Pendular behavior of public transport networks

In this paper, we propose a methodology that bears close resemblance to the Fourier analysis of the first harmonic to study networks subjected to pendular behavior. In this context, pendular behavior is characterized by the phenomenon of people's dislocation from their homes to work in the morning and people's dislocation in the opposite direction in the afternoon. Pendular behavior is a relevant phenomenon that takes place in public transport networks because it may reduce the overall efficiency of the system as a result of the asymmetric utilization of the system in different directions. We apply this methodology to the bus transport system of Brasília, which is a city that has commercial and residential activities in distinct boroughs. We show that this methodology can be used to characterize the pendular behavior of this system, identifying the most critical nodes and times of the day when this system is in more severe demanded.

Figure 1

(a) Administrative boroughs of the DF, numbered from 1 to 30, and (b) the 3673 bus stops of the public bus transport system. Number 1 refers to Brasília, which houses the federal government and is the center of the metropolitan area. Most of the population lives in other boroughs, mainly in the southwest of Brasília. The rectangle encompasses most of the residential and job positions and is the area analyzed in Fig. 10.

Figure 2

Population distribution among the DF's boroughs. The geographical positions of the boroughs are shown in Fig. 1. The job positions are concentrated in Brasília and the population of commuting workers is concentrated in Ceilandia, Samambaia, and Taguatinga, which are found on the lower left side of the maps in Figs. 1 and 10.

Figure 3

A public transport system is composed of routes that visit the same sequence of stops during each trip: (a) transport system, (b) $L$ space, (c) $C$ space, (d) $P$ space, and (e) $B$ space. The hypothetical system shown in (a) is composed of three routes $r_1,\cdots ,r_3$ and twelve stops $s_1,\cdots ,s_{12}$. Section 2 discusses the four spaces that can be used when translating the system in a network. Networks (b)–(e) are the representation of those spaces of the transport system shown in (a).

Figure 4

Normalized distribution along the day of every time a bus of the DF public transport system went through a stop on 13 October 2014, a typical weekday. In (a) the size of the bars is proportional to the number of buses stopping at each half-hour interval. In (b) each stop is weighted by the number of passengers in the bus. The peaks related to the morning and the afternoon commuting are visible in both distribution, although they are much more evident in the passengers' motion.

Figure 5

A point along the circle of radius $r=1$ is assigned to each bus visiting a stop, the angle defined by the hour of the visit. Three stops are shown, namely, at 9, 12, and 15 h, from which the c.m. was calculated. In (a), every stopping of a bus has the same weight. In (b), the number of passengers is used as weight. The c.m. is attracted by heavier trips. The hour $h$ is connected to the angle $\theta$ by Eq. (1).

Figure 6

(a) The c.m. of a continuous distribution is easily calculated and can be used to estimate the c.m. of a regular distribution of trips along the hours defining the angles ${\theta }_1$ and ${\theta }_2$. The ${\vec{R}}_{\text{c.m.}}$ points toward the mean value of these angles and its length (b) is given by Eq. (7). Larger values of $R_{\text{c.m.}}$ indicate more concentrated distributions.

Figure 7

Fifty routes of the DF public transport system with more trips in 13 October 2014, a typical weekday. The number of trips and the mean number of passengers per trip are shown for each route. The average and the standard deviation of the mean number of passengers per trip among these 50 routes are 34.25 and 14.85, respectively.

Figure 8

Distribution of $M$ [Eq. (3)], $h_{\text{c.m.}}$ [Eq. (5b)], and $R_{\text{c.m.}}$ [Eq. (5a)] of the network edges. (a) The top plots are from the trip-weighted network and (b) the bottom plots are from the passengers-weighted network. The sizes of the bars are proportional to the number of edges. A significant number of edges with $R_{\text{c.m.}}=1$ exist in the passenger-weighted network and are related to roads where most passengers commute in a short time interval even though no such concentration is observed in the trip-weighted network.

Figure 9

Distribution of $M$ [Eq. (3)], $h_{\text{c.m.}}$ [Eq. (5b)], and $R_{\text{c.m.}}$ [Eq. (5a)] of the network edges. (a) The top plots are from the trip-weighted network and (b) the bottom plots are from the passenger-weighted network. The sizes of the bars are proportional to the number of trips in the top plots and to the number of passengers in the bottom plots. A comparison with Fig. 8 shows that the weighting can significantly affect the distribution, with the main effect being the reduced importance of edges with few trips or passengers.

Figure 10

Pendularity of the most populated area of DF, highlighted in Fig. 1. Pendularity is measured by (a) and (d) the morning $h_{\text{c.m.}}$ and (b) and (e) the afternoon $h_{\text{c.m.}}$ and (c) and (f) the concentration of trips is measured by $R_{\text{c.m.}}$. The widths of the edges are proportional to the edge weight. The color of pendular edges is magenta and the color of the nonpendular edges is cyan. Plots (a)–(c) use the number of trips as weight and plots (d)–(f) use the number of passengers as weight. (a) Morning $h_{\text{c.m.}}$, trips; (b) afternoon $h_{\text{c.m.}}$, trips; (c) $R_{\text{c.m.}}$, trips; (d) morning $h_{\text{c.m.}}$, passengers; (e) afternoon $h_{\text{c.m.}}$, passengers; and (f) $R_{\text{c.m.}}$, passengers.