scholarly journals Random Networks with Quantum Boolean Functions

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 792 ◽  
Author(s):  
Mario Franco ◽  
Octavio Zapata ◽  
David A. Rosenblueth ◽  
Carlos Gershenson
Keyword(s):  
Boolean Functions ◽  
Boolean Network ◽  
Complex Dynamics ◽  
Boolean Networks ◽  
Stable Complex ◽  
Complex Dynamic ◽  
Random Boolean Networks ◽  
Quantum Simulator ◽  
Novel Model ◽  
Unstable Dynamics

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

scholarly journals On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1035
Author(s):  
Ilya Shmulevich
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Boolean Functions ◽  
Boolean Network ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Asymptotic Formulas ◽  
Small Perturbations ◽  
Monotone Boolean Functions ◽  
Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

CHAOTIC MEAN FIELD DYNAMICS OF A BOOLEAN NETWORK WITH RANDOM CONNECTIVITY

2007 ◽  
Vol 18 (09) ◽  
pp. 1459-1473 ◽  
Author(s):  
MALIACKAL POULO JOY ◽  
DONALD E. INGBER ◽  
SUI HUANG
Keyword(s):  
Regulatory Networks ◽  
Boolean Network ◽  
Mean Field ◽  
Period Doubling ◽  
Structural Heterogeneity ◽  
Boolean Networks ◽  
Analytical Treatment ◽  
Random Boolean Networks ◽  
The Mean ◽  
Parameter Values

Random Boolean networks have been used as simple models of gene regulatory networks, enabling the study of the dynamic behavior of complex biological systems. However, analytical treatment has been difficult because of the structural heterogeneity and the vast state space of these networks. Here we used mean field approximations to analyze the dynamics of a class of Boolean networks in which nodes have random degree (connectivity) distributions, characterized by the mean degree k and variance D. To achieve this we generalized the simple cellular automata rule 126 and used it as the Boolean function for all nodes. The equation for the evolution of the density of the network state is presented as a one-dimensional map for various input degree distributions, with k and D as the control parameters. The mean field dynamics is compared with the data obtained from the simulations of the Boolean network. Bifurcation diagrams and Lyapunov exponents for different parameter values were computed for the map, showing period doubling route to chaos with increasing k. Onset of chaos was delayed (occurred at higher k) with the increase in variance D of the connectivity. Thus, the network tends to be less chaotic when the heterogeneity, as measured by the variance of connectivity, was higher.

ASYNCHRONOUS RANDOM BOOLEAN NETWORK MODEL WITH VARIABLE NUMBER OF PARENTS BASED ON ELEMENTARY CELLULAR AUTOMATA RULE 126

2006 ◽  
Vol 20 (08) ◽  
pp. 897-923 ◽  
Author(s):  
MIHAELA T. MATACHE
Keyword(s):  
Cellular Automata ◽  
Network Model ◽  
Boolean Network ◽  
Random Variable ◽  
Variable Number ◽  
Boolean Networks ◽  
Random Boolean Networks ◽  
Elementary Cellular Automata ◽  
Boolean Network Model ◽  
Point Analysis

A Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which can vary from one node to another, is considered. This is a generalization of previous results obtained for a constant number of parent nodes, by Matache and Heidel in "Asynchronous Random Boolean Network Model Based on Elementary Cellular Automata Rule 126", Phys. Rev. E71, 026 232, 2005. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. The Boolean rule is a generalization of rule 126 of elementary cellular automata, and is assumed to be the same for all the nodes. We provide a model for the probability of finding a node in state 1 at a time t for the class of generalized asynchronous random Boolean networks (GARBN) in which a random number of nodes can be updated at each time point. We generate consecutive states of the network for both the real system and the models under the various schemes, and use simulation algorithms to show that the results match well. We use the model to study the dynamics of the system through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the GARBN's dynamics range from order to chaos, depending on the type of random variable generating the asynchrony and the parameter combinations.

scholarly journals Minimum complexity drives regulatory logic in Boolean models of living systems

2021 ◽  
Author(s):  
Ajay Subbaroyan ◽  
Olivier C. Martin ◽  
Areejit Samal
Keyword(s):  
Biological Networks ◽  
Boolean Functions ◽  
Regulatory Networks ◽  
Statistical Physics ◽  
Boolean Networks ◽  
Average Sensitivity ◽  
Boolean Models ◽  
Complexity Measures ◽  
Random Boolean Networks ◽  
Update Rules

The properties of random Boolean networks as models of gene regulation have been investigated extensively by the statistical physics community. In the past two decades, there has been a dramatic increase in the reconstruction and analysis of Boolean models of biological networks. In such models, neither network topology nor Boolean functions (or logical update rules) should be expected to be random. In this contribution, we focus on biologically meaningful types of Boolean functions, and perform a systematic study of their preponderance in gene regulatory networks. By applying the k[P] classification based on number of inputs k and bias P of functions, we find that most Boolean functions astonishingly have odd bias in a reference biological dataset of 2687 functions compiled from published models. Subsequently, we are able to explain this observation along with the enrichment of read-once functions (RoFs) and its subset, nested canalyzing functions (NCFs), in the reference dataset in terms of two complexity measures: Boolean complexity based on string lengths in formal logic which is yet unexplored in the biological context, and the average sensitivity. Minimizing the Boolean complexity naturally sifts out a subset of odd-biased Boolean functions which happen to be the RoFs. Finally, we provide an analytical proof that NCFs minimize not only the Boolean complexity, but also the average sensitivity in their k[P] set.

scholarly journals Classification of Discrete Dynamical Systems Based on Transients

