Boolean networks are frequently used to model biological systems, such as gene regulatory networks. Canalizing functions have been suggested as suitable update rules for genetic networks and have been studied rigorously. Mathematically, the biology-inspired concept of canalization means that a single input can determine the output of a function , irrespective of all the other inputs. Here, we relax this definition and explore the broader class of collectively canalizing Boolean functions as well as the k-set canalizing proportion, which is the proportion of sets of k inputs which canalize a Boolean function. We exhibit how the number and type of these sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function. We further show how the k-set canalizing proportions connect the concept of canalization and the well-studied concept of the average sensitivity of a Boolean function  .
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 .
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Boolean networks are frequently used to model biological systems, such as gene regulatory networks. Canalizing functions have been suggested as suitable update rules for genetic networks and have been studied rigorously. Mathematically, the biology-inspired concept of canalization means that a single input can determine the output of a function
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
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irrespective of all the other inputs. Here, we relax this definition and explore the broader class of collectively canalizing Boolean functions as well as the k-set canalizing proportion, which is the proportion of sets of k inputs which canalize a Boolean function. We exhibit
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
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show how the k-set canalizing proportions
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. 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