International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Matrix Analysis of Synchronous Boolean Networks

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80200, Jeddah 21589, Saudi Arabia.

Adnan Ahmad Alsogati
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80200, Jeddah 21589, Saudi Arabia.

DOI https://doi.org/10.33889/IJMEMS.2021.6.2.036

Received on July 11, 2020
  ;
Accepted on October 05, 2020

Abstract

The synchronous Boolean network (SBN) is a simple and powerful model for describing, analyzing, and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by employing a matrix method that relies on relating the network transition matrix to its function matrix via a self-inverse state matrix. A recursive ordering of the underlying basis vector leads to a simple recursive expression of this state matrix. Hence, the transition matrix is computed via multiplication of binary matrices over the simplest finite (Galois) field, namely the binary field GF(2), i.e., conventional matrix multiplication involving modulo-2 addition, or XOR addition. We demonstrate the conceptual simplicity and practical utility of our approach via an illustrative example, in which the transition matrix is readily obtained, and subsequently utilized (via its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors) to correctly predict both the transient behavior and the cyclic behavior of the network. Our matrix approach for computing the transition matrix is superior to the approach of scalar equations, which demands cumbersome manipulations and might fail to predict the exact network behavior. Our approach produces result that exactly replicate those obtained by methods employing the semi-tensor product (STP) of matrices, but achieves that without sophisticated ambiguity or unwarranted redundancy.

Keywords- Synchronous Boolean networks, Transition matrix, Function matrix, Recursive ordering, Self-inverse state matrix, Galois field GF (2), Characteristic equation, Minimal equation, 1-eigenvectors, 0-eigenvectors.

Citation

Rushdi, A. M. A., & Alsogati, A. A. (2021). Matrix Analysis of Synchronous Boolean Networks. International Journal of Mathematical, Engineering and Management Sciences, 6(2), 598-610. https://doi.org/10.33889/IJMEMS.2021.6.2.036.

Conflict of Interest

The authors assert that no conflict of interest exists.

Acknowledgements

This work is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), Jeddah, Saudi Arabia. Therefore, the authors acknowledge, with thanks, the DSR for their financial and technical support. The first-named author (AMAR) is gratefully indebted to Dr. Rufaidah Rushdi, of Kasr Al-Ainy Faculty of Medicine (Cairo University, Arab Republic of Egypt) for stimulating discussions concerning cellular biological networks.

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