2021 ◽  
pp. 1-26
Author(s):  
Barbora Hudcová ◽  
Tomáš Mikolov
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Complex Dynamics ◽  
Computation Time ◽  
Boolean Networks ◽  
Random Boolean Networks ◽  
Average Computation Time ◽  
Design Systems ◽  
Computational Systems

Abstract In order to develop systems capable of artificial evolution, we need to identify which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical systems. The method is based on classifying the asymptotic behavior of the average computation time in a given system before entering a loop. We were able to identify a critical region of behavior that corresponds to a phase transition from ordered behavior to chaos across various classes of dynamical systems. To show that our approach can be applied to many different computational systems, we demonstrate the results of classifying cellular automata, Turing machines, and random Boolean networks. Further, we use this method to classify 2D cellular automata to automatically find those with interesting, complex dynamics. We believe that our work can be used to design systems in which complex structures emerge. Also, it can be used to compare various versions of existing attempts to model open-ended evolution (Channon, 2006; Ofria & Wilke, 2004; Ray, 1991).

scholarly journals Effects of Antimodularity and Multiscale Influence in Random Boolean Networks

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Luis A. Escobar ◽  
Hyobin Kim ◽  
Carlos Gershenson
Keyword(s):  
Complex Dynamics ◽  
Boolean Networks ◽  
Downward Causation ◽  
Statistical Properties ◽  
The Other ◽  
Random Boolean Networks ◽  
Dynamical Behaviors ◽  
Modular Networks ◽  
Other Hand ◽  
The One

We investigate the effects of modularity, antimodularity, and multiscale influence on random Boolean networks (RBNs). On the one hand, we produced modular, antimodular, and standard RBNs and compared them to identify how antimodularity affects the dynamical behaviors of RBNs. We found that the antimodular networks showed similar dynamics to the standard networks. Confirming previous results, modular networks had more complex dynamics. On the other hand, we generated multilayer RBNs where there are different RBNs in the nodes of a higher scale RBN. We observed the dynamics of micro- and macronetworks by adjusting parameters at each scale to reveal how the behavior of lower layers affects the behavior of higher layers and vice versa. We found that the statistical properties of macro-RBNs were changed by the parameters of micro-RBNs, but not the other way around. However, the precise patterns of networks were dominated by the macro-RBNs. In other words, for statistical properties only upward causation was relevant, while for the detailed dynamics downward causation was prevalent.

scholarly journals ROBUSTNESS OF TRANSCRIPTIONAL REGULATION IN YEAST-LIKE MODEL BOOLEAN NETWORKS

2010 ◽  
Vol 20 (03) ◽  
pp. 929-935 ◽  
Author(s):  
MURAT TUĞRUL ◽  
ALKAN KABAKÇIOĞLU
Keyword(s):  
Transcriptional Regulation ◽  
Boolean Functions ◽  
Boolean Network ◽  
Regulation Of Gene Expression ◽  
Boolean Networks ◽  
Transcription Level ◽  
Yeast Saccharomyces Cerevisiae ◽  
Boolean Network Model ◽  
Global Topology ◽  
Model Networks

We investigate the dynamical properties of the transcriptional regulation of gene expression in the yeast Saccharomyces Cerevisiae within the framework of a synchronously and deterministically updated Boolean network model. With a dynamically determinant subnetwork, we explore the robustness of transcriptional regulation as a function of the type of Boolean functions used in the model that mimic the influence of regulating agents on the transcription level of a gene. We compare the results obtained for the actual yeast network with those from two different model networks, one with similar in-degree distribution as the yeast and otherwise random, and another due to Balcan et al., where the global topology of the yeast network is reproduced faithfully. We, surprisingly, find that the first set of model networks better reproduce the results found with the actual yeast network, even though the Balcan et al. model networks are structurally more similar to that of yeast.

scholarly journals Construction Method of Probabilistic Boolean Networks Based on Imperfect Information

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 268
Author(s):  
Katsuaki Umiji ◽  
Koichi Kobayashi ◽  
Yuh Yamashita
Keyword(s):  
Gene Regulatory Networks ◽  
Boolean Functions ◽  
Regulatory Networks ◽  
Imperfect Information ◽  
Boolean Network ◽  
Construction Method ◽  
Boolean Networks ◽  
Probabilistic Boolean Network ◽  
Gene Regulatory ◽  
Selection Probabilities

A probabilistic Boolean network (PBN) is well known as one of the mathematical models of gene regulatory networks. In a Boolean network, expression of a gene is approximated by a binary value, and its time evolution is expressed by Boolean functions. In a PBN, a Boolean function is probabilistically chosen from candidates of Boolean functions. One of the authors has proposed a method to construct a PBN from imperfect information. However, there is a weakness that the number of candidates of Boolean functions may be redundant. In this paper, this construction method is improved to efficiently utilize given information. To derive Boolean functions and those selection probabilities, the linear programming problem is solved. Here, we introduce the objective function to reduce the number of candidates. The proposed method is demonstrated by a numerical example.

scholarly journals The Influence of Canalization on the Robustness of Boolean Networks

2016 ◽  
Author(s):  
C. Kadelka ◽  
J. Kuipers ◽  
R. Laubenbacher
Keyword(s):  
Boolean Functions ◽  
Boolean Network ◽  
Network Models ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Molecular Networks ◽  
Weighted Sum ◽  
Computationally Efficient ◽  
Computational Tools ◽  
The Impact

AbstractTime- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k-canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c-sensitivity and provides formulas for the activities and c-sensitivity of general k-canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c-sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

Sign in / Sign up

Export Citation Format

Share Document