Abelian Networks I. Foundations and Examples

In Dhar’s model of abelian distributed processors, finite automata occupy the vertices of a graph and communicate via the edges. A local commutativity condition ensures that the output of such a network does not depend on the order in which the automata process their inputs. In this paper (the first of a series) we consider the halting problem for such networks and the crticial group, an invariant that governs the behavior of the network on large inputs. Our main results are 1. A finite abelian network halts on all inputs if and only if its Laplacian is positive definite; 2. The critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network; 3. The critical group is 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.