Broad edge of chaos in strongly heterogeneous Boolean 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.

arXiv comment8 pages, 3 figures

References in corpus (29)

• Statistical mechanics of complex networks
• Assortative mixing in networks
• Dynamical and correlation properties of the Internet
• Random Boolean Network Models and the Yeast Transcriptional Network
• Synchronization on small-world networks
• Ising Model on Networks with an Arbitrary Distribution of Connections
• Genetic networks with canalyzing Boolean rules are always stable
• Topological Evolution of Dynamical Networks: Global Criticality from Local Dynamics
• Resilience to damage of graphs with degree correlations
• Ferromagnetic ordering in graphs with arbitrary degree distribution
• Number and length of attractors in a critical Kauffman model with connectivity one
• Ferromagnetic Phase Transition in Barabasi-Albert Networks
• Stable and unstable attractors in Boolean networks
• Stability of the Kauffman Model
• Finite-size scaling in complex networks
• Self-organized critical neural networks
• Synchronization transition in scale-free networks: Clusters of synchrony
• Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study
• The Modular Structure of Kauffman Networks
• First- and second-order phase transitions in scale-free networks
• Ising model in scale-free networks: A Monte Carlo simulation
• Evolution of scale-free random graphs: Potts model formulation
• Canalizing Kauffman networks: non-ergodicity and its effect on their critical behavior
• Percolation transition in networks with degree-degree correlation
• Scaling in critical random Boolean networks
• Emergent Criticality from Co-evolution in Random Boolean Networks
• Scaling in a general class of critical random Boolean networks
• Exhaustive percolation on random networks
• Spontaneous Self-Assembly of Transcription Factor Based Gene Regulation Networks

Cited by in corpus (10)

• The effect of network topology on the stability of discrete state models of genetic control
• Stability of Boolean networks: The joint effects of topology and update rules
• Critical Boolean networks with scale-free in-degree distribution
• Evolution of regulatory networks towards adaptability and stability in a changing environment
• Giant component in a configuration-model power-law graph with a variable number of links
• Scaling laws in critical random Boolean networks with general in- and out-degree distributions
• Interplay between degree and Boolean rules in the stability of Boolean networks
• Dynamics of attractor transitions in Boolean networks under noise
• Mean-Field and Non-Mean-Field Behaviors in Scale-free Networks with Random Boolean Dynamics
• The order-chaos phase transition for a general class of complex Boolean networks