On the sensitivity to noise of a Boolean function

Abstract

In this paper we generate upper and lower bounds for the sensitivity to noise of a Boolean function using relaxed assumptions on input choices and noise. The robustness of a Boolean network to noisy inputs is related to the average sensitivity of that function. The average sensitivity measures how sensitive to changes in the inputs the output of the function is. The average sensitivity of Boolean functions can indicate whether a specific random Boolean network constructed from those functions is ordered, chaotic, or in critical phase. We give an exact formula relating the sensitivity to noise and the average sensitivity of a Boolean function. The analytic approach is supplemented by numerical results that illustrate the overall behavior of the sensitivities as various Boolean functions are considered. It is observed that, for certain parameter combinations, the upper estimates in this paper are sharper than other estimates in the literature and that the lower estimates are very close to the actual values of the sensitivity to noise of the selected Boolean functions.

Harmonic analysis of Boolean networks: determinative power and perturbations

“…The expectation in (18) is with respect to the distribution of X A . Inserting q(X A ) as given by ( 17) in (19) concludes the proof.…”

“…For the X i being equally but possibly nonuniformly distributed and a slightly different noise model, it was found in [18] that as(f ) still upper bounds the noise sensitivity. This result was generalized to product distributed X in [19].…”

“…= E[h(q(X A ))], (19) where h(•) is the binary entropy function as defined in (7). To obtain (18), we used (17).…”

“…Next, we prove the upper bound in Theorem 2. Applying the upper bound on the binary entropy function h(p) ≤ (p(1 − p)) 1/ ln (4) , given in Theorem 1.2 of [26], on (19) yields…”