Abelian networks III. The critical group

TL;DR

This paper explores the critical group of abelian networks, revealing its structure, action on recurrent states, and generalizing key algorithms, thereby deepening understanding of network behavior on large inputs.

Contribution

It introduces a detailed structure of the critical group for irreducible abelian networks, generalizes Dhar's burning algorithm, and estimates network running time.

Findings

Critical group acts freely and transitively on recurrent states

Critical group is a quotient of a free abelian group by a Laplacian-related subgroup

Generalized Dhar's burning algorithm for abelian networks

Abstract

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.