Similarity and economy of scale in urban transportation networks and optimal transport-based infrastructures

Transportation networks critically impact urban efficiency but are constrained by costs and physical limitations. Traditional assessments often analyze topological properties a posteriori, relating them to efficiency, cost, or robustness, which limits insight into underlying design principles. An alternative is an a priori optimization setup that defines a cost function and searches for optimal network topologies. However, many existing methods either require an initial infrastructure backbone, explore a limited set of shapes, or rely on heuristics and bilevel formulations that are difficult (NP-hard) to solve. A principled metric for comparing observed and simulated networks is commonly lacking. This work investigates whether observed urban transportation networks can be explained by simple optimal transport principles and an economy-of-scale mechanism. We pose the question: which network design principles yield simulated networks most similar to real ones when only a few origin–destination points are given? We propose a continuous-space network design framework that builds networks from scratch and quantitatively compares them with real infrastructures, tuning a single parameter to interpolate between operating and infrastructure cost regimes.

Prior studies on transportation network design include heuristic-based methods, bilevel formulations addressing operator-passenger trade-offs (NP-hard), and biologically inspired models mimicking Physarum networks. Many optimize flows on an existing backbone rather than designing from scratch, or restrict to predetermined geometries. Physarum-inspired works showed comparable transportation properties to real systems (e.g., Tokyo rail) but did not establish rigorous network similarity metrics. Other approaches examine scaling laws, connectivity, and efficiency, suggesting underlying optimality, yet each property captures only facets of design mechanisms. Recent optimal transport formulations (Dynamic/Monge–Kantorovich dynamics) and related network extraction methods motivate a continuous-space, optimization-driven approach that can yield network-like structures and support principled graph comparison via optimal transport distances.

• Problem setup: Consider a 2D urban area with a small set of origin (sources) and destination (sinks) points representing entry/exit of passengers. No initial backbone is assumed; nodes and edges are selected by optimization in continuous space.
• Dynamics (DMK-inspired): Two fields evolve on the 2D domain: transport potential and conductivity (transport density). The flux follows Fick–Poiseuille law q = −μ∇u. A dynamic system enforces mass conservation and adapts conductivities in response to fluxes toward a stationary equilibrium. Under assumptions, the equilibrium minimizes a Lyapunov functional.
• Cost functional: The equilibrium minimizes L(u, μ) = ∫(1/2)|∇u|^2 dx + (β/2) ∫ μ^2/(β−1) dx. The first term represents operating cost (power dissipation/Dirichlet energy); the second is infrastructure cost. For β > 1, a principle of economy of scale promotes consolidating flows on fewer, larger edges, inducing branched structures. Tuning β controls the trade-off: β≈1 yields shortest-path-like structures; β>1 yields branched transport.
• Network extraction: From continuous equilibrium fields, apply Nextrout to extract a discrete graph (nodes, edges, weights proportional to conductivities) and filter redundancies.
• Input selection (O–D): To limit inputs, choose few origins and one destination. Primary criterion uses network centralities from the observed network: low-degree nodes as origins and the highest-degree (or highest betweenness) node as destination. As an alternative (used for case studies), destinations can be selected using an attractiveness density combining population and POI (from OSM) via an H3 tiling; centrality-based selection approximates this.
• Data and preprocessing: Analyze multiple modes (rail, subway, tram) across 18 cities and one national rail network. Focus on largest connected components with low loop ratio (L_ratio < 0.2), consistent with the model’s propensity for loopless structures. Map geographic coordinates to [0,1]^2, deduplicate near-coincident nodes, and run multiple simulations per O–D set over β ∈ [1,2].
• Evaluation metrics: Compare observed and simulated networks via: (i) cost TL (total number of edges), (ii) flow-weighted total path length l = Σ_e |u_e| l_e, where u_e are DMK-inferred flows on edges, (iii) Gini(T) of traffic T_e=|u_e| to assess flow concentration, (iv) density of branching points DBP (fraction of degree-3 nodes). Flows for both observed and simulated graphs are obtained by a discrete DMK dynamics (e.g., at β=1.5 for consistency).
• Wasserstein similarity for automatic β selection: Build the union graph of observed G1 and simulated G2, assign edge weights (Euclidean distances), and define source/sink vectors from edge-indicator differences. Run a discrete OT-like dynamics on the union to obtain conductivities and define W1(G1, G2) = Σ_e w_e u_e. This distance measures the minimum “effort” to morph one graph into the other; lower is more similar. Use the β minimizing W1 to automatically select the most similar simulated network.

