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Simple dynamics on graphs

Abstract : 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 $f:A^n\to A^n$ with $G$ as interaction graph has a ``complex'' dynamics ? If $|A|\geq 3$ we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph $G$ there exists a system $f:A^n\to A^n$ with $G$ as interaction graph that converges toward a unique fixed point in at most $\lfloor\log_2 n\rfloor+2$ steps. 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 toward a unique fixed point in polynomial, linear or constant time.
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Maximilien Gadouleau, Adrien Richard. Simple dynamics on graphs. Theoretical Computer Science, Elsevier, 2016, 628, pp.62-77. ⟨10.1016/j.tcs.2016.03.013⟩. ⟨hal-01298053⟩

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