  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 characteristics widely employed
for river networks: (1) Hack's law of length ($L$)-area ($A$) scaling [$L
\propto A^{h}$], and (2) exceedance probability distribution of upstream
contributing area $(\delta)$ [$P(A\geq \delta) \sim a \delta^{-\epsilon}$]. For
the smallest UDNs ($<2 \>\text{km}^2$), length-area scales linearly ($h\sim
1$), but power-law scaling emerges as the UDNs grow. While $P(A\geq \delta)$
plots for river networks are abruptly truncated, those for UDNs display
exponential tempering [$P(A\geq \delta) \>\text{=}\> a
\delta^{-\epsilon}\exp(-c\delta)$]. 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 $\epsilon$ for large UDNs
tends to be slightly larger 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 non-random branching.
