# Exploring the Dynamic Organization of Random and Evolved Boolean Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}(t) (i = 1…N), which can take either continuous or discrete values, is associated to every node i of the network at time t, and a deterministic law (e.g., a differential or finite difference equation) determines the time behavior of x

_{i}. If this equation contains at least one term which depends upon x

_{j}then there is a link from node j to node i.

## 2. Materials and Methods

#### 2.1. The Relevance Index

_{1}, x

_{2}… x

_{n}} describing a system whose status changes along a collections of m observations, and a subset S of U composed of k elements. Its integration I(S), semidefinite positive, is defined as:

_{j}terms denoting the cardinality of the alphabet of the j-th variable belonging to the subset S to be analyzed.

_{3}= 3.0 and θ

_{5}= 5.0, which in the case of Gaussian statistics correspond, respectively, to p-values close to 1.3⋅10

^{−3}and 2.7⋅10

^{−7}.

#### 2.2. Random Boolean Networks

_{i}(t), the state of the whole network can be represented by the vector X(t) = [x

_{1}(t), x

_{2}(t), … x

_{N}(t)]. In “classic” RBNs, each node has the same incoming connectivity, while multiple connections and auto-connections are prohibited. Each node is associated with a Boolean function, which represents the response to the signals (proteins—not represented in the model) coming from the upstream nodes. In this paper, in case of random generation of the Boolean functions, for each entry, we extract a 1 with probability p (a quantity called “bias”)—and consequently the frequency of the values equal to 0 is (p − 1).

#### 2.3. Genetic Algorithm

## 3. Results

#### 3.1. Random Avalanches and Avalanches in Random Boolean Networks

_{5}threshold, we are focusing our attention on the most dynamically active elements, while the θ

_{3}threshold has so far been used as a level below which it is no longer correct to carry out mergers (create larger collective variables) [12].

_{5}threshold. We can note that, despite the fact that the avalanches in RND systems involve a very high number of nodes, the nodes present within a RS are significantly less than the nodes present within a RS in RBN systems.

_{3}threshold (Figure 2b), with the obvious remark that the total number of nodes belonging to at least one RS increases; the average and median RSs’ sizes also increase, but the ordering between the classes remains unchanged. The same holds in all our experiments: therefore, for the sake of simplicity, in the following, we will always use the θ

_{5}threshold.

#### 3.2. Static and Dynamic Characteristics in Evolved Systems

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The average number of performed avalanches (

**red**) and the average number of nodes involved in at least one avalanche (

**green**), for the four classes of system under examination (50 systems composed by 50 nodes in each classes).

**Figure 2.**The average number of nodes present within the RSs identified in random avalanches and in avalanches occurring in critical RBN, the average number of nodes excluded from them, and the average (MaxGr_Avr) and median (MaxGr_Median) size of the maximum size RS. (

**a**) RI analysis with threshold θ

_{5}; (

**b**) RI analysis with threshold θ

_{3}.

**Figure 3.**The observed frequency of the Boolean function present in evolved systems vs. the frequency characteristic of RBN class (uniform distribution, red line). The vertical dotted line separates on the left the Boolean functions which to an input of double “1” correspond to an output equal to “0”, and on the right the Boolean functions that correspond an output equal to “1”. The red ellipses highlight the cases of significant deviations from the hypothesis of uniform distribution (at the level of 1%). (

**a**) Fit1 class systems; (

**b**) Fit2 class systems.

**Figure 4.**(

**a**) The average number of nodes present within the RSs identified in the systems under examination, the average number of nodes excluded from them, and the average (MaxGr_Avr) and median (MaxGr_Median) size of the maximum size RS (RI threshold θ

_{5}). (

**b**) The detailed RSs’ size distribution (RI threshold θ

_{5}).

**Figure 5.**The fraction of RSs, on the total of all RSs identified in the 50 systems (for each type) under examination, with size equal to at least the value on the X axis, linear (

**a**) and logarithmic (

**b**) scale. The dynamic organization of RBNs involves a number of large RSs that is much higher than that of large RSs randomly present in random avalanches. The action of evolution modified this dynamic organization in different directions for different fitness (amplifying or reducing the dimensions of the RSs).

**Table 1.**The parameter values used during simulations. In order to characterize the systems under analysis, when necessary (searching for attractors, determination of the Derrida parameter), we used 10,000 initial conditions. The used BNs are composed by nodes having two inputs each: RBNs with this topology have a critical regime if their bias is equal to 0.5.

System | Parameter | Value |
---|---|---|

GA | number of generations | 100 |

GA | number of RBNs in population | 100 |

GA | crossover probability | 0.7 |

GA | mutation probability per node | 0.02 |

GA | number of best individuals directly transmitted to next generation (elitism) | 3 |

BN | number of nodes per BN | 50 |

BN | number of inputs per node | 2 |

BN | average initial bias in initial population | 0.5 |

BN | number of initial conditions per BN | 10,000 |

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**MDPI and ACS Style**

d’Addese, G.; Magrì, S.; Serra, R.; Villani, M.
Exploring the Dynamic Organization of Random and Evolved Boolean Networks. *Algorithms* **2020**, *13*, 272.
https://doi.org/10.3390/a13110272

**AMA Style**

d’Addese G, Magrì S, Serra R, Villani M.
Exploring the Dynamic Organization of Random and Evolved Boolean Networks. *Algorithms*. 2020; 13(11):272.
https://doi.org/10.3390/a13110272

**Chicago/Turabian Style**

d’Addese, Gianluca, Salvatore Magrì, Roberto Serra, and Marco Villani.
2020. "Exploring the Dynamic Organization of Random and Evolved Boolean Networks" *Algorithms* 13, no. 11: 272.
https://doi.org/10.3390/a13110272