Figure 3

Zipf and Lorenz curve of some trips. (a) Zipf plots of several trips of different bus routes. The Zipf plot is constructed by sorting decreasingly the values of \(\Delta t^{*}\). The values of \(\Delta t^{*}\) are obtained dividing the course of each trip into constant spaces of \(\Delta s^{*}= 10\mbox{ m}\). Thus, in (a), the y-axis corresponds to the values \(\Delta t^{*}\) and the x-axis is the rank of each \(\Delta t^{*}\) both in logarithmic scale. It has ranking 1 the highest value \(\Delta t^{*}\), ranking 2 the second highest value \(\Delta t^{*}\) and so on. The Zipf plot shows a pervasive fat-tail characteristic for the interpolated time values \(\Delta t^{*}\), where the dashed line corresponds to the line with slope equals −0.5 in log–log space. (b) The Lorenz curves \(L(s)\) of the trips shown in (a), where each color represents the same trip. The equality line \(E(s)\) is shown as a dashed line and is used to calculate the Gini coefficient values for each trip, namely, being twice the area between \(E(s)\) and \(L(s)\). Each curve gives a Gini coefficient, one for each trip, representing the heterogeneity of the interpolated times. The continuous vertical line indicates where the 80/20 ratio of the Pareto Principle occurs