Fractals and Scaling: Toward a theory of urban scaling

So here’s where things stand. There is some fairly robust statistical evidence that socio-economic outputs of cities scale super linearly with an exponent greater than 1 So we’ve seen income, GDP, number of new patents quantities like these tend to scale with an exponent greater than 1 And many of these exponent values are clustered around 1.15 Many infrastructural properties of cities, total length of roads length of water pipes, number of gas stations tends to scale sub linearly, less quickly than linearly If you double the size of the city, you don’t double length of the roads. It’s less than double. So the exponent here less than 1 And these exponent values tend to clustered around 0.85 So there is some empirical regularities that one observes The question then is can we explain these empirical regularities Why do we see this pattern? And I am gonna discuss some thought along these lines in this video. So the situation was similar for Kleiber’s law. It was well established that the metabolic rate depended on scaled with a creature’s mass to the three quarters power. And than that beg an explanation and the West Brown Enquist scaling theory provides a pretty convincing explanation for why that particular pattern, that behavior is observed for the metabolic rate of animals. is there a similar thing for cities, in my opinion the answer is not yet but I think we are getting close. So, relatively recent paper by Luis Bettencourt in 2013 puts forth possible explanation for observed scaling in cities And this is still ongoing work, I think it’s not settled, it’s not in my opinion fully developed But it’s promising and it’s certainly in my opinion interesting to think about So in this video, I wanna describe some of the elements towards this theory some ways you might think about cities that could explain some of the scaling laws that we see. So our starting point again is going to think about networks So, for the vascular system, we said that the animals have to be served by a space-filling vascular network, So this is a obviously very crude drawing of this It started with a heart and then it branches in the three in the three in the three And this has to be space or volume filling because every part of you needs to be served by the farthest out unit of this. There needs to be capillaries everywhere in your body in order to deliver oxygen, nutrients and to take wastes away. So an initial thought is to think about a city as also being served by a network. And I think that makes sense, we need side walks and then roads we need water and electricity and all of those delivered to us by networks. However the networks for urban infrastructure may be self-similar fractal the way the vascular system is but it’s not a tree. So this is tree because it goes and branches, goes and branches It doesn’t have any link sort of cross like this. This is a very rough artist’s rendition or not very artistic rendition of what urban infrastructure might look like. So the idea here is this could really big road may be a high way and then there are medium size roads, so these things here in blue and then there are smaller roads here in red or whatever color it is and so on and so on. So we would see some sort of fractal or self-similar structure grids laid on top of grids laid on top of grids. But it doesn’t have this tree sort of behavior to it. So we would describe it a little bit differently, just mathematics of it Some other assumptions that go into this theory So for the metabolic systems we assume that capillaries are the same size and all creatures and then I think it’s a pretty well accepted assumption based on the basic sort of biochemical properties that capillaries have to perform and we can make a similar assumption here It’s probably not as on as solid footing but it seems fairly reasonable that say for a water network distribution network of water pipes the smallest network the smallest unit there, isn’t a capillary but it might be a faucet. and it seems reasonable to say that the size of your faucet, size of faucets and houses independent of the size of the city So in New York City or Mexico City some the biggest cities around faucets are probably about this big. I live in a really small town, my faucet is about this big. So faucets are about the same size everywhere. There might be a fundamental unit for transportation. If we’re talking about sidewalks, it might be the width of a single person, right, there is no, not in small towns you don’t have smaller sidewalk, smaller than width of a person. Everybody doorway might be about the same. So anyway, so that’s another assumption that might go into a theory of urban infrastructure and then another network property I wanna think about is that if you imagine things flowing across here again we are gonna have some conservation of flow flow conservation and optimization thereof went into thinking about this and here also flow would have to be conserved. if these are cars, the road network, the amount of cars that flow in has to equal the amount of cars that flow out. Their speed might be different things, might be traffic, they will be pack together But flow in is gonna have to equal flow out. So these all give us, these fairly reasonable assumptions give us some mathematical tools that we can use to begin to think about how and why we might see scaling in these urban systems. There are some other assumptions or ways of thinking about cities that enter into the possible theory or explanation for scaling and cities that Luis put forth in this 2013 paper. The first is the idea that cities exist to mix people up. That is city is mixing that if you’re in the same city, we’re in the same city we can travel to see each other, the infrastructural network exist to do that May be we live in the different side of the city. we don’t see each other everyday But we could travel to see each other And that extends almost from the definition of the city A city is an agglomeration of people, a bunch of people living together That can at least, that can all kinda travel to from again from one side to other could at least in principal interact spatially. Again the entire city doesn’t mix itself up everyday, but it’s a mass of people that are somehow connected connected by some sort of a transportation network. So another assumption is that this transportation network, well, there is some cost associated with it as well cost might be related to socio-economic outputs and that would scale not with the area of the city but would transversal length of the city how far it is to get from the one side to another. So that sets up some scaling relationships there. Additionally another consideration when thinking about cities that helps lead to some scaling results is that these infrastructure networks that I described previously grow incrementally so you might have a network like this and then a new house comes and it’s in between two other houses and then it’s gonna need water and electricity and maybe a path or a road to it. So networks rather than sort of growing like trees, they grow incrementally and often sort of get in, get filled in from the middle. So that leads to networks that get more dense over time. That’s definitely a difference from the tree like networks that we have for vascular systems. So that’s another consideration to think about in terms of how these networks to play off of each other. And then the last assumption or key idea this is in some ways the most fundamental is that socio-economic outputs are related to interactions. So that what makes a city a productive place economically culturally is not just a mere effect that there a whole bunch of people living close together but those people interact, they share ideas, cultures, language, food, all sort of stuff and that in turn leads to certain types of productivity new ideas in terms of new patents, additional economic activity and so on. So one of the base assumptions is that socio-economic outputs are proportional to the amount of local interactions people would have. So we take those ideas and work with a mathematically It’s possible to come up with arguments for scaling relationships for cities. And in particular there is an argument that suggests that socio-economic outputs patents, GDP and so on should scale as the population size to the seven sixth the power So that’s a number that’s a little bit larger than 1, 1.15, 1.16 that kinda tends to be around many of these exponents cluster. Similarly infrastructure network properties length of roads, areas of roads things like that will scale as population size to the five sixth, that’s the number less than 1 About around .85 which is sort of around what we see. So as I’ve said this theory is still young. I think it’s sort of the first steps towards a more solid theory And I’m really interested to see how the theory progresses, as people think more about the mathematics behind it and assumptions behind it where it leads over the next several years. To me it’s interesting to think about I think it holds great promise It’s not as solid as West Brown Enquist theory may not be able to be because cities are different than metabolic systems. But it’s an interesting sort of powerful provocative ideas and shows I think some of what happens some of the good things that can happen interesting thought provoking things that can happen when some of these ideas of scaling and complex systems are applied to these complicated social systems like cities.


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