  The dynamical properties of finite dynamical systems (FDSs) have been
investigated in the context of coding theoretic problems, such as network
coding, and in the context of hat games, such as the guessing game and
Winkler's hat game. The instability of an FDS is the minimum Hamming distance
between a state and its image under the FDS, while the stability is the minimum
of the reciprocal of the Hamming distance; they are both directly related to
Winkler's hat game. In this paper, we study the value of the (in)stability of
FDSs with prescribed interaction graphs. The first main contribution of this
paper is the study of the maximum stability for interaction graphs with a loop
on each vertex. We determine the maximum (in)stability for large enough
alphabets and also prove some lower bounds for the Boolean alphabet. We also
compare the maximum stability for arbitrary functions compared to monotone
functions only. The second main contribution of the paper is the study of the
average (in)stability of FDSs with a given interaction graph. We show that the
average stability tends to zero with high alphabets, and we then investigate
the average instability. In that study, we give bounds on the number of FDSs
with positive instability (i.e fixed point free functions). We then conjecture
that all non-acyclic graphs will have an average instability which does not
tend to zero when the alphabet is large. We prove this conjecture for some
classes of graphs, including cycles.
