Based on the analysis of the edge distributions of the supply and demand networks, it was necessary to verify their modularity, in other words, we needed to verify if the networks have sub-networks with a high coefficient of grouping and low connectivity with other sub-networks. In order to compute the modularity of the supply and demand networks it was used the Blondel et al. (2008) [33] algorithm.
Figure 5 depicts the relation between the number of communities and their respective modularities for the supply and demand networks. The orange circles mark the modularity of the supply network. The yellow symbols mark the modularity of the demand network. The modularity supply network is larger than the demand network in virtually all configurations. This result reveals that the supply network seems to be designed to cater for shorter movements than what in fact people need. The modularities of networks tend to converge when they are divided into fewer communities, with a greater approximation of values when the networks are divided into exactly ten communities, with a modularity of ≈0.83.
The modularity values found in the supply network indicate that the edges within the communities of this network are stronger than the edges found in the communities of the demand network, possibly because the users of the bus system travel longer stretches than expected (in the supply network), increasing the weight of local edges and preventing community detection algorithms from finding configurations with the level of modularity found in the supply network.
In order to identify the bus lines that are overloaded, lines that have a greater demand than they can support, and bus lines where there is loss of resources, which are lines that have a very large supply for their demand, the configurations with ten communities of both networks were used. These configurations have ≈0.83 of modularity and the algorithm used [33] removed 241 edges of the supply network and 380 edges in the demand network in the detection of the ten communities. An overload index and a waste index were verified to identify the problematic stretches in the network.
In Fig. 6a there are illustrated stretches of the supply network where it is overloaded, these stretches are traversed by bus lines that are possibly with crowded vehicles. These stretches were identified from the edges removed by the method of detecting communities proposed by Blondel et al. (2008) [33]. A weak edge removed (by the method of detecting communities) from the supply network can mean a supply bottleneck, since this edge connects inter-communal movements that have already been proved to be predominant [3]. In all the edges that were removed, the overload index was calculated IS=ws/wsmax−wd/wdmax. Edges with IS<0 are sections where supply is greater than demand, and potentially where bottlenecks are. It was also observed that bus lines that have a bottleneck usually take people to the city center (highlighted by the red circle).
In Fig. 6b, network stretches are shown where there are potentially a waste of resources, i.e., empty buses. These stretches were identified from the edges removed by the method of detecting communities, now in the demand network. Community detection methods remove weak connections between components with high cluster coefficient, which in the case of the demand network are sub-regions of intense passenger flow. In this context, the edges eliminated by the method in question represent stretches from the network of low passenger movement, where potential resource waste may occur. In a similar way to what was done in the supply, we calculate the waste index ID=wd/wdmax−ws/wsmax, where edges with ID<0 represent segments where proportionally demand is greater than supply, and consequently where resources are being wasted.
The proposed index, IS, identified that the lines 503, 029, 014, 605, 725, 379, 087, 325, 316, 013, 130, 815, 030, 077, 011, 650, 755, 361, 612, and 075 have stretches in their itineraries where the vehicles are crowded, while ID identified that lines 394, 315, 024, 220, 605, 709, 087, 013, 050, 311 and 317 are at some point in their itinerary virtually empty. While validating on location with professionals from ETUFOR (Urban Transport Company of Fortaleza) we were informed that it is known that on some days of the week the lines 503, 029, 014, 605, 725, 379, 087, 325, 316, 013, 130, 815, 030, 077, 075 and 011 are crowded in the highlighted stretches. As for the lines 650, 755, 361 and 612, the professionals were surprised that they were full, and informed us that they would be evaluated in future actions. Regarding to the lines with few passengers the professionals chose not to comment. This was justified by the fact that it is difficult to know if a line has been used by few passengers at a certain time, since this fact does not generate any complaints from the users.