Abelian networks III: The critical group

作者：Bond Benjamin; Levine Lionel^*

来源：Journal of Algebraic Combinatorics, 2016, 43(3): 635-663.

DOI：10.1007/s10801-015-0648-4

• Abelian distributed processors
• Asynchronous computation
• Burning algorithm
• Chip firing
• Commutative monoid action
• Laplacian lattice
• Sandpile group
• Script algorithm

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar's burning algorithm to abelian networks and estimate the running time of an abelian network on an arbitrary input up to a constant additive error.

• 出版日期2016-5