  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.
