Stability of Linear Boolean Networks

TL;DR

This work investigates the stability and dynamical structure of linear Boolean networks by leveraging Derrida curves and algebraic decompositions. It demonstrates a phase transition at an average in-degree of $K_c=1$, making linear networks particularly prone to chaotic behavior, with unstable networks hosting many long attractors. The authors derive explicit theoretical results: the proportion of attractor states equals $1/2^{r}$ where $r$ is the nilpotent component dimension, the expected number of fixed points is $2 - 2^{-n}$, and the fixed-point space is typically 1-dimensional in the Boolean case, with a high probability of invertibility structure in bijective linear maps. These findings provide exact formulas for attractor distributions, fixed points, and bijectivity, offering analytic tools that extend beyond binary updates and informing stability analyses in broader linearized or semi-tensor representations of Boolean networks.

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.

Figures (3)

• Figure 1: Wiring diagram for F(x) in Example \ref{['linear_func_eg']}
• Figure 2: Derrida plot for linear networks with $N=20$ nodes and (a) fixed in-degree distribution of $k=1,2,3$ and (b) scale-free out-degree distribution with a fixed parameter $\gamma$ chosen to match the average degrees of the networks in (a). That is, lines with the same color in (a) and (b) show results from linear networks with the same average degree. Each curve is averaged across $50$ random networks. The shaded area signifies the standard deviation. A black dashed line, which coincides with the Derrida curve for linear networks with fixed in-degree of 1 in (a), highlights the critical threshold.
• Figure 3: (a,b) Number and (c,d) length of attractors for linear networks with $N=20$ nodes and (a,c) a fixed in-degree $k$, and (b,d) scale-free out-degree distribution with a parameter $\gamma$ chosen to match the average degrees of the networks in (a). For each parameter value, $50$ random networks were generated. Orange lines depict the median, each box extends across the interquartile range (IQR), whiskers extend to the lowest data point still within 1.5 IQR of the lower quartile, and the highest data point still within 1.5 IQR of the upper quartile, and black circles show outliers. The length of the whiskers was computed after log2-transformation of the data.

Theorems & Definitions (43)

• Example 2.1
• Example 2.2
• Lemma 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Theorem 4.1
• proof
• Corollary 4.2
• proof