Designing efficient and optimal urban transportation networks is a complex challenge. Current approaches often analyze network properties *a posteriori* or rely on *a priori* optimization with pre-existing infrastructure. This study addresses the limitation of existing methods by proposing a novel approach that generates transportation networks from scratch, starting only from a limited set of origin and destination points. The approach leverages optimal transport theory and the principle of economy of scale, mimicking the behavior of biological systems like the slime mold *Physarum polycephalum*, which optimizes its network structure for efficient resource acquisition. The research aims to determine if urban transportation networks exhibit underlying topologies similar to those following optimality principles defined in optimal transport theory, and to develop a rigorous quantitative definition of network similarity. This is important because understanding the underlying principles governing transportation network design can lead to more efficient and cost-effective systems.

Existing research on transportation network design often focuses on *a posteriori* analysis of topological properties or *a priori* optimization of existing infrastructure. While optimization approaches exist, most rely on a pre-existing backbone, hindering the ability to design entirely new networks. Heuristic methods are common but may not generate equilibrium solutions. Bilevel formulations, considering passenger and network manager objectives, are NP-hard to solve. Biologically inspired models, like those mimicking slime mold networks, offer an alternative but lack a principled similarity metric for comparing simulated and real networks. This study builds upon this existing work by proposing a method that generates networks from scratch without an initial backbone, using a principled optimization framework based on optimal transport theory, and introducing a rigorous similarity measure.

The proposed model uses a continuous optimal transport framework inspired by the slime mold *Physarum polycephalum*. It considers a 2D urban area with origin and destination points (POIs), aiming to connect them by minimizing a cost functional that combines infrastructure and operating costs. The model uses two main quantities: transport density (conductivity) and transport potential. The dynamics are governed by two partial differential equations, whose stationary solution minimizes a Lyapunov cost functional. This functional incorporates a parameter, β, that balances infrastructure and operating costs, reflecting the principle of economy of scale. The algorithm, called Nextrout, extracts optimal network structures from the solution. Origin and destination points are selected using centrality measures from existing networks (degree and betweenness centrality) or a population and point-of-interest density measure. The algorithm was applied to 18 city networks and one national rail network, focusing on networks with few loops (*L_ratio* < 0.2). Network similarity is assessed using various metrics, including cost, total path length, traffic distribution (Gini coefficient), and density of bifurcations. A novel Wasserstein similarity measure, based on optimal transport, provides a principled way to automatically select the optimal β value for each network.

The study found that the optimal transport model successfully describes the macroscopic structure of diverse urban transportation networks. Simulated networks generated by Nextrout, using minimal input information, exhibited significant similarity to real-world networks in several cases, as measured by multiple similarity metrics (Fig. 3, 4). Tuning the parameter β allows simulating different network topologies, from shortest path-like structures to branched ones (Fig. 1, 5), reflecting a trade-off between infrastructure and operating costs. The Wasserstein similarity measure effectively selects the β value that yields a simulated network most similar to the real one (Fig. 5). Comparison of key network properties (cost, total path length, traffic distribution, and bifurcation density) showed that simulated networks often displayed similar characteristics to real networks (Fig. 4, 8). Analysis of the New York subway system, a complex network, suggested that treating each line as an independent network, and selecting POIs based on POI density, provided reasonable similarity. The study of the initial French railway system in 1850 demonstrated the model's applicability to early network development phases. Simulated networks, selected by the Wasserstein measure, consistently showed higher similarity across various properties than networks selected using other criteria (Supplementary Information S1). Finally, by varying β, the model allows exploring a wider range of network properties than observed in real networks, suggesting potential improvements in terms of cost, total path length, and traffic distribution (Fig. 8).

The findings support the hypothesis that simple optimality principles, based on optimal transport and economy of scale, can explain the macroscopic structure of many urban transportation networks. The proposed model offers a computationally efficient way to generate and compare networks, providing a valuable tool for transportation planning. The Wasserstein similarity measure provides a principled method for selecting optimal parameters and comparing simulated and real networks. The ability to simulate different network topologies by varying β allows for exploring design trade-offs and identifying potential improvements in network properties. The model’s success with diverse networks, including the initial French railway system, highlights its robustness and applicability to various scenarios.

This study presents a novel method for simulating urban transportation networks based on optimal transport theory and economy of scale. The model successfully generates networks that exhibit remarkable similarity to real-world systems, using minimal input information. The introduced Wasserstein similarity measure provides a principled approach to compare simulated and observed networks. Future research could focus on extending the model to include loops, time-dependent demands, and multiple transportation modes, leading to a more comprehensive understanding of optimal urban transportation network design. Investigating co-evolutionary dynamics between urban networks and population distribution is also an important avenue for future research.

The model's current limitation is its inability to directly generate networks with loops, which are present in many real-world systems. The assumption of static networks also simplifies reality. Future work should address these limitations by incorporating stochasticity, time-dependence, and more sophisticated modeling techniques. Furthermore, the model does not explicitly consider geographical constraints, which can influence network design in real-world scenarios. While the centrality-based origin/destination selection provides a reasonable approximation, integrating more detailed urban features could potentially improve model accuracy.