Beyond Boolean: Ternary networks and dynamics
Yu-Xiang Yao1, Jia-Qi Dong1, Jie-Ying Zhu2
1Lanzhou Center for Theoretical Physics and Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China.
Chaos (Woodbury, N.Y.)
|September 1, 2022
Summary
Random ternary networks extend Boolean dynamics for complex systems. This research analytically defines boundaries between ordered and disordered states, revealing pivotal node behaviors for richer modeling in biology and beyond.
Area of Science:
- Complex Systems Dynamics
- Computational Biology
- Network Theory
Background:
- Boolean networks are foundational for modeling gene regulatory and other complex systems with binary states.
- Binary state limitations necessitate more sophisticated models for systems with non-binary dynamics, particularly in biological contexts.
- Existing models may not capture the full spectrum of complex system behaviors.
Purpose of the Study:
- To introduce and analyze random ternary networks as an extension of Boolean networks.
- To investigate the dynamical properties and phase transitions in ternary systems.
- To provide a framework for quantitatively describing richer dynamical behaviors beyond binary limitations.
Main Methods:
- Development of random ternary network models with ternary discretized variables.
- Analytical determination of the boundary between ordered and disordered dynamics in parameter space.
- Numerical verification of key dynamical events, such as the emergence of additional fixed points.
Main Results:
- Ternary dynamics exhibit both ordered and disordered regimes, characterized by a positive Lyapunov exponent.
- The boundary between ordered and disordered dynamics is analytically derivable.
- Pivotal nodes show distinct behaviors in different parameter regions, with boundaries coinciding with dynamical phase transitions.
Conclusions:
- Ternary networks offer a significant expansion of the Boolean network paradigm.
- This framework enables a more quantitative description of complex dynamics in systems with non-binary states.
- The findings provide new insights into system behavior and phase transitions in complex networks.
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