Chip-firing (Bak-Tang-Wiesenfeld inspired) is an abelian sandpile model on a graph. Each vertex v has chips[v] chips. When chips[v] ≥ deg(v), vertex v
fires: it sends one chip to each neighbor. A sink vertex absorbs chips. The process is abelian — order of firings doesn't matter.
v fires ⟺ chips[v] ≥ deg(v) | chips[v] -= deg(v) | chips[w] += 1 for w~v
The
sandpile group is isomorphic to the critical group of the graph. Avalanche sizes follow power-law distributions in the self-organized critical state. The number of firings of each vertex in a complete firing sequence equals the number of spanning trees of certain subgraphs.