Iterated maps for annealed Boolean networks.

Boolean networks are used to study the large-scale properties of nonlinear systems and are mainly applied to model genetic regulatory networks. A statistical method called the annealed approximation is commonly used to examine the dynamical properties of randomly generated Boolean networks that are created with selected statistical features. However, in the literature there are several variations of the annealed approximation. These approximations cannot be interchangeably used in all cases due to different background assumptions. In this paper, we present the so-called four-state model, derive the different approximations from this model, and make the differences and connections between these approximations explicit. As an application of the presented results, we study the properties of the Boolean networks that are constructed with random functions, canalizing functions, and regulatory functions found in the biological literature.