Fractals and Scaling: some initial results in urban scaling


The starting point for urban scaling is similar to the starting point for metabolic scaling. In both we asked about how properties of something depend on their size. For urban scaling the size it will be interested in is their population and we’ll look at different properties. wages, GDP, length of roads, amount of electricity used and so on. So in this video I want to just take an empirical look at this what does the data suggest about this question. So I’ll do so by showing you several plots And as usual I’ll put the reference with the plots down here. So here is the first one. This is looking at population, down here. And this is total wages and this is for US cities And in this context the city is taken to be a metropolitan statistical area, So it might not be exactly the same as a formal city boundary, often might includes suburbs. If you have two cities are right next to each other they will be considered the part of the same metropolitan area. So any event, here is the data, I think we got around 300 data points here And this is on a log-log plot And we can definitely see a linear trend. We can calculate the slope, which we know as the exponent in a power law And that gives us a Beta at 1.12 So note here, this is larger than 1, So what this means is if we have a small city and then we do compare to that city that twice as large we might ask, well, how would the wages, the total wages, total amount of money made in both of those cities compared. And you might think well, the city that has twice as many people should have wages that are twice as much. What this says, it’s actually more than that, it’s faster than linear, super linear. So that if you double the population on average according to this trend, you would more than double the total wages It would go by to do though the 1.12 not to do the 1 Alright so, that’s sort of interesting I think and others have thought. Because we might expect that it would be linear doubling population with double wages, but that’s definitely now what we see of course that can’t help but notice that there is an awful a lot fuzz around this line. so there’s a very clear trend that’s pretty hard to deny but it’s not an exact relationship like a physical law might be there is even more scattered I think than for most of the metabolic scaling plots. So there’s a lot of variation among cities as well. And there is a clear trend. And as we talked about in metabolic scaling the trend can be interesting and the deviations from the trend can be interesting and those two statements don’t need to be in competition to each other. Both can be interesting. In this case I think both are interesting. Ok, let’s look at a few other results. And there are lots of lots of data sets like this But I’ll show you a few more. Alright, again we have population on the horizontal axis. A log-log scale This is log not a wages but it’s GDP, gross domestic product And these are for Chinese cities so this is measured in million Yuan. And again we can see there is a very clear trend. It’s certainly not a flat line. Beta is 1.12 But for this data set there is even more variation about that trend. But again there definitely is a trend line. This plot here is for Germany, German cities Again this is a log of GDP, gross domestic product measured in Euros, very clearly trend here. Beta in this case is 1.10 and some variations about the trend but not as much as for China. In both cases though this exponent is larger than 1 This is statistically significantly so indicating that log of GDP or GDP grows faster than linearly with population. So again in both these cases if you double population, you more than double the GDP of the city Alright, let’s look a one more this sort of plots So here this is now the total road miles in the city How many roads are there measured in miles And again this is a log-log plot, population here and in this case the exponent is 0.85 So that means the growth is slower than linear. If you double the size of a population on average, you don’t double the length the roads It’s actually less than double it, through to the 0.85 So let me also explain what these lines are This line here, this is the darkest line is a line with a slope of 1 And what this is showing is that this data themselves are clearly there is trend clearly less than 1 These two here, one of these lines is a fit line with that data. The other is a line from the theory. So it’s sort of theoretical fit that I’ll explain in a subsequent video. So note again here we see a quantity road miles that’s not scaling linearly. But in this case the exponent is less than 1 And here’s one more GDP plot This is for US cities again we’re seeing faster than linear growth. This black line would indicate linear growth that’s a slope of 1 The measured data, the measured exponent is 1.13 That’s faster than linear. And there are actually two lines here. One is the measured exponent. The other is that predicted by theory. So there’s an urban scaling group at the Santa Fe Institute lead by Luis Bettencourt, Geoffrey West and many others. They produced a series of papers and are continuing to do so with a lots of lots of plots like this. So there are many, a lot more data we can look at but for this video the main observation is that there is evidence of scaling, some sort of linear relationship on a log-log plot. In some cases less than linear. In some cases more than linear. And there is a fair amount of fuzz around this, It’s not an exact relationship, it’s a trend. But there is still a fair amount of variation around this trend.




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