Random Boolean Networks – Statistics of Attractors and their Basins of Attraction

Our research focuses on studying the statistics of attractors in different phases of Kauffman networks, primarily how the cycle length influences the size of the basins of attraction. We could observe that these two have a monotone increasing functional dependence on each other in the chaotic phase. Moreover, when properly normalized, the average basin size for a fixed cycle length stays the same for different system sizes; thus is an intensive property. We tested this for random boolean networks varying from 6 to 20 nodes. On the critical line and in the frozen phase, they seem not to be independent of each other, but the averages of the basin sizes vary quite strongly. Cycles with a length of the least common multiple of at least two prime numbers tend to have smaller basins of attraction. Vice versa, in such networks, we find a greater number of attractors. This has to be the case since the averaged normalized basin size is inversely proportional to the number of attractors. Therefore, we assume the existence of sub-attractors, which are attractors of disjunct node sets. The parallel cycling through sub-attractors gives rise to the longer ones and can partially explain why such systems have more attractors on average.