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Table 4.18 Values of the
global frustration
D
, attractor
numbers and periods for
positive circuits of order
8 with Boolean transitions
identity or negation
D
(frustration)
Attractor number
Attractor period
0
2
1
2
7
8
4
3
4
4
16
8
6
7
8
8
1
2
Proposition 1 is used to estimate the evolution of the robustness of a network,
because from Demongeot and Waku (
2012a
,
b
) and Demongeot and Demetrius
(
submitted
) it results that the quantity
E
E
μ
—
E
attractor
, called the evolutionary
entropy serves as robustness parameter (Ben Amor et al.
2008
; Demongeot
et al.
2009a
,
b
,
2010a
; Demongeot and Waku
2012a
,
b
; Elena et al.
2008
; Lesne
2008
; Gunawardena
2010
), being related to the capacity a getBren has to return to
μ
¼
, the equilibrium measure, after endogenous or exogenous perturbation.
E
attractor
can be evaluated by the quantity:
E
attractor
¼Σ
k¼
1
;m
2
n
μð
C
k
Þ
log
μð
C
k
Þ;
where
m
is the number of attractors and
C
k
¼
A
k
is the union of the attractor
A
k
and of its attraction basin
B
(
A
k
). A systematic calculation of
E
attractor
allows
quantifying the increase in complexity of a network ensuring a dedicated regulatory
function in different species: for example, the increase of the inhibitory sources
with multiple targets in up-trees converging on a conserved subgraph of a genetic
network (e.g., the core regulating the cell cycle in
C. elegans
,
D. melanogaster,
and
mammals, cf. Fig.
4.18
) causes a decrease of its attractor number (Demongeot and
Waku
2012a
; Demongeot and Demetrius (
submitted
); Caraguel et al.
2010
), hence
an increase of its evolutionary entropy, showing that the robustness of a network is
in this case positively correlated with its connectivity (i.e., the ratio between the
numbers of interactions and genes in the network).
Propositions 2 and 3 give examples of extreme cases, where the networks are
either discrete (Cinquin and Demongeot
2005
) or continuous (Cinquin and
Demongeot
2002
) potential (or gradient) systems, generalizing previous works on
continuous or discrete networks in which authors attempt to explicit Waddington
and Thom chreode's potential-like (Waddington
1940
; Thom
1972
), or Hamilto-
nian (Demongeot and Demetrius (
submitted
); Demongeot et al.
2011b
): in
(Demongeot et al.
2012
) for example, it is proposed a method for calculating the
number of attractors in case of circuits with Boolean transitions identity or nega-
tion. These results about attractors counting constitute a partial response to the
discrete version of the XVIth Hilbert's problem and can be approached by using the
Hamiltonian energy levels. For example, for a positive circuit of order 8, it is easy
to prove that, in case of parallel updating, we have only even values for the global
frustration
D
(they are odd for a negative circuit), corresponding to different values
B
(
A
k
)
[
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