Stability of Linear Boolean Networks (2209.02044v2)

Abstract: Stability is an important characteristic of network models that has implications for other desirable aspects such as controllability. The stability of a Boolean network depends on various factors, such as the topology of its wiring diagram and the type of the functions describing its dynamics. In this paper, we study the stability of linear Boolean networks by computing Derrida curves and quantifying the number of attractors and cycle lengths imposed by their network topologies. Derrida curves are commonly used to measure the stability of Boolean networks and several parameters such as the average in-degree K and the output bias p can indicate if a network is stable, critical, or unstable. For random unbiased Boolean networks there is a critical connectivity value Kc=2 such that if K<Kc networks operate in the ordered regime, and if K>Kc networks operate in the chaotic regime. Here, we show that for linear networks, which are the least canalizing and most unstable, the phase transition from order to chaos already happens at an average in-degree of Kc=1. Consistently, we also show that unstable networks exhibit a large number of attractors with very long limit cycles while stable and critical networks exhibit fewer attractors with shorter limit cycles. Additionally, we present theoretical results to quantify important dynamical properties of linear networks. First, we present a formula for the proportion of attractor states in linear systems. Second, we show that the expected number of fixed points in linear systems is 2, while general Boolean networks possess on average one fixed point. Third, we present a formula to quantify the number of bijective linear Boolean networks and provide a lower bound for the percentage of this type of network.

• The paper demonstrates that linear Boolean networks transition from order to chaos at an average in-degree of 1, highlighting their inherent instability.
• It employs Derrida curves and polynomial algebra to quantify attractors and cycle lengths, linking network topology with dynamic stability.
• The findings offer theoretical and numerical insights that pave the way for advanced analysis and control methods in Boolean network research.

Stability of Linear Boolean Networks: An Overview

The paper “Stability of Linear Boolean Networks” by Chandrasekhar et al. explores the behavior and properties of linear Boolean networks (LBNs), focusing on their stability. Stability is a crucial aspect of network models that influences other important characteristics such as controllability. The authors investigate the stability of LBNs by analyzing Derrida curves, quantifying the number of attractors, and assessing cycle lengths based on network topologies.

Key Findings

1. Derrida Curves and Stability Assessment:
   • The paper demonstrates the use of Derrida curves, which are commonly applied to measure the stability of Boolean networks.
   • The authors highlight that for random unbiased Boolean networks, there exists a critical connectivity value $K_c = 2$, distinguishing ordered and chaotic regimes.
   • For linear networks, which are posited as the most unstable due to their lack of canalization, the phase transition from order to chaos occurs at an average in-degree of $K_c = 1$.
2. Number of Attractors and Cycle Lengths:
   • The paper indicates that unstable networks exhibit a high number of attractors with long limit cycles. Conversely, stable and critical networks have fewer attractors with shorter limit cycles.
   • They present theoretical results to quantify the dynamical properties of linear networks, including:
     • A formula for the proportion of states already present in the attractors, indicating that this proportion is $\frac{1}{2^r}$, where $r$ represents the dimension of the nilpotent component.
     • The expected number of fixed points in linear systems is calculated to be 2, while general Boolean networks possess an average of one fixed point.
     • An estimation of the number of bijective linear Boolean networks and a lower bound for the percentage of such networks are provided.

Methodological Approach

• Derrida Curves:
  • The authors simulate networks with different degrees and configurations to compute Derrida curves. These plots help to infer whether a network operates in an ordered, chaotic, or critical regime.
  • Interestingly, the results show that linear networks with a fixed in-degree of $k = 1$ are critical. Beyond this, the networks exhibit chaotic behavior, reinforcing the instability inherent in linear functions.
• Counting Attractors:
  • The method involves generating functions and using properties of polynomial algebra over finite fields. The Frobenius normal form is leveraged to calculate the rank and nullity of specific matrices, ultimately leading to the count of attractors of various lengths.

Numerical and Theoretical Insights

• The paper presents a series of numerical examples and graphs illustrating how the topology of wiring diagrams in LBNs affects their stability.
• The numerical examples corroborate theoretical predictions, such as the manifestation of more and longer attractors in networks with higher average in-degrees.
• Theoretical results encompass derivations and proof techniques that provide deeper insights into the dynamics of LBNs. For example, the paper includes a proof for the expected number of fixed points in linear systems, underlying its theoretical rigor.

Implications and Future Work

The implications of this research extend beyond the confines of LBNs:

• Practical Applications: Given that biological and biochemical networks often leverage Boolean models, understanding the specific behaviors of linear networks is pertinent. Although most biological systems tend to utilize canalizing functions, insights from LBNs can aid in the analysis of more complex, non-linear systems.
• Theoretical Expansion: Future research could further explore the linear representation of non-linear Boolean networks. The connection between linear systems and representations in higher-dimensional spaces, as indicated by semi-tensor product representation, opens avenues for robust analytical and control methods in Boolean networks.

Conclusion

This paper provides a comprehensive examination of the stability of linear Boolean networks. The combination of Derrida curve analysis, attractor quantification, and theoretical proofs builds a solid foundation for understanding the dynamics of LBNs. While the inherent instability due to non-canalization is evident, the mathematical frameworks and methods introduced are instrumental for future explorations in both linear and nonlinear Boolean networks.

For a detailed exploration, readers are encouraged to delve into the paper, particularly for its meticulous derivations and the thoughtful blend of numerical and theoretical analyses. These elements collectively contribute to a nuanced understanding of LBNs and their stability properties.