Upper bound for the stability of Boolean networks

TL;DR

The paper addresses how to quantify and bound the stability of Boolean networks by linking basin coherence to basin size and, finally, to network entropy. The authors develop a rigorous, asymptotically tight framework based on Harper arrays and coding-theoretic ideas to prove an upper bound on basin coherence and extend it to the entire network, revealing a negative linear relationship between network coherence and basin entropy in the limit, i.e., $\psi_F \le 1 - \mathcal{E}_F/N + O(1/N)$. They show that the bound is tight asymptotically and provide constructive networks that approach it, alongside numerical evidence from random networks. This work provides a normalization for comparing stability across networks with different basin distributions and offers insights into how biological networks might balance robustness with phenotypic complexity.

Abstract

Boolean networks, inspired by gene regulatory networks, were developed to understand the complex behaviors observed in biological systems, with network attractors corresponding to biological phenotypes or cell types. In this article, we present a proof for a conjecture by Williadsen, Triesch and Wiles about upper bounds for the stability of basins of attraction in Boolean networks. We further extend this result from a single basin of attraction to the entire network. Specifically, we demonstrate that the asymptotic upper bound for the robustness and the basin entropy of a Boolean network are negatively linearly related.

Figures (6)

• Figure 1: The state space for the synchronous 2-node Boolean network $f(x,y)=(y,x)$. The states $(0,0)$ and $(1,1)$ are fixed points, while the states $(0,1)$ and $(1,0)$ form a limit cycle of length $2$.
• Figure 2: Example of a 3-node Boolean network with two basins of sizes $5$ (blue nodes) and $3$ (orange nodes), respectively. The state $(0,0,0)$ is the only state with coherence $1$. The states $(0,0,1), (0,1,0), (0,1,1)$, and $(1,1,1)$ have coherence $2/3$, while the states $(1,0,0), (1,0,1)$, and $(1,1,0)$ have coherence $1/3$. The coherence of the blue basin is $2/3$, while the coherence of the orange basin is $4/9$. Therefore, the coherence of the entire network is $\frac{1}{8} (5\cdot \frac{2}{3} + 3\cdot \frac{4}{9}) = \frac{14}{24}$.
• Figure 3: Example of a 4-node Boolean network with three basins of size $9$ (blue), $4$ (orange) and $3$ (white), respectively. The basins in blue and white are both Harper arrays, whereas the basin in orange is not a Harper array.
• Figure 4: Illustration why three basins of sizes $15$ (blue), $13$ (orange), and $4$ (white) within a 5-dimensional hypercube cannot all be Harper arrays. The basin of size $15$ forms a Harper array, which leaves the vertex circled in red unassigned. This vertex must be included in either the basin of size $13$ or $4$. In this configuration, it is assigned to the basin of size $13$; however, this assignment causes the orange basin to not form a Harper array. If instead assigned to the basin of size $4$, this basin could not be a Harper array.
• Figure 5: Coherence versus entropy for 50,000 random 12-node Boolean networks with a constant in-degree of $2$. The dashed red line represents the asymptotic upper bound derived in Eq. \ref{['eq:mprline_without_bigO']}.To emphasize the low entropy regime, where the difference between maximally observed coherence and the upper bound is smaller and potential violations of the bound may occur, data corresponding to networks with entropy greater than 0.8 have been excluded from the analysis.