Broad edge of chaos in strongly heterogeneous Boolean networks

TL;DR

This paper analyzes the stability of gene regulation Boolean networks, revealing how heterogeneity in network connectivity influences the transition between stable and chaotic dynamics, with implications for biological robustness.

Contribution

It provides an analytical and numerical study of how in-degree distribution heterogeneity affects the phase boundary in Boolean networks, highlighting the role of diverging second moments.

Findings

Hamming distance behavior is universal near the phase boundary for finite second moments.

Power-law in-degree distributions with 2<γ<3 cause a broader phase boundary.

Heterogeneous connectivity enhances robustness and evolvability in biological networks.

Abstract

The dynamic stability of the Boolean networks representing a model for the gene transcriptional regulation (Kauffman model) is studied by calculating analytically and numerically the Hamming distance between two evolving configurations. This turns out to behave in a universal way close to the phase boundary only for in-degree distributions with a finite second moment. In-degree distributions of the form $P_d(k)\sim k^{-\gamma}$ with $2<\gamma<3$, thus having a diverging second moment, lead to a slower increase of the Hamming distance when moving towards the unstable phase and to a broadening of the phase boundary for finite $N$ with decreasing $\gamma$. We conclude that the heterogeneous regulatory network connectivity facilitates the balancing between robustness and evolvability in living organisms.