Dynamics of Boolean Networks: An Exact Solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

Figure 1

Evolution of the magnetization [$m\equiv m(t)$] and correlation [$C\equiv C(t+t_w,t_w)$] functions with time $t$ is governed by noiseless dynamics (11). Theoretical results (lines) are plotted against the results of MC simulations (symbols) with $N={10}^5$. Each MC data point is averaged over 10 runs. Error bars are smaller than symbol size. Top: Evolution of $m$ (a) and $C$ (b) for $h\in \mathbb{R}$. In (b) we plot $C$ for $h=0.5$ and $k=3$. Bottom: Evolution of $m$ (c) and $C$ (d) for $h\in \mathbb{Z}$. In (d) we plot $C$ for $h=0$ and $k=2$.