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Simple dynamics on graphs.

Gadouleau, Maximilien and Richard, Adrien (2016) 'Simple dynamics on graphs.', Theoretical computer science., 628 . pp. 62-77.


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→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 that converges toward a unique fixed point in at most ⌊log2⁡n⌋+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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2016 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:08 March 2016
Date deposited:30 August 2016
Date of first online publication:10 March 2016
Date first made open access:10 March 2017

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