The book *Fractal Urban System* triggered my interest in analyzing urban systems. Without the access to this type of data sets, I decided to use the satellite images released by NASA. In this study, I found that human activities (which are indicated by the lightness of the studied area)increase faster than city size.

**1. The central place model of Shanghai**

W.Christaller (1933) suggested that cities generally confirm to the “central place model”, i.e., most of varibales concerning human activities distributed sparsely in the remote zone than at the center of cities. This pattern can be quantified by a negative relatiosnhip between population-related variable l and the distance from the center r. Or, by intergating both variables on r, we get the relatiosnip beween L and A, total activty and area.

Figure 1

I downloaded a night satellite photo of Shanghai from NASA, cut the photo to exclude Suzhou (a city closed to Shanghai) and then changed the color scheme of the photo from RGB to gray scale. The former scheme expresses every pixel with three numbers vary from 0 to 255, whereas the latter only uses a number between 0-1 (the brighter the pixel is, the greater the value is). This transformation lead to a loss of information, but is necessary to our analysis, because we are only concerned about the brightness, which indicates the strength of human activities.

With brightness data at hand, we now want to find the “center” of Shanghai to examine the aforementioned model. The CBD (yellow point in Figure 1) is apparently a good candidate. For comparison I also calculated the “center of mass” of Shanghai (red point in Figure 1). In doing this I assume that the city is a multi-particle system, with the mass of each point denoted by the lightness of the pixel.

Now we can examine the relation between *L* and *A*. Specifically, for a light point *i*, I calculated *ri*, its Euclidean distance from the center. Obviously, *Ai* is the sqaure of *ri*, while the sum of the lightness of the points closer to the center is *Li*. In Figure 2, (d) shows the negative replationship between *r* and *l*. Therefore the central place model is confirmed. (a) and (b) show the sub-linear increase of *A* with *L* measured from two version of city center, respectively. Both of them are shown in log-log axes (C) combines (a) and (b) together and displays the data points in linear axes. the two data displayed together to linear coordinates.

**2. The allometric growth of U.S. cities**

Bettencourt et al(2007) investigated the allometric growth pattern of cities. Allometric means non-linear and usually particularly refer to power-law relationships. They suggested that most of variables related to human activities scales with population across cities of different size. Here I am going to examine this assumption on the relationship between *S* and *L*.

Figure 3

First of all, we need a photo including many cities. I downloaded a photo of the U.S. (Figure 3a) and converted it into gray scale (Figure 3b). Next, I considered a parameter *k* between 0-1, when the brightness of a pixel is less than *k*, I “turned-off” it by setting its gray scale value to 0. Figure 3c demonstrate the case when *k* = 0.5. Then I defined *S* as the number of pixels within disjoint clusters and *L* as the sum of lightness over all pixels within the clusters. Figure 3d shows that when *k* = 0.5 there are totally 1322 clusters.

Figure 4

I found that when *k* = 0.5, there is a scaling relationship between S and L as given by Fig 4b. The slope of the OLS regression line fitted in log-log axes is 1.07, r-squared is 0.99. This is a

strong pattern across almost all U.S. cities (or urban clusters). It means that (1) the human activities grows faster than city size, and (2) large cities are scaled versions of small cities. In a previous study on the accelerating growth of online tagging communities, I proposed to explain this sup-linear scaling pattern by the power-law distribution of human activities of a scaling exponent smaller than 2. Here I found that my theory also fits the cities data, as given by Figure 4a.

**3. The univeral scaling pattern across cities worldwide**

Figure 5

It is obviously that the value of the allometric exponent might be affected by k, if k changes, it is likely that the scaling relatiosnship will also change. Therefore, I investigated the

relationship between k and exponent gamma across three systems, including the world, the United States, and Shanghai. Figure 5 shows that the number of clusters (which are shown in the left down corner of figures) generally decreases when *k* increases. The value of *k* are 0.1,0.3,0.6,and 0.9 in the figure columns of figures, respectively.

Figure 6

I found that, as shown in Figure 6b, (a) the value of gamma is always greater than 1 (all data points have r-square greater than 0.98), and (2) the value of gamma decrease as k increase. I also investigate the distribution of the pixel lightness across three systems and found both of the pixels in the worldwide and U.S. photos confirms to power law distribution.

**4. An simple network model on the growth of cities**

Figure 7

In the field of systems science, there is a well-known saying that if one truely “understand” a system, he should be able to replicate it. It turns out that the observed patterns can be generated by a very simple network model. I used Netlogo scripts to set up a agent model that leads to scaling relationship both in space (Figure 7) and time (Figure 8).

Figure 8