Invariance under the order of updating in automata networks

Abstract

The dynamics of automata networks can be computed according to three possible orders of updating: parallel, serial order, and random. The use of small networks enabled us to state the problem precisely and accurately, especially for random updating. Invariance theorems are proven for most general cases, showing that (i) fixed attractors are common to all orders of updating and (ii) limit cycles are found which are common to all unspecified modes of random updating. While they are common to other modes of updating, the converse is not true. We discuss the use of large networks of networks to assess the variations induced by the order of updating on the stability of the attractors and on the relative size of the basins of attraction.