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Fig. 4.2. Basin of attraction of a dynamical attractor of a random Boolean network. Network states
(circles) and transitions between them are shown, which eventually reach a periodic attractor cycle.
Some network states do not have any precursor state (garden-of-Eden states). Most states are
transient states and form tree-like patterns of transient flows towards the attractor (adapted from
Wuensche (1994)).
does not assume any particular functionality or connectivity for the nodes. In a
RBN with N nodes and K inputs the state of each node in the network at time
t+1
is determined by the states of its
K
inputs at time
t
through a randomly generated
Boolean function. Typically the Boolean functions do not change throughout the
lifetime of the network. The Boolean function for each node maps each of the 2
K
possible input state combinations to an output state of 0 or 1 and can be
represented with a look-up table. The expected attractor length of an RBN
depends on the topology of the network. Early studies on random networks found
that below a critical connectivity (average number of incoming links per node)
K
c
= 2 the network decouples into many disconnected regions, resulting in short
transients and short attractors in its dynamics. The attractors are robust to small
changes in initial conditions or to small perturbations in the state and hence in
this regime the system can be classified as ordered. Above the connectivity
K
c
any local signal will initiate an avalanche of activity that may propagate
throughout most of the system. This sensitivity to small perturbations led to the
classification of this regime as chaotic. In this regime transients as well as
