This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for regulatory networks, and the class of functions studied here captures a key feature of biological network dynamics, namely that a subset of one or more variables, under certain conditions, can dominate the value of a Boolean function, to the exclusion of all others. These functions have rich mathematical properties to be explored. The paper shows how the number and type of such sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function. We further connect the concept of collective canalization with the well-studied concept of the average sensitivity of a Boolean function. The relationship between Boolean functions and the dynamics of the networks they form is important in a wide range of applications beyond biology, such as computer science, and has been studied with statistical and simulation-based methods. But the rich relationship between structure and dynamics remains largely unexplored, and this paper is intended as a contribution to its mathematical foundation.

翻译：本文研究了集体管道布尔函数的数学性质，这类函数源于系统生物学中的应用。布尔网络作为调控网络的建模框架日益流行，本文研究的函数类别捕捉了生物网络动态的一个关键特征：即在某些条件下，一个或多个变量子集能够排除其他所有变量，主导布尔函数的值。这些函数具有丰富的数学性质有待探索。本文展示了这类子集的数量和类型如何影响函数行为，并定义了任一布尔函数管道强度的新度量。我们进一步将集体管道概念与布尔函数平均敏感度这一经典研究概念相联系。除生物学外，布尔函数与其所构成网络动态之间的关系在计算机科学等广泛领域具有重要意义，且已通过统计和仿真方法得到研究。然而，结构与动态之间的丰富关系仍鲜有探索，本文旨在为其数学基础做出贡献。