• Across multiple cities and modes (18 urban networks plus a national railway), the OT-based dynamics (Nextrout) produced simulated networks that often resemble real infrastructures using only a few O–D inputs and no initial backbone.
• Tuning β interpolates between shortest-path-like and branched topologies. In Rome, increasing β from 1 to 2 transitions from a single path to branching and back to a consolidated path; β around 1.5 yielded a branch qualitatively matching the real network, while β=2 reduced similarity despite lower infrastructure cost.
• Quantitatively, simulated networks frequently matched or approached real networks across metrics: TL (cost), flow-weighted total path length l, traffic inequality Gini(T), and DBP. Examples include Adelaide rail (similar l, cost, and traffic), and Bordeaux and Nantes trams (similar cost and traffic).
• The Wasserstein-based selection criterion for β often identified simulated networks with transportation properties more aligned with real networks than selections based on individual topological metrics, according to additional validation (Supplementary Information).
• Example case (Grenoble tram, N=80): the Wasserstein distance W1 across β indicated best similarity at β=2; disruptions such as missing small branches increased W1 (e.g., around β≈1.6).
• New York subway (complex, loopy): By treating major lines separately with a shared destination (selected via POI density), simulated line-level networks showed similar TL and DBP in several cases (e.g., red and green lines displayed comparable branching patterns), while the orange line exhibited substantial structural differences, highlighting limits of the current optimality assumptions for highly complex, loopy structures.
• Historical French rail (circa 1850): Simulations of major components (loopless stage) showed varying degrees of similarity. The largest component (including Paris) matched DBP but had higher TL in the observed network due to earlier branch splits and detours; other components showed closer alignment, with some differences plausibly due to geographical constraints not modeled.
• Coverage of the design space: Simulated networks span a wider range of metric values than real ones, enabling alternatives with lower l at comparable TL and lower traffic inequality for similar costs. Networks selected via minimal Wasserstein tend to have lower cost and fewer bifurcations (lower DBP).

The study addresses whether simple optimal transport principles and an economy-of-scale mechanism can explain the macroscopic structure of urban rail-like networks. Results across diverse cities and modes show that many real networks are consistent with branched optimal transport solutions derived from minimal inputs, suggesting simple universal rules may underlie their backbone design. The single parameter β effectively captures trade-offs between operating and infrastructure costs and modulates branching, aligning simulated structures with real ones in several cases. The Wasserstein-based similarity furnishes a principled, automatic way to select the β that yields a globally similar network across multiple properties, overcoming the ambiguity of single-metric matching. Where discrepancies arise (e.g., complex, loopy networks or detours around obstacles), the method highlights departures from the assumed optimality principles and provides candidate alternative designs with improved properties, offering actionable insights for planners.

This work introduces a continuous-space, optimal-transport-based framework that designs transportation networks from scratch using minimal O–D inputs and no backbone, guided by a cost functional balancing operating and infrastructure costs (economy of scale). The approach reproduces key features of real urban rail, tram, and subway networks across multiple cities, and supplies a principled graph similarity via a Wasserstein distance for automatic model selection. Simulated networks often match or improve upon real networks in core transportation metrics, enabling exploration of design trade-offs and potential enhancements. Future directions include extending the model to capture loops (e.g., via stochastic or time-varying demands, multicommodity formulations), integrating geographic constraints and multilayer interactions, and modeling growth dynamics to study phased development and co-evolution with population and land use.

• The OT-based dynamics and extraction procedure retrieve primarily loopless backbones; applicability is limited for networks with substantial loop density.
• Networks are treated as static; temporal evolution and phased construction are not explicitly modeled.
• Geographic and physical constraints (e.g., rivers, terrain) are not included, potentially causing path detour mismatches with real infrastructures.
• O–D selection is limited and relies on proxies (centralities or POI-based attractiveness); comprehensive demand modeling is not performed.
• The continuous-space mapping and discretization steps may introduce fine-grained mismatches when comparing to real geospatial layouts.