Topological invariants in microscopic transport on rough landscapes: Morphology, hierarchical structure, and Horton analysis of riverlike networks ofvortices

River basins as diverse as the Nile, the Amazon, and the Mississippi satisf y certain topological invariants known as Horton's laws. Do these macroscop ic (up to 10(3) km) laws extend to the micron scale? Through realistic simu lations, we analyze the morphology and statistical properties of networks o f vortex flow in flux-gradient-driven superconductors. We derive a phase di agram of the different network morphologies, including one in which Horton' s laws of length and stream number are obeyed-even though these networks ar e about 10(9) to 10(15) times smaller than geophysical river basins. [S0031 -9007(98)08235-0].