Kauffman Boolean model in undirected scale free networks

Abstract

We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)\sim k^{-\gamma }$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<\gamma <3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $\gamma <3$, the position of the critical line in Kauffman model seems to be an important exception from the rule. Second, since gene regulatory networks are characterized by scale-free topology with $\gamma <3$, the observation that in finite-size networks the mentioned transition moves towards smaller $\gamma$ is an argument for Kauffman model as a good starting point to model real systems. We also explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena.

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• Statistical mechanics of complex networks
• Assortative mixing in networks
• Percolation in Directed Scale-Free Networks
• Percolation transition in networks with degree-degree correlation
• Damage Spreading and Criticality in Finite Random Dynamical Networks
• Unanimity Rule on networks

Cited by in corpus (3)

• Critical Boolean networks with scale-free in-degree distribution
• More than two equally probable variants of signal in Kauffman networks as an important overlooked case, negative feedbacks allow life in chaos
• Scaling laws in critical random Boolean networks with general in- and out-degree distributions