Parallel Dynamical Systems over Graphs and Related Topics: A Survey

Abstract

In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called aparallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes.

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

“…The interactions among elements of a phenomenon do typically not occur simultaneously. When it occurs, it is said that the model updates synchronously or parallelly (see [25,26,[28][29][30][31][32][33][34][35][36]); otherwise it is said that the model updates asynchronously or sequentially (see [20,27,[37][38][39][40][41]). In this last case, an update order is needed to specify the sequence in which the states of the elements evolve.…”

An Algorithm for Counting the Fixed Point Orbits of an AND-OR Dynamical System with Symmetric Positive Dependency Graph

“…A dynamical system (DS) [1][2][3][4] or Boolean Network (BN) [5][6][7] S is a couple S = (G, F) where G is a directed graph called the dependency graph, F is a set of functions F = { f v |v ∈ V(G)} as defined below. We also define a |V(G)|-dimensional vector O of states.…”

Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata

“…In fact, cellular automata with two-state cells is a special kind of Boolean network where all the nodes use the same function and the links are all arranged in a regular bounded integer lattice structure. Boolean networks have attracted great attention in many different areas such as bioinformatics [7], computational processes [8], graph dynamical systems [9], and parallel discrete dynamical systems [10,11]. This paper shows the subtle relation between binomial sequences and cellular automata.…”