  The dynamical properties of finite dynamical systems (FDSs) have been
investigated in the context of coding theoretic problems, such as network
coding and index coding, and in the context of hat games, such as the guessing
game and Winkler's hat game. In this paper, we relate the problems mentioned
above to properties of FDSs, including the number of fixed points, their
stability, and their instability. We first introduce the guessing dimension and
the coset dimension of an FDS and their counterparts for digraphs. Based on the
coset dimension, we then strengthen the existing equivalences between network
coding and index coding. We also introduce the instability of FDSs and we study
the stability and the instability of digraphs. We prove that the instability
always reaches the size of a minimum feedback vertex set. We also obtain some
non-stable bounds independent of the number of vertices of the graph. We then
relate the stability and the instability to the guessing number. We also
exhibit a class of sparse graphs with large girth that have high stability and
high instability; our approach is code-theoretic and uses the guessing
dimension. Finally, we prove that the affine instability is always
asymptotically greater than or equal to the linear guessing number.
