Deciphering Heterogeneous Scaling Properties in Urban Drainage Networks

Urban drainage networks (UDNs) constitute essential infrastructure for mitigating urban flooding hazards and public health risks by collecting and conveying stormwater and wastewater. Accordingly, much of the existing literature has focused on improving UDN functional performance, typically through detailed process-based modeling approaches. While these methods have substantially advanced UDN design and operational analysis, their intensive data requirements and high computational costs have highlighted the need for complementary perspectives that can infer UDNs’ functional characteristics directly from network topology. Flows within UDNs are predominantly gravity-driven and organized along converging drainage paths, analogous to those observed in natural river networks. This physical resemblance implies fundamental commonalities in the structural organization of UDNs and rivers, motivating the characterization of UDN topology using scaling laws originally found from river networks in a concise and physically grounded manner. Building on this perspective, recent studies have shown that UDNs exhibit self-similarity analogous to that of natural river networks. These findings naturally prompt further inquiry into: (1) To what extent do scaling properties represent a homogeneous feature of UDN topologies? (2) If not, how should heterogeneity in scaling properties be interpreted through functional or physical perspectives? To address these questions, we analyzed ~220 UDNs (~4,000 km) constructed in Seoul (~605 km², ~9.7M people), South Korea, a representative megacity in Asia. Three classical scaling features identified in river networks were applied to UDNs: (i) power function in the area-length relationship (scaling exponent h), (ii) power function in the area exceedance probability distribution (scaling exponent ε), and (iii) a set of Hortonian scaling ratios (i.e., bifurcation, length, and area ratios). For the interpretation of scaling properties in UDNs, we considered ~20 descriptive indicators covering topographic, geometric, structural, and socio-economic domains. We found heterogeneity in the studied UDNs’ scaling properties. First, the power-law scaling exponents, h and ε, exhibited broader distributions and smaller values (0.2 < h < 0.9; 0.05 < ε < 0.41), respectively, compared to those typically reported for natural river networks (0.5 < h < 0.7; 0.40 < ε < 0.46). This reflects the influence of urban constraints, including road layouts and building distributions. Moreover, only about half of the analyzed UDNs simultaneously satisfied the three Hortonian scaling ratios, indicating that hierarchical scaling is not a universal property for UDNs. This heterogeneity in hierarchical scaling is closely linked to network size, drainage efficiency, structural maturity, and specific socio-economic characteristics. Our findings are expected to not only provide a fundamental basis for coupling the structural and functional characteristics of UDNs but also offer a conceptual foundation for bridging the gap between network topology and functional resilience in UDNs.

Acknowledgements

This work was supported by the Creative-Pioneering Researchers Program through Seoul National University and by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2025-00523350).