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.

中文翻译：

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共同处理布尔函数

本文研究了共同处理布尔函数的数学特性，布尔函数是一类从系统生物学应用中产生的函数。布尔网络是一种越来越流行的监管网络建模框架，这里研究的函数类别捕捉了生物网络动力学的一个关键特征，即一个或多个变量的子集在某些条件下可以支配布尔函数的值，排除所有其他人。这些函数具有丰富的数学特性有待探索。该论文展示了此类集合的数量和类型如何影响函数的行为，并为任何布尔函数的运河强度定义了新的度量。我们进一步将集体运河的概念与经过充分研究的布尔函数的平均灵敏度概念联系起来。布尔函数与其形成的网络动力学之间的关系在生物学以外的广泛应用中非常重要，例如计算机科学，并且已经使用基于统计和模拟的方法进行了研究。但是结构和动力学之间的丰富关系在很大程度上仍未得到探索，本文旨在为其数学基础做出贡献。