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The dynamical properties of finite dynamical systems (FDSs) have beeninvestigated in the context of coding theoretic problems, such as networkcoding, and in the context of hat games, such as the guessing game andWinkler's hat game. The instability of an FDS is the minimum Hamming distancebetween a state and its image under the FDS, while the stability is the minimumof the reciprocal of the Hamming distance; they are both directly related toWinkler's hat game. In this paper, we study the value of the (in)stability ofFDSs with prescribed interaction graphs. The first main contribution of thispaper is the study of the maximum stability for interaction graphs with a loopon each vertex. We determine the maximum (in)stability for large enoughalphabets and also prove some lower bounds for the Boolean alphabet. We alsocompare the maximum stability for arbitrary functions compared to monotonefunctions only. The second main contribution of the paper is the study of theaverage (in)stability of FDSs with a given interaction graph. We show that theaverage stability tends to zero with high alphabets, and we then investigatethe average instability. In that study, we give bounds on the number of FDSswith positive instability (i.e fixed point free functions). We then conjecturethat all non-acyclic graphs will have an average instability which does nottend to zero when the alphabet is large. We prove this conjecture for someclasses of graphs, including cycles.

On the stability and instability of finite dynamical systems with prescribed interaction graphs