Geodesic
A Variation on Chip-Firing: the diffusion game
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 1.

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We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research. 
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     author = {Duffy, C. and Lidbetter, T. F. and Messinger, M. E. and Nowakowski, R. J.},
     title = {A {Variation} on {Chip-Firing:} the diffusion game},
     journal = {Discrete mathematics & theoretical computer science},
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Duffy, C.; Lidbetter, T. F.; Messinger, M. E.; Nowakowski, R. J. A Variation on Chip-Firing: the diffusion game. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 1. doi : 10.23638/DMTCS-20-1-4. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-1-4/

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