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Canalizing Boolean Functions Maximize the Mutual Information [article]

Johannes Georg Klotz, David Kracht, Martin Bossert, Steffen Schober
2012 arXiv   pre-print
Using Fourier analysis we show that canalizing functions maximize the mutual information between an input variable and the outcome of the function.  ...  One measure to quantify this ability is the well known mutual information.  ...  In this paper we combine two approaches to show that canalizing functions maximize the mutual information for a given expectation value of the functions output.  ... 
arXiv:1207.7193v2 fatcat:oz2rsei22ncvnoyrajd6teef2u

CommentsComments on "Canalizing Boolean Functions Maximize Mutual Information"

Thomas A. Courtade
2015 IEEE Transactions on Information Theory  
In their recent paper "Canalizing Boolean Functions Maximize Mutual Information," Klotz et al. argued that canalizing Boolean functions maximize certain mutual informations by an argument involving Fourier  ...  Index Terms-Boolean functions, mutual information.  ...  showed that canalizing Boolean functions maximize certain mutual informations by way of an argument involving Fourier analysis on the hypercube [2] .  ... 
doi:10.1109/tit.2014.2375183 fatcat:xielln3vlnelnhlknfmzmekqaq

Maximizing local information transfer in Boolean networks

T Haruna, K Nakajima
2018 New Journal of Physics  
We study a Boolean network model such that rules governing the time evolution of states are not given a priori but emerge from the maximization process of local information transfer and are stabilized  ...  We argue that the stabilized rules have generic properties of real-world gene regulatory networks, being both critical and highly canalized.  ...  Informax seeks the optimal function between its input and output by maximizing the average mutual information subject to given external constraints.  ... 
doi:10.1088/1367-2630/aadbc3 fatcat:6nkivdw7hncshngloe7rzrgff4

The maximum mutual information between the output of a binary symmetric channel and a Boolean function of its input [article]

Septimia Sarbu
2017 arXiv   pre-print
We prove the Courtade-Kumar conjecture, which states that the mutual information between any Boolean function of an n-dimensional vector of independent and identically distributed inputs to a memoryless  ...  Let f:{0,1}^n →{0,1} be an n-dimensional Boolean function. Then, MI(f(X),Y) ≤ 1-H(p).  ...  They show that this mutual information between a function f that produces an output with fixed mean, µ = E [f (X)], and one input variable, X i , is maximized, if the function f is canalizing in the variable  ... 
arXiv:1604.05113v2 fatcat:lqvy2mwvv5e7zdkuatdvodqkee

Guiding the Self-organization of Random Boolean Networks [article]

Carlos Gershenson
2010 arXiv   pre-print
Random Boolean networks (RBNs) are models of genetic regulatory networks.  ...  It has recently been found that RBNs near the critical regime maximize information storage and coherent information transfer (Lizier et al, 2008) , as well as maximize Fisher information (Wang et al,  ...  Boolean functions. n o p q r s ... ...  ... 
arXiv:1005.5733v2 fatcat:7l4fswkszfaclbjqxepisnxexu

Logical Reduction of Biological Networks to Their Most Determinative Components

Mihaela T. Matache, Valentin Matache
2016 Bulletin of Mathematical Biology  
The determinative power of a node is obtained by a summation of all mutual information quantities over all nodes having the chosen node as a common input, thus representing a measure of information gain  ...  We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain.  ...  In a subsequent paper by Klotz et al. (2014) , it is shown that canalizing Boolean functions maximize the mutual information under the same assumption as in Heckel et al. (2013) .  ... 
doi:10.1007/s11538-016-0193-x pmid:27417985 pmcid:PMC4993808 fatcat:bqjjg5shlvclvjutqptngpihbu

The maximum mutual information between the output of a discrete symmetric channel and several classes of Boolean functions of its input [article]

Septimia Sarbu
2017 arXiv   pre-print
This conjecture states that the mutual information between any Boolean function of an n-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and  ...  Let f:{0,1}^n →{0,1} be an n-dimensional Boolean function. Then, MI(f(X),Y) ≤ 1-H(p).  ...  They show that this mutual information between a function f that produces an output with fixed mean, µ = E [f (X)], and one input variable, X i , is maximized, if the function f is canalizing in the variable  ... 
arXiv:1701.05014v2 fatcat:jhplzh4f25br7ks6vexm7a6ieu

