A Variation on Chip-Firing: the diffusion game

We introduce a natural variant of the parallel chip-firing game, called thediffusion game. Chips are initially assigned to vertices of a graph. At everystep, all vertices simultaneously send one chip to each neighbour with fewerchips. As the dynamics of the parallel chip-firing game occur on a finite setthe process is inherently periodic. However the diffusion game is not obviouslyperiodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, theremay exist time steps where some vertices have a negative number of chips. Weinvestigate the process, prove periodicity for a number of graph classes, andpose some questions for future research.Comment: 18 pages, 3 figure