Boolean networks with variable number of inputs (K).

Abstract

We studied a random Boolean network model with a variable number of inputs K per element. An interesting feature of this model, compared to the well-known fixed-K networks, is its higher orderliness. It seems that the distribution of connectivity alone contributes to a certain amount of order. In the present research, we tried to disentangle some of the reasons for this unexpected order. We also studied the influence of different numbers of source elements (elements with no inputs) on the network's dynamics. An analysis carried out on the networks with an average value of K=2 revealed a correlation between the number of source elements and the dynamic diversity of the network. As a diversity measure we used the number of attractors, their lengths and similarity. As a quantitative measure of the attractors' similarity, we developed two methods, one taking into account the size and the overlapping of the frozen areas, and the other in which active elements are also taken into account. As the number of source elements increases, the dynamic diversity of the networks does likewise: the number of attractors increases exponentially, while their similarity diminishes linearly. The length of attractors remains approximately the same, which indicates that the orderliness of the networks remains the same. We also determined the level of order that originates from the canalizing properties of Boolean functions and the propagation of this influence through the network. This source of order can account only for one-half of the frozen elements; the other half presumably freezes due to the complex dynamics of the network. Our work also demonstrates that different ways of assigning and redirecting connections between elements may influence the results significantly. Studying such systems can also help with modeling and understanding a complex organization and self-ordering in biological systems, especially the genetic ones.