Boolean Networks in Inference and Dynamic Modeling of Biological Systems at the Molecular and Physiological Level - Biological Networks - page 63

attractor cycles tend to become quite long, their lengths increaseing exponentially
with the number of nodes involved. A prominent feature of the dynamics of
networks close to the critical connectivity is the relatively small number and
intermediate length of attractors compared with the 2 N possible states of the
network (Fig. 4.3). This feature motivated the hypothesis that a similar
mechanism potentially could stabilize the macro-states of cellular regulation such
as cell types (Kauffman 1993). While the notion of criticality is only well defined
for random networks, it has long been argued that an intermediate range of
activity is particularly suitable for efficient information processing.
Fig. 4.3. The full state space of a random Boolean network with N = 13 nodes: 2 13 = 4192 initial
states each flow into one of 15 attractors (adapted from Wuensche (1994)). The basin of attraction
marked with an arrow is the one shown in Fig. 4.2.
The attractors of gene regulatory networks can correspond to distinct cellular
states such as differentiation, proliferation or to cell types; the attractors of a
signal transduction network correspond to the expected response(s) to the
presence or absence of a given signal (Platt and Reece 1998; Irie, Mattoo et al.
2004; Boczko, Cooper et al. 2005). Kauffman argues that life must exist on the
edge of chaos such that the networks representing real genetic regulatory
networks operate at the boundary between order and chaos. Phase transitions