Boolean networks synchronism sensitivity and XOR circulant networks convergence time (1208.2767v1)

Abstract: In this paper are presented first results of a theoretical study on the role of non-monotone interactions in Boolean automata networks. We propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours according to two axes. The first one consists in supporting the idea that non-monotony has a peculiar influence on the sensitivity to synchronism of such networks. It leads us to the second axis that presents preliminary results and builds an understanding of the dynamical behaviours, in particular concerning convergence times, of specific non-monotone Boolean automata networks called XOR circulant networks.

• The paper demonstrates that non-monotone interactions induce significant synchronism sensitivity, influencing network trajectories and attractor stability.
• It classifies synchronism sensitivity into levels, showing that level 2 networks undergo profound structural changes including the creation and destruction of attractors.
• The analysis reveals that 2-xor circulant networks, particularly with odd density configurations, converge in at most n steps, highlighting non-trivial dynamic behavior.

Analysis of Boolean Networks Synchronism Sensitivity and XOR Circulant Networks Convergence Time

This paper provides a rigorous investigation into Boolean automata networks, focusing on the effects of non-monotone interactions on network dynamics, particularly concerning sensitivity to synchronism. The paper introduces and examines the behaviour of a specific class of non-monotone networks known as xor circulant networks. The authors, Mathilde Noual, Damien Regnault, and Sylvain Sené, present a thorough theoretical exploration to elucidate the role of non-monotony and provide foundational results on the convergence time of these networks.

Key Contributions

1. Synchronism Sensitivity in Boolean Networks: The paper distinguishes Boolean automata networks by their sensitivity to different updating modes—specifically, synchronous versus asynchronous updates. This sensitivity is vital as it affects the network's trajectory and asymptotic behaviour. The authors identify minimal structural conditions that induce synchronism sensitivity, particularly emphasizing the role of non-monotone interactions.
2. Classification of Synchronism Sensitivity:
   • Level 0: Networks where synchronous transitions act merely as shortcuts or benign deviations.
   • Level 1: Networks showing increased liberty of evolutions for transient configurations or an expansion of unstable attractors.
   • Level 2: Networks where synchronism introduces substantial structural changes, including the creation and destruction of attractors.

Notably, the paper asserts that minimal synchronous sensitive networks of level 2 are inherently non-monotone. This classification and the accompanying structural findings provide significant insights into the dynamics of complex systems modeled as Boolean networks.

1. Exploration of XOR Circulant Networks: The second major focus, xor circulant networks, are systematically analyzed for their dynamical properties, particularly convergence times. These networks are characterized by their circulant matrix structures that determine interactions between multiple automata.

Numerical and Theoretical Results:

• Basic Properties:
  • The paper confirms that for any $k$-xor circulant network of size $n$, certain configurations like $(0, \ldots, 0)$ and $(1, \ldots, 1)$ have predictable stability based on $k$'s parity.
  • The analysis shows that configurations of density $\frac{1}{n}$, which are unit vectors by nature, achieve maximal convergence times, establishing a relationship between configuration density and dynamic stability.
• Specific Convergence Results:
  • The authors prove that $2$-xor circulant networks of sizes $n = 2^p$ converge to a stable configuration in at most $n$ steps. Specifically, configurations with odd densities achieve the maximal convergence time, showcasing non-trivial dynamic behaviours.

Implications and Future Directions:

The results have far-reaching implications for understanding genetic regulatory networks and other systems modeled using Boolean networks. The analysis of synchronism sensitivity provides a framework for understanding how simultaneous transitions in such networks can lead to fundamentally different dynamic behaviours. This understanding can be applied to synthesis and control in synthetic biology, where the timing of gene expression can critically influence system-level outcomes.

Future work should consider:

• Extending these principles to broader classes of non-monotone networks.
• Investigating the dynamical behaviours of xor circulant networks under varying constraints and extending analyses to non-circulant structures.
• Practical applications in synthetic biology where precise understanding and control of genetic networks are crucial.

In conclusion, this paper lays foundational work for further studies into the dynamics of Boolean networks, with a significant emphasis on the impact of non-monotone interactions and network synchronism. The provided theoretical insights and numerical results offer a robust basis for deeper explorations into the behaviour of complex, discrete dynamical systems.