Abelian networks II. Halting on all inputs
  Abelian networks are systems of communicating automata satisfying a local commutativity condition. We show that a finite irreducible abelian network halts on all inputs if and only if all eigenvalues of its production matrix lie in the open unit disk. 
Benjamin Bond and Lionel Levine
cs.FL cond-mat.stat-mech math.CO