Extremely chaotic Boolean networks

TL;DR

This paper demonstrates that Boolean networks can exhibit chaotic behavior even under conditions typically associated with order, and characterizes the structural features of such chaotic networks.

Contribution

It reveals that certain structural constraints do not prevent chaos in Boolean networks and characterizes the structure of networks with large periodic orbits.

Findings

Boolean networks can be chaotic despite ordered-inducing conditions.

Networks with large periodic orbits tend to have many variables copying others.

Chaotic networks can resemble small Turing machines in structure.

Abstract

It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic regime, even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include using at most two inputs for every variable, using biased and canalyzing regulatory functions, and restricting the number of negative feedback loops. We also prove for n-dimensional Boolean networks that if in addition the number of outputs for each variable is bounded and there exist periodic orbits of length c^n for c sufficiently close to 2, any network with these properties must have a large proportion of variables that simply copy previous values of other variables. Such systems share a structural similarity to a relatively small Turing machine acting on one or several tapes.