# On the Lyapunov Exponent of Monotone Boolean Networks

## 1. Introduction

## 2. Definitions and Preliminaries

#### 2.1. Monotone Boolean Functions

#### 2.2. The Structure of Special Monotone Boolean Functions

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. Main Results

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

- $\tilde{x}\in {E}^{n,\left(n-3\right)/2},$$f\left(\tilde{x}\right)=1,$${x}_{j}=1$
- $\tilde{x}\in {E}^{n,\left(n-3\right)/2},$$f\left(\tilde{x}\right)=0,$${x}_{j}=0$
- $\tilde{x}\in {E}^{n,\left(n+1\right)/2},$$f\left(\tilde{x}\right)=1,$${x}_{j}=1$
- $\tilde{x}\in {E}^{n,\left(n+1\right)/2},$$f\left(\tilde{x}\right)=0,$${x}_{j}=0$

- $\tilde{x}\in {E}^{n,\left(n-1\right)/2},$$f\left(\tilde{x}\right)=1,$${x}_{j}=1$
- $\tilde{x}\in {E}^{n,\left(n-1\right)/2},$$f\left(\tilde{x}\right)=0,$${x}_{j}=0$
- $\tilde{x}\in {E}^{n,\left(n+3\right)/2},$$f\left(\tilde{x}\right)=1,$${x}_{j}=1$
- $\tilde{x}\in {E}^{n,\left(n+3\right)/2},$$f\left(\tilde{x}\right)=0,$${x}_{j}=0$

## Funding

## Acknowledgments

## Conflicts of Interest

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f | Boolean function |

n | number of variables of f |

$\partial f\left(\tilde{x}\right)/\partial {x}_{j}$ | partial derivative of f with respect to ${x}_{j}$ |

$\tilde{x}=\left({x}_{1},\dots ,{x}_{n}\right)$ | vector of n binary values |

${\alpha}_{j}^{f}$ | activity of variable ${x}_{j}$ |

${s}^{f}$ | average sensitivity of a Boolean function f |

$\rho (\tilde{\alpha},\tilde{\beta})$ | Hamming distance between $\tilde{\alpha}$ and $\tilde{\beta}$ |

$M\left(n\right)$ | set of monotone Boolean functions of n variables |

${M}_{0}\left(n\right)$ | set of special monotone Boolean functions of n variables |

${E}^{n}$ | Boolean n-cube or ${\left\{0,1\right\}}^{n}$ with vertices $\tilde{x}$ |

${E}^{n,k}$ | $\{\tilde{x}\in {E}^{n}|\rho (\tilde{x},\tilde{0})=k\}$ or the kth layer of ${E}^{n}$ |

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