Simple dynamics on graphs

Does the interaction graph of a finite dynamical system can force this system to have a “complex” dynamics ? In other words, given a finite interval of integers A, which are the signed digraphsG such that every finite dynamical system f : An → An with G as interaction graph has a “complex” dynamics ? If |A| ≥ 3 we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph G there exists a system f : An → An with G as interaction graph such that f ⌊log2 n⌋+2 is a constant. The boolean case |A| = 2 is more difficult, and we provide partial answers instead. We exhibit large classes of unsigned digraphs which admit boolean dynamical systems which converge in linear time. We also prove that any symmetric digraph, and any graph with a loop on each vertex admits a boolean dynamical system which converges in constant time.