Allometry is the the study of the change in proportion of various parts of an organism as a consequence of growth. It was first applied to living organisms to study the relation of volume to shape to behaviour.
A first very interesting result was obtained by Kleiber who found out in the 1930ies that the metabolic rate (in other words the use of energy) of living things was proportional to the volume squared to 3/4. Hence, the metabolic rate growth in a sub linear way, the bigger you get the more efficient you are in the use of energy: a bacteria uses much less energy than a mouse who uses much less energy than a whale but the mouse uses energy more efficiently than a bacteria and a whale uses energy more efficiently than a mouse. How is that?
You know that mathematicians are always interested in why some numbers pop up, and so they tried to understand why this 3/4 linearity (or better sub linearity, since you have linearity when you have 1/1).
A quick observation was that the metabolic rate equates to heat, and a body produces heat based on its volume whilst it dissipates heat (if you don't want to boil the dissipation should be equal to the production...) based on its surface. Now, that would lead to a 2/3 sub linearity (do your math and by approximating -physicists love approximation... and mathematicians love modelling...- a body to a sphere you see that the ratio growth of body surface to volume equates to 2/3, since surface grows by the square of the radius whilst volume grows by the cube of the radius).
2/3 is closer to 3/4 than 1/1 but it is not the same. So what could be the reason of this amazing constant ratio over a 27 order of magnitude growth (that is the span between the volume of a bacteria and a volume of a blue whale)?
Mathematicians looked at the way heat is diffused and found out that you can model it by looking at the circulation of molecules, that in most animals depends on the circulation of blood, that in turns depends on the vessels (veins and arteries). Now, independently of the species the vessel infrastructure supporting blood circulation repeats over and over in the various parts of the body: it has a fractal structure. And, behold!, in a fractal structure the ratio as you increase the magnification, is 3/4. Voilà.
Now, interestingly, cities can be modelled as living organisms, and they have a sort of vessels infrastructures to disseminate "heat" which in the case of cities is economic wealth. These infrastructures, that used to be roads, and then were complemented by water pipes, sewage pipes and more recently by telecommunications wires have a fractal structure, just like the veins in a human body. And indeed, statistics show that the ratio between the "effort" (the metabolic rate) to keep a city going and its production of wealth is in the order of 3/4!
In other words, this means that the large a city is, the more effective it is, provided, of course, that it has the feeding infrastructures it takes to keep the city working as a single organisms.
If these infrastructures break down, they are not pervasive, than also the city splits into several organisms, and becomes less and less efficient as these individual organisms become smaller and smaller!
Now, this is a very important observation, since it clearly points out the crucial role of infrastructures at city level in terms of city efficiency and savvy use of energy, which translate into its capacity of generating "wealth".
However, if statistics show that pervasive infrastructures result in a Kleiber curve relating cities of different sizes, it also shows that different quality of infrastructures (e.g having or not having a broadband infrastructure) result in different efficiencies levels. Again, the rule of the 3/4 applies: cities having simile infrastructure "quality" yields a wealth that is super linear to their size (it increase faster than the size of the city,, actually in the ration 4hat /3 which is equivalent to say tthe effort dedicated to "run" the city scales sub linearly in the ratio 3/4). In other terms, the better the infrastructure (the more efficient it is) the more wealth can be produced but the size factor remains with a scale ration of 3/4!
Now, I stumbled onto an interesting research by Hygor Piaget Melo and his colleagues at the Federal University of Ceará in Brasil who studied several thousands of cities in Brasil and in the USA from the point of view of quality of life, as it could be measured by parameters like the number of homicide (security), number of car accidents (safety) and suicides (happiness).
According to their study the homicide rates is super linear, that is bigger cities have a number of homicides that is larger than the scaling factor, whilst the number of traffic incident is isometrical to the scaling factor (a city that has double the size will have double number of traffic accident) but the numb rod suicide is sub linear, that is, the bigger the city the less the percentage of suicides. It seems that a larger city can provide more reasons to live than a smaller one, hence offering alternatives to those in grave depression state.
You can read the article to get more insights.
What I would like to do is to come back to the crucial role of infrastructures. The 3/4 ratio is basically a consequence of the fractal nature of the infrastructures. However, if we move from wired infrastructures to wireless one, and if we move from a communications infrastructure to a communications fabric where data take the upper hand we no longer have a fractal structure (it is less hierarchical, more flat and distributed).
A city working as a centrally regulated organisms, is different from a city that is self aware. And as we move towards the internet of things through a bottom up awareness creation, as opposed to a top down regulation, we are basically changing the working model of a city.
It would be interesting to measure the impact of the scaling factor. My bet is that we will probably find that size will no longer play a significant roll in the overall efficiency (within ranges, of course) and that a small city might be as "efficient" as a big one!
This is something that we should investigate when looking at making cities "smarter". We need to understand what technology to deploy, and how to deploy it, so that the overall efficiency is no longer depending on the city size and that should allow us to evolve cities that are both smarter, and safer, happier for its dwellers.
You may also want to read a little summary of an interview I had on the theme of Smart, Happier City and watch the related video clip.