Simple dynamics on graphs

Can the interaction graph of a finite dynamical system force this system to have a “complex” dynamics? In other words, given a finite interval of integers A, which are the signed digraphs G   such that every finite dynamical system f:An→Anf:An→An with G   as interaction graph has a “complex” dynamics? If |A|≥3|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→Anf:An→An with G   as interaction graph that converges toward a unique fixed point in at most ⌊log2⁡n⌋+2⌊log2⁡n⌋+2 steps. The boolean case |A|=2|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 toward a unique fixed point in polynomial, linear or constant time.