  Automata networks are mappings of the form f : Q Z $\rightarrow$ Q Z , where
Q is a finite alphabet and Z is a set of entities; they generalise Cellular
Automata and Boolean networks. An update schedule dictates when each entity
updates its state according to its local function fi : Q Z $\rightarrow$ Q. One
major question is to study the behaviour of a given automata networks under
different update schedules. In this paper, we study automata networks that are
invariant under many different update schedules. This gives rise to two
definitions, locally commutative and globally commu-tative networks. We
investigate the relation between commutativity and different forms of locality
of update functions; one main conclusion is that globally commutative networks
have strong dynamical properties, while locally commutative networks are much
less constrained. We also give a complete classification of all globally
commutative Boolean networks.
