We show that urban drainage networks (UDNs) evolve to the state topologically similar to rivers: UDNs exhibit self-similarity as they grow beyond a small threshold size, like river networks. We present the gradual emergence of two power laws for UDNs - Hack’s law and the size distribution - during decadal growth of two distinct cities. Although an initial UDN reflects the strong influence of engineering design, expanding UDN along with urban growth leads to a scale-invariant topology. The power laws emerge during growth with exponents similar to those seen for rivers. The inevitability of the self-similar topology has significant implications for managing urban drainage infrastructure in the rapidly urbanizing world, with increasing demands on reliable provision of critical services to growing populations.

We investigated the scaling and topology of engineered urban drainage networks (UDNs) in two cities, and further examined UDN evolution over decades. UDN scaling was analyzed using two power law scaling characteristics widely employed for river networks: (1) Hack’s law of length (L)-area (A) [L / Ah] and (2) exceedance probability distribution of upstream contributing area (d) [PðA  dÞ  ad2E]. For the smallest UDNs (<2 km2), length-area scales linearly (h  1), but power law scaling (h  0.6) emerges as the UDNs grow. While PðA  dÞ plots for river networks are abruptly truncated, those for UDNs display exponential tempering [PðA  dÞ5ad2Eexp ð2cdÞ]. The tempering parameter c decreases as the UDNs grow, implying that the distribution evolves in time to resemble those for river networks. However, the power law exponent E for large UDNs tends to be greater than the range reported for river networks. Differences in generative processes and engineering design constraints contribute to observed differences in the evolution of UDNs and river networks, including subnet heterogeneity and nonrandom branching.