Topological Invariants in Microscopic Transport on Rough Landscapes: Morphology, Hierarchical Structure, and Horton Analysis of Riverlike Networks of Vortices

River basins as diverse as the Nile, the Amazon, and the Mississippi satisfy certain topological invariants known as Horton's laws. Do these macroscopic (up to 10^3 km) laws extend to the micron scale? Through realistic simulations, we analyze the morphology and statistical properties of networks of vortex flow in flux-gradient-driven superconductors. We derive a phase diagram of the different network morphologies, including one in which Horton's laws of length and stream number are obeyed -- even though these networks are about 10^9 to 10^15 times smaller than geophysical river basins.