Emergent heterogeneous scaling regimes in the functional topology of urban drainage networks

Urban drainage networks (UDNs) are essential public infrastructure systems responsible for the conveyance of stormwater and wastewater. With intensifying urbanization, UDNs have evolved into structurally complex systems whose functional organization is not adequately resolved by conventional graph-based representations. This limitation necessitates a conceptual shift from structure-centric analyses toward functionality-informed topological approaches. One such approach represents UDN layouts within a dual-mapping domain, in which pipe segments conventionally treated as edges are redefined as nodes, while their intersections are encoded as edges. Transformations into the dual-mapping domain enable network-scale comparisons, and facilitate to uncover emergent features of diverse individual complex networks, even further co-evolutionary self-similar characteristics among various networks. These features are captured in the power-law relationship between the dual-node degree k and its probability P(k) with an exponent γ, i.e., P(k) ~ k ^-γ. Nonetheless, little is known about the physical and mechanistic interpretation of the scaling exponent γ, although its values have been extensively reported across different infra-networks. In addition, dual-mapped representation commonly relies on the Hierarchical Intersection Continuity Negotiation (HICN) method; however, as this method was originally designed for road networks, it fails to capture the convergent and flow-directed nature of UDNs. Consequently, unmodified application of the HICN method can result in unintended merges, disconnections and non-reproducible dual representations. To address these conceptual and methodological limitations, this study adopts the Horton-Strahler order as a constant hierarchical criterion and integrates flow-aligned continuity criterion for merging pipe segments. We further elaborate the understanding of the UDNs’ dual-degree distribution by (1) analytically deriving γ as a function of Horton’s bifurcation and segment ratios in ideal Hortonian networks and (2) interpreting this relationship through fractal dimension to enable quantitative links between scaling and topology. Our analytical results are validated using ~ 200 UDNs in Seoul, Republic of Korea, alongside synthetic drainage networks simulated by Gibbs-model. We find two distinct topological architectures of the dual-mapped UDNs that exhibit either a single or double power-law scaling. While ~50% of the Seoul UDNs and the synthetic networks exhibited self-similarity consistent with ideal Hortonian networks, the dual-node degree distributions of the remaining networks were better described with double power-law characteristics. This double power-law behavior serves as a critical indicator of network heterogeneity, quantitatively reflecting engineering factors such as variations in pipe-type composition, sub-catchment density, and redundancy in critical conduits. Overall, the proposed method significantly improves reproducibility and strengthens the physical interpretability of complex-network indicators, offering a robust tool for monitoring UDN evolution under the pressures of urban expansion.

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).