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.

1.
S. A.
Kauffman
,
The Origins of Order
(
Oxford University Press
,
New York
,
1993
), pp.
173
235
.
2.
I.
Shmulevich
and
S. A.
Kauffman
,
Phys. Rev. Lett.
93
,
048701
(
2004
).
3.
I.
Shmulevich
,
E. R.
Dougherty
, and
W.
Zhang
,
Proc. IEEE
90
,
1778
(
2002
).
4.
V.
Kaufman
,
T.
Mihaljev
, and
B.
Drossel
,
Phys. Rev. E
72
,
046124
(
2005
).
5.
K.
Klemm
and
S.
Bornholdt
,
Phys. Rev. E
72
,
055101
(
2005
).
6.
L.
Raeymaekers
,
J. Theor. Biol.
218
,
331
(
2002
).
7.
M.
Aldana
and
P.
Cluzel
,
Proc. Natl. Acad. Sci. U.S.A.
100
,
8710
(
2003
).
8.
C.
Huepe
and
M.
Aldana-González
,
J. Stat. Phys.
108
,
527
(
2002
).
9.
S.
Wolfram
,
A New Kind of Science
(
Wolfram Media
,
Champaign
,
2002
).
10.
C. S.
Goodrich
and
M. T.
Matache
,
Physica A
379
,
334
(
2007
).
11.
S.
Bilke
and
F.
Sjunnesson
,
Phys. Rev. E
65
,
016129
(
2001
).
12.
G. L.
Beck
and
M. T.
Matache
,
Physica A
387
,
4947
(
2008
).
13.
E.
Friedgut
,
Combinatorica
18
,
27
(
1998
).
14.
S.
Schober
,
Proceedings of the Fifth International Workshop on Computational Systems Biology (WCSB)
,
2008
(unpublished), pp.
173
176
.
15.
I.
Benjamini
,
G.
Kalai
, and
O.
Schramm
,
Publ. Math. (Debrecen)
90
,
5
(
1999
).
16.
K. E.
Gustafson
and
D. K. M.
Rao
,
Numerical Range
(
Springer-Verlag
,
New York
,
1997
).
17.
M.
Talagrand
,
Ann. Probab.
22
,
1576
(
1994
).
18.
M.
Ben-Or
and
N.
Linial
, in
Randomness and Computation
, edited by
S.
Micali
(
Academic
,
New York
,
1989
), pp.
91
115
.
19.
S.
Schober
and
M.
Bossert
, “
Analysis of random Boolean networks using the average sensitivity
,” e-print arXiv:0704.0197S [nlin.CG].
You do not currently have access to this content.