Critical line in undirected Kauffman boolean networks - the role of percolation

TL;DR

This paper demonstrates that accurately locating the critical line in undirected Kauffman boolean networks requires considering percolation phenomena, with mean field formulas derived and validated against simulations.

Contribution

It introduces a mean field formula for the critical line in undirected networks and highlights the shielding effect of small clusters on chaos.

Findings

Critical line depends on percolation phenomena.

Mean field formula matches numerical simulations.

Small clusters shield chaotic behavior.

Abstract

We show that to correctly describe the position of the critical line in the Kauffman random boolean networks one must take into account percolation phenomena underlying the process of damage spreading. For this reason, since the issue of percolation transition is much simpler in random undirected networks, than in the directed ones, we study the Kauffman model in undirected networks. We derive the mean field formula for the critical line in the giant component of these networks, and show that the critical line characterizing the whole network results from the fact that the ordered behavior of small clusters shields the chaotic behavior of the giant component. We also show a possible attitude towards the analytical description of the shielding effect. The theoretical derivations given in this paper quite tally with numerical simulations done for classical random graphs.