Physarum Can Compute Shortest Paths

Physarum polycephalum is a slime mold that is apparently able to solve shortest path problems. A mathematical model has been proposed by Tero, Kobayashi and Nakagaki [Journal of Theoretical Biology, 244, 2007, pp. 553--564] to describe the feedback mechanism used by the slime mold to adapt its tubular channels while foraging two food sources s0 and s1. We prove that, under this model, the mass of the mold will eventually converge to the shortest s0-s1 path of the network that the mold lies on, independently of the structure of the network or of the initial mass distribution. This matches the experimental observations by Tero et al. and can be seen as an example of a "natural algorithm", that is, an algorithm developed by evolution over millions of years.