Universal scaling laws of collective human flow patterns in urban regions

Detail observation of human locations became available recently by the development of information technology such as mobile phones with GPS (Global Positioning System). We analyzed temporal changes of global human flow patterns in urban regions based on mobile phones’ GPS data in 9 large cities in Japan. By applying a new concept of drainage basins in analogous to river flow patterns, we discovered several universal scaling relations. These include, the number of moving people in a drainage basin of diameter L is proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^3$$\end{document}L3 in the morning rush hour, which is surprisingly different from reasonable intuition of proportionality to the 2 dimensional area, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2. We show that this unexpected 3 dimensional feature is related to the strong attraction of the city center to become a 3 dimensional structure due skyscrapers.

1 Basic properties of the mobile phone GPS data 1

.1 Returners and Travelers
To show that the data is reliable, we measure the displacement by each person on weekdays and compare with previous results [1][2][3][4]. We first introduce categorization of users into 2 groups; returners whose location in the morning is the same as at midnight, and travelers whose locations are different in the morning and at midnight. Users that send data 30 times or more per day and the difference between their home and the last stop is within 100 m are regarded as returners, and 10 km or more are regarded as travelers. Users who do not belong to either of these cases are excluded from our analysis.

Displacement distribution
We analyzed the distributions of travel distances of individuals in Fig. S2a. We categorize the users into 2 types, returners whose location at 5 a.m. is the same as that at 12 p.m. midnight, and travelers whose locations are different in the morning and in the midnight. The faction of returners is about 88%. In Fig. S2a Total means the summation of distances of location changes in one day, and the Straight line means the maximum distance between the location at 5 a.m. morning (home) and the farthest place from home in the same day. We find that there is a long tail in the travel distance distributions of returners which can be approximated by a power law with an exponent about -1.5, and Fig. S2b indicates that the total travel distance is about 2.8 times of the straight line distance because any travel route has some zig-zag. As seen in Fig

Data trimming
We describe the data trimming which we applied to the original data set. Fig. S3 shows all cities probability density function of the speed, which is defined by the square root of the sum of squares of velocity components in longitude and latitude, in semi-log plot. It is seen that the density decreases sharply around 320 km/h, which agrees with the maximum speed of trains in Japan, and we remove those data larger than this value as abnormal values (namely, we neglect travelers by airplanes). Based on the governmental information [5], data observed on the uninhabitable area such as on the ocean, river or mountain were also categorized as abnormal and removed from the data [5].

Correlation of the averaged velocity as a function of distance
We calculate the correlation of velocities of squares in distance r km at the same time, C(r), which is defined by the inner product of the averaged velocities as follows: where < v x, T, k > is the mean value of v x, T, i, j, k averaged over all (i, j), ∥ v T, i, j, k ∥ is the the norm, and < > r denotes an average taken over all Fig. S4 we find that the correlation is positive for distance less than 30 km in the morning, while the correlation vanishes except very short distance in the afternoon. From this follows that the velocity directions in the afternoon is nearly random. 1330-1400 0730-0800 Figure S4: Correlation between velocities in squares at distance r. Blue dots are for the morning rush hour, and red dots are for the afternoon. In an artificial case the velocities are randomized, the correlation is 0 theoretically for any non zero distance.

Evening flow patterns around Tokyo and size distributions
We showed, in the main text, only morning and afternoon patterns. In addition, we show here, in Fig. S5a the evening flow pattern. As expected we can clearly see that many arrows are directed just the opposite compared with the morning pattern. In Fig. S5b, we show the top 15 basins of Tokyo area for the evening. In the evening, all large basins are directed towards outside the city center implying that people go home in suburban regions. The size distributions of basins for the evening are plotted in Fig. S5c. It is seen that the distributions are closer to the morning rush hour with limitation to the largest value, which can be well approximated by truncated power laws.

Drainage basins around Osaka and Nagoya metropolitan area
In addition to Tokyo metropolitan area, we show here, in Fig. S6, examples of flow pattern of Osaka and Nagoya metropolitan areas for the morning rush hour, afternoon, and evening.

Drainage basins around Tokyo metropolitan area on holidays
We now show in Fig. S7, the flow patterns on holidays and compare them with weekdays in Fig.  2 in the main text. At first glance, it looks that there is no difference between drainage basin patterns on holidays (shown in Figs. S7a, S7b and S7c) and weekdays, but the CDF shows a large difference. In Fig. S7d, we see in the distributions that the sizes of large drainage basins are much smaller than the case of weekdays, implying missing of commuter rush hour. On the other hand, due to the difference in lifestyle between holidays and weekdays, the human flow during the daytime on holidays clearly deviates from the distribution for random patterns indicating attractive flows to suburban cities.