On the Stability and Instability of Finite Dynamical Systems with Prescribed Interaction Graphs

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.

On the Rank and Periodic Rank of Finite Dynamical Systems

A finite dynamical system is a function $f : A^n \to A^n$ where $A$ is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.