Canalizing Boolean Functions Maximize Mutual Information

Klotz, Johannes; Kracht, David; Bossert, Martin; Schober, Steffen

doi:10.1109/tit.2014.2304952

Cited by 17 publications

(13 citation statements)

References 35 publications

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“…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 ).…”

Section: Introductionmentioning

confidence: 90%

“…(2014) it is shown that MI is maximized for canalizing functions. However, real networks do not consist of a single type of Boolean function.…”

Section: Final Commentsmentioning

confidence: 99%

“…( 2013 ), Klotz et al. ( 2014 ), or Kahn et al. ( 1988 ) where it is essential in obtaining formulas for (and other related quantities relevant in the study of BNs) in terms of the Fourier coefficients of [see, e.g., (Kahn et al.…”

Section: Influence Sensitivity and Strengthmentioning

confidence: 99%

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Logical Reduction of Biological Networks to Their Most Determinative Components

Matache

2016

Bull Math Biol

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Boolean networks have been widely used as models for gene regulatory networks, signal transduction networks, or neural networks, among many others. One of the main difficulties in analyzing the dynamics of a Boolean network and its sensitivity to perturbations or mutations is the fact that it grows exponentially with the number of nodes. Therefore, various approaches for simplifying the computations and reducing the network to a subset of relevant nodes have been proposed in the past few years. We consider a recently introduced method for reducing a Boolean network to its most determinative nodes that yield the highest information gain. 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 obtained by the knowledge of the node under consideration. The determinative power of nodes has been considered in the literature under the assumption that the inputs are independent in which case one can use the Bahadur orthonormal basis. In this article, we relax that assumption and use a standard orthonormal basis instead. We use techniques of Hilbert space operators and harmonic analysis to generate formulas for the sensitivity to perturbations of nodes, quantified by the notions of influence, average sensitivity, and strength. Since we work on finite-dimensional spaces, our formulas and estimates can be and are formulated in plain matrix algebra terminology. We analyze the determinative power of nodes for a Boolean model of a signal transduction network of a generic fibroblast cell. We also show the similarities and differences induced by the alternative complete orthonormal basis used. Among the similarities, we mention the fact that the knowledge of the states of the most determinative nodes reduces the entropy or uncertainty of the overall network significantly. In a special case, we obtain a stronger result than in previous works, showing that a large information gain from a set of input nodes generates increased sensitivity to perturbations of those inputs.

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“…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 ).…”

Section: Introductionmentioning

confidence: 90%

“…(2014) it is shown that MI is maximized for canalizing functions. However, real networks do not consist of a single type of Boolean function.…”

Section: Final Commentsmentioning

confidence: 99%

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Logical Reduction of Biological Networks to Their Most Determinative Components

Matache

2016

Bull Math Biol

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“…In particular, E[X], H(X), and I(X; Y) denote expectation, entropy, and mutual information, respectively. Motivated by problems in computational biology [4], Kumar…”

Section: Introduction and Mainmentioning

confidence: 99%

Dictator functions maximize mutual information

Pichler¹,

Piantanida²,

Matz³

2018

Ann. Appl. Probab.

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Let (X, Y) denote n independent, identically distributed copies of two arbitrarily correlated Rademacher random variables (X, Y). We prove that the inequality I(f (X); g(Y)) ≤ I(X; Y) holds for any two Boolean functions: f, g : {−1, 1} n → {−1, 1} (I( • ; •) denotes mutual information). We further show that equality in general is achieved only by the dictator functions f (x) = ±g(x) = ±xi, i ∈ {1, 2, . . . , n}.

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“…The evolution of canalizing Boolean networks was studied in [SD07]. Fourier analysis has shown that canalizing Boolean networks maximize mutual information [KKBS14]. An exact formula was derived for the number of Boolean canalizing functions in [JSK04].…”

Section: Introductionmentioning

confidence: 99%

Stratification and enumeration of Boolean functions by canalizing depth

Macauley

2016

Physica D: Nonlinear Phenomena

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ABSTRACT. Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively "picked off", until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. 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. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.

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Canalizing Boolean Functions Maximize Mutual Information

Cited by 17 publications

References 35 publications

Logical Reduction of Biological Networks to Their Most Determinative Components

Logical Reduction of Biological Networks to Their Most Determinative Components

Dictator functions maximize mutual information

Stratification and enumeration of Boolean functions by canalizing depth

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