Origin of the scaling law in human mobility: Hierarchy of traffic systems

Uncovering the mechanism leading to the scaling law in human trajectories is of fundamental importance in understanding many spatiotemporal phenomena. We propose a hierarchical geographical model to mimic the real traffic system, upon which a random walker will generate a power-law-like travel displacement distribution with tunable exponent, and display a scaling behavior in the probability density of having traveled a certain distance at a certain time. The simulation results, analytical results, and empirical observations reported in D. Brockmann et al. [Nature (London) 439, 462 (2006)] agree very well with each other.

Figure 1

(a) Illustration of an eight-step travel in a three-layer system ($N=3$, $M=5$ and $K=2$), where orange (with oblique lines), green (light gray), and blue (dark gray) nodes respectively denote the first-, second-, and third-layer cities. For clarity, we do not plot the connections between cities in the same layer, except the one between the two first-layer cities, $A$ and $B$. (b) All the connections in the domain of a second-layer city and its surrounded third-layer cities. (c) All the connections in the domain of the first-layer city $B$ and its surrounded second-layer cities.

Figure 2

(a) The trajectory of a walker in 5000 consecutive steps. (b) Displacement distribution $P(L)$, averaged by $1000$ independent runs, each of which lasts ${10}^5$ time steps. The parameters are $N=5$, $M=9$, $K=9$, and $S=100$. In (a), $r=2.0$. The red (light gray) solid lines denote the fitting functions, $f(L)=0.070L^{-3.11}{e}^{-0.0L}$ for $r=1.0$, and $f(L)=0.038L^{-1.63}{e}^{-0.015L}$ for $r=2.0$.

Figure 3

Displacement distribution $P(L)$ for different $M$, obtained by 1000 independent runs, each of which lasts ${10}^5$ time steps. Panels (a) and (b) respectively correspond to the cases of $r=1.0$ and $r=2.0$. The parameters are $N=5$, $K=9$, and $S=100$. The red (light gray) dashed lines denote the power laws with the corresponding exponents.

Figure 4

Displacement distribution $P(L)$ for different $N$ with (a) $r=1.0$ and (b) $r=2.0$, and for different $K$ with (c) $r=1.0$ and (d) $r=2.0$. Other parameters are $M=9$ and $S=100$. All the data points are averaged by 1000 independent runs, each of which lasts ${10}^5$ time steps. The red (light gray) dashed lines denote the power laws with the corresponding exponents.

Figure 5

(a) $\beta$ vs $r$ when $M=9$, and (b) $\beta$ vs $M$ when $r=2.0$. Circles are simulation results and solid lines denote the analytical solution $\beta =3-4{\text{ln}}_{M}r$. Other parameters are $N=5$, $S=100$, and $K=9$.

Figure 6

The probability $W(d,t)$ of having traveled a distance $d$ at time $t$. The parameters are $N=5$, $M=9$, $K=9$, $S=100$, and $r=2.0$. This plot is obtained by averaging 1000 independent runs, each of which lasts ${10}^4$ time steps. The black dashed line, as a guide of eyes, is of slope 1.