A comparative analysis of underground and bus transit networks through graph theory

The aim of this paper is to study the topographical features of a transportation infrastructure through graph theory. First, we construct a planar, connected, and simple graph for each considered infrastructure; then, we compute some normalized indices associated to the graph, namely largest eigenvalue, gap, a Betti number, and codimension. The set of indices proposed in this paper is new for this application. These indices are computed from either the adjacency matrix or the edge ideal of the graph, and so they depend on the overall topology of the graph itself; furthermore, since the normalized indices are scale-free, they allow us a more effective comparison between different transportation infrastructures. Two scenarios are considered in order to understand advantages and limits of the proposed approach: the first scenario concerns a set of underground networks of certain large cities in the world, whereas the second one concerns a set of bus transit networks of several medium-sized cities in Italy. Indices calculated for both scenarios show two types of results. First, they show that the proposed indices are able to estimate the different topologies of the considered networks: networks with the same number of vertices and of edges but not with the same graph have different indices. Second, they show that the values of the indices in the two scenarios not only belong to the same curve separately but fit well also into the same curve: the transportation networks, no matter whether underground or bus transit, seem to be controlled by similar mechanisms.