Abstract
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.
Highlights
Many entities organise themselves as complex networks, where each entity has a finitely valued state and a function which the electronic journal of combinatorics 26(3) (2019), #P3.32 updates the value of the state
Since entities influence each other, this local update function depends on the states of some of the entities. Such a network is called a Finite Dynamical System (FDS), with special cases or variants appearing under different names, such as Boolean Networks [12, 20], Boolean Automata Networks [15], Multi-Valued Networks [4], etc
The main problem when studying an FDS is to determine its dynamics, such as the number of its fixed points, or how the trajectory of a state depends on the initial state
Summary
Many entities (such as genes, neurons, persons, computers, etc.) organise themselves as complex networks, where each entity has a finitely valued state and a function which the electronic journal of combinatorics 26(3) (2019), #P3.32 updates the value of the state. The structure of an FDS f : An → An can be represented via its interaction graph G(f ), which indicates which update functions depend on which variables. Winkler’s hat game was studied in [5, 8, 7, 19, 6, 13, 1] This hat game, and a dual version where the players aim to guess the colour incorrectly, can be formalised in terms of the stability and the instability of FDSs with prescribed interaction graphs [7]. This time, the average stability is relatively easy to handle, while the instability is more interesting We relate the latter problem to the number of fixed-point free functions with a prescribed interaction graph.
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