Dynamics of Directed Boolean Networks under Generalized Elementary Cellular Automata Rules, with Power-Law Distributions and Popularity Assignment of Parent Nodes

This study provides an analysis of the dynamics of fixed-size directed Boolean networks governed by generalizations of elementary cellular automata rules 22 and 126, under a power-law distribution of parent nodes and a popularity parent assignment. The analysis shows the existence of a two-piece chaotic attractor for smaller values of the power-law parameter which evolves into a “cloud”-like attractor for larger values of the parameter. Values of the parameter for which the system exhibits an ordered behavior are indicated as well. The dynamics are investigated using space-time diagrams, delay plots, bifurcation diagrams, and Lyapunov exponent computations. It is also shown that the children (out)links do not obey a power-law distribution; more precisely, numerical investigations indicate that the children links have a Gaussian-like distribution.