Beyond Boolean: Ternary networks and dynamics

Yu Xiang Yao, Jia Qi Dong, Jie Ying Zhu, Liang Huang, Duan Qing Pei, Ying Cheng Lai

Research output: Contribution to journalArticlepeer-review


Boolean networks introduced by Kauffman, originally intended as a prototypical model for gaining insights into gene regulatory dynamics, have become a paradigm for understanding a variety of complex systems described by binary state variables. However, there are situations, e.g., in biology, where a binary state description of the underlying dynamical system is inadequate. We propose random ternary networks and investigate the general dynamical properties associated with the ternary discretization of the variables. We find that the ternary dynamics can be either ordered or disordered with a positive Lyapunov exponent, and the boundary between them in the parameter space can be determined analytically. A dynamical event that is key to determining the boundary is the emergence of an additional fixed point for which we provide numerical verification. We also find that the nodes playing a pivotal role in shaping the system dynamics have characteristically distinct behaviors in different regions of the parameter space, and, remarkably, the boundary between these regions coincides with that separating the ordered and disordered dynamics. Overall, our framework of ternary networks significantly broadens the classical Boolean paradigm by enabling a quantitative description of richer and more complex dynamical behaviors.

Original languageEnglish (US)
Article number083117
Issue number8
StatePublished - Aug 1 2022

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics


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