Stratification and enumeration of Boolean functions by canalizing depth

Qijun He, Matthew Macauley
2016 Physica D : Non-linear phenomena  
Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions.  ...  This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth.  ...  Fourier analysis has shown that canalizing Boolean networks maximize mutual information [KKBS14] , implying that canalizing Boolean networks contain more information than general Boolean networks.  ... 
doi:10.1016/j.physd.2015.09.016 fatcat:j7iy667s5ndsvlwex5e6cywwfu

Guiding the self-organization of random Boolean networks

Carlos Gershenson
2011 Theory in biosciences  
Random Boolean networks (RBNs) are models of genetic regulatory networks.  ...  It has recently been found that RBNs near the critical regime maximize information storage and coherent information transfer (Lizier et al. 2008) , as well as maximize Fisher information (Wang et al.  ...  -Increase the number of canalizing functions.  ... 
doi:10.1007/s12064-011-0144-x pmid:22127955 pmcid:PMC3414703 fatcat:h3tp6gjvhzgkdjevgeubslzoye

Dynamical Instability in Boolean Networks as a Percolation Problem

Shane Squires, Edward Ott, Michelle Girvan
2012 Physical Review Letters  
Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially.  ...  We now introduce a third case, in which we consider Boolean networks with canalizing functions.  ...  The method used for our previous results can be extended to canalizing functions, but because the truth table elements in a canalizing function are not generated independently, we need to consider a new  ... 
doi:10.1103/physrevlett.109.085701 pmid:23002759 fatcat:6vivndscyjbrfh5jlfjcs3leua

Stratification and enumeration of Boolean functions by canalizing depth [article]

Qijun He, Matthew Macauley
2015 arXiv   pre-print
Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions.  ...  This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth.  ...  The evolution of canalizing Boolean networks was studied in [SD07] . Fourier analysis has shown that canalizing Boolean networks maximize mutual information [KKBS14] .  ... 
arXiv:1504.07591v1 fatcat:ktit5t73angy7cydgjmk72minm

Identification of Boolean Network Models From Time Series Data Incorporating Prior Knowledge

Thomas Leifeld, Zhihua Zhang, Ping Zhang
2018 Frontiers in Physiology  
Using vector form of Boolean variables and applying a generalized matrix multiplication called the semi-tensor product (STP), each Boolean function can be equivalently converted into a matrix expression  ...  Results: We propose a new approach to identify Boolean networks from time series data incorporating prior knowledge, such as partial network structure, canalizing property, positive and negative unateness  ...  (B) Identified Boolean network without canalizing information. (C) Identified Boolean network with prior knowlege.  ... 
doi:10.3389/fphys.2018.00695 pmid:29937735 pmcid:PMC6002699 fatcat:ngj7sxalarf5zmaztqwqnqtexu

Harmonic analysis of Boolean networks: determinative power and perturbations

Reinhard Heckel, Steffen Schober, Martin Bossert
2013 EURASIP Journal on Bioinformatics and Systems Biology  
We argue that the mutual information (MI) between a given subset of the inputs X = X_1, ..., X_n of some node i and its associated function f_i(X) quantifies the determinative power of this set of inputs  ...  Consider a large Boolean network with a feed forward structure.  ...  We used Fourier analysis of Boolean functions to study the mutual information between a function f (X) and a set of its inputs X A , as a measure of determinative power of X A over f (X).  ... 
doi:10.1186/1687-4153-2013-6 pmid:23642003 pmcid:PMC3748841 fatcat:p6emh3r4l5e4hbqf6lm2q26h7i

Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

Thomas A. Courtade, Gowtham R. Kumar
2014 IEEE Transactions on Information Theory  
Index Terms-Boolean functions, mutual information, extremal inequality, isoperimetric inequality.  ...  We pose a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs.  ...  Intuitively, Conjecture 2 concerns the structure of maximally-informative Boolean functions, while Conjecture 3 handles the inequality component of Conjecture 1.  ... 
doi:10.1109/tit.2014.2326877 fatcat:ckxbprw4abgvtpt4gs62rdbpxi

Which Boolean Functions are Most Informative? [article]

Gowtham R. Kumar, Thomas A. Courtade
2013 arXiv   pre-print
We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let X^n be i.i.d.  ...  For any Boolean function b:{0,1}^n→{0,1}, we conjecture that I(b(X^n);Y^n)≤ 1-H(α). While the conjecture remains open, we provide substantial evidence supporting its validity.  ...  This work is supported in part by the Air Force grant FA9550-10-1-0124 and by NSF Center for Science of Information under grant agreement CCF-0939370.  ... 
arXiv:1302.2512v2 fatcat:o77abau4ovg7vimlxvhwjxydx4
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