See Dynamical Instability in Boolean Networks as a percolation Problem, Boolean network

New paper: Network Structure and Activity in Boolean Networks

Activities and Sensitivities in Boolean Network Models

Boolean functions in which few variables have high importance and most other variables have low importance play a role in eliciting order from Boolean networks.

We should mention in passing that much of the discussion in this Letter can be formulated in terms of spectral methods or harmonic analysis on the $n$ cube.

Boolean function derivative

Activity

Sensitivity

For a random Boolean function with bias $p$ (so that each bit in the truth table is $1$ with probability $p$ and $0$ otherwise), the probability that two Hamming neighbors are different is equal to $2p(1-p)$, since one can be $1$ (with probability $p$) and the other 0 (with probability $1-p$), and vice versa.

From this one can see that $E[\alpha_i^f] = 2p(1-p)$, and $E[s^f] = K2p(1-p)$, where $E$ means expectation value w.r.t. the prob. distribution of the truth tables. We can then conclude that highly biased functions ($p$ far away from 0.5) are expected to have low average sensitivity.

For a Boolean function $f$, a canalizing variable is a variable that determines (canalizes) the value of $f$ if it has a given value. See the article for more precise definition. A random Boolean function with a single canalizing variable, it is shown here that the expected activity of the canalizing variable is $1/2$, while that of the rest of the variables is $1/4$

The average sensitivity (when averaged over all the functions in the network) appears to be a good parameter for predicting whether the dynamics of the Boolean network are ordered or chaotic. This can be observed by looking at Derrida curves.