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Transportation and distribution networks are a class of spatial networks thathave been of interest in recent years. These networks are often characterizedby the presence of complex structures such as central loops paired withperipheral branches, which can appear both in natural and man-made systems,such as subway and railway networks. In this study, we investigate theconditions for the emergence of these non-trivial topological structures in thecontext of human transportation in cities. We propose a minimal model forspatial networks generation, where a network lattice acts as a spatialsubstrate and edge velocities and distances define an effective temporaldistance which quantifies the efficiency in exploring the urban space. Complexnetwork topologies can be recovered from the optimization of joint networkpaths and we study how the interplay between a flow probability between twonodes in space and the associated travel cost influences the resulting optimalnetwork. In the perspective of urban transportation we simulate these flows bymeans of human mobility models to obtain Origin-Destination matrices. We findthat when using simple lattices, the obtained optimal topologies transitionfrom tree-like structures to more regular networks, depending on the spatialrange of flows. Remarkably, we find that branches paired to large loopsstructures appear as optimal structures when the network is optimized for aninterplay between heterogeneous mobility patterns of small range travels andlonger range ones typical of commuting. Finally, we show that our framework isable to recover the statistical spatial properties of the Greater London Areasubway network.

Emergence of complex network topologies from flow-weighted optimization of network efficiency