Abstract
Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.
Highlights
Boolean networks are complex dynamical systems that were proposed as models of genetic regulatory networks [1,2] and have since been used to model a range of complex phenomena
In the thermodynamic limit, meaning that the number of nodes goes to infinity, the phase transition curve for Random Boolean networks (RBNs) is given by λ = log(2Kp (1 − p)) = 0, where p is the so-called bias of the random
Since | M0 (n)| ∼ | M (n)|, we can focus our attention on functions in M0 (n) and derive the average sensitivity of a typical function from M0 (n)
Summary
Boolean networks are complex dynamical systems that were proposed as models of genetic regulatory networks [1,2] and have since been used to model a range of complex phenomena. In the thermodynamic limit, meaning that the number of nodes goes to infinity, the phase transition curve for RBNs is given by λ = log(2Kp (1 − p)) = 0, where p is the so-called bias of the random. Under a synchronous updating scheme, whereby all Boolean functions get updated simultaneously at each time step, this phase transition curve separates two qualitatively distinct dynamical regimes. Absent additional constraints on the Boolean functions, such as the classes of canalizing functions [5,23,24,25] or Post classes [26], random monotone Boolean networks quickly enter the disordered regime relative to n, but slower than RBNs. For example, for n = 4, monotone Boolean networks have the expected average sensitivity ŝ f = 1.125, which is already slightly in the chaotic regime (λ > 0).
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