Nested Canalizing Functions and Their Networks
The concept of a nested canalizing Boolean function has been studied over the last decade in the context of understanding the regulatory logic of molecular interaction networks, such as gene regulatory networks. Such networks are predominantly governed by nested canalizing functions. Derrida values are frequently used to analyze the robustness of a Boolean network to perturbations. This paper introduces closed formulas for the calculation of Derrida values of networks governed by Boolean nested canalizing functions, which previously required extensive simulations. Recently, the concept of nested canalizing functions has been generalized to include multistate functions, and a recursive formula has been derived for their number, as a function of the number of variables. This paper contains a detailed analysis of the class of nested canalizing functions over an arbitrary finite field. In addition, the concept of nested canalization is further generalized and closed formulas for the number of such generalized functions, as well as for the number of equivalence classes under permutation of variables, are derived. The latter is motivated by the fact that two nested canalizing functions that differ only by a permutation of the variables share many important properties.