We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.
Cite
@article{arxiv.1508.04262,
title = {Chip-firing on general invertible matrices},
author = {Johnny Guzman and Caroline Klivans},
journal= {arXiv preprint arXiv:1508.04262},
year = {2015}
}