Fig. 1: Problem setup for the subway network of Rome.

a Given a set of real latitude-longitude coordinates denoting origins (green) and destinations (red), we aim to build a network structure that resembles well the observed public transportation network connecting those points, as in (b). c–e Intuitive ways to build a network structure by connecting origins and destinations, versus networks extracted with our optimal transport-based method in f–h. The only known information is the set of six origins (O) and one destination (D). We capture different optimization mechanisms by tuning the β parameter: in (f), the network is the shortest path-like structure, while in (g) and (h) we show examples of branched transportation schemes. This information is added to the population density across multiple urban areas (2019)60, where darker (lighter) colors indicate higher (lower) densities. L denotes the total length of the network, measured in kilometers.