251
6 Indore 17576.37000 19.90 883.234673 4.244929 9.786653 4.314697
3 Bengaluru 131675.76680 84.30 1561.990116 5.119506 10.609170 6.397461
8 NAGPUR 90644.85000 24.10 3761.197095 4.957343 11.876976 4.590961
```

The code to plot the data is:

The plot looks like:

Let’s take *Agartala* and *Indore* for comparison. Population of Agartala is 4.38 * 10⁶, and of Indore is 20 * 10⁶. Which means Indore is almost 5 times as big as Agartala, population wise. Log of electricity per capita for Agartala is 2⁶, and for Indore is 2¹⁰. 2¹⁰ / 2⁶ is 16. Which means Indore is almost 16 times more creative than Agartala, despite being 5 times as big.

## Conclusion

In this post we saw that if a city is n times as bigger that the other, it is > n times more creative. It means a bigger city churns out more patents than it’s counterpart.

The levels of innovation required for reducing crime, installing new electric poles, constructing new roads and bike lanes, waste disposal system, etc. is higher in a bigger city.

The law that governs metabolism of energy in bacteria to plants also governs how a city expands. And I am sure it is plausible that if more research is conducted, this law would also be visible in how galaxies and supernovas expand.

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