Definition: A boolean network consists in a group of *N* nodes or elements *{N*
_{
1
}, *N*
_{
2
}, *...* ,*N*
_{
N
}
*}* so that

*σ*
_{
i
}
*∈ {True, False}*

is the state of the *N*
_{
i
} node. The state True or False of each node is determined by the initial conditions along with the Boolean rules that determine the state of *N*
_{
i
} determined by the state of its regulators {*N*
_{
i1
}, *N*
_{
i2
} ,*...*, *N*
_{
ik
}}.

The boolean operations are used as follows: if two or more elements can induce the activation of a node in an independent way, we combine both with the logical function OR, if two or more components cannot induce the activation in an independent way, we associate to both the logical operator AND and finally, the operator NOT will be associated to the inversion of the state of the element.

Our method of analysis consists in finding the states of the network that are most frequently visited and the identification of the variable elements since they show the activated or deactivated network sub-pathways and their biological influence on the global network behavior. As detailed below.

The algorithm describing the network logics is updated sequentially changing the states of the nodes as a result of their interactions. However, some elements with no prior information about what determine their states are updated with a random boolean state what defines a hybrid synchronous/asynchronous dynamics that is different from that used in reference [2]. To illustrate the situation of random elements consider, for instance, node GCR1 that inhibits GPA1 in Fig. 1. At each update GCR1 receives a random state. So, although the interactions in the network are deterministic the random updates of some nodes imply that the network evolves randomly in time. Denoting by

*Σ*
_{
t
}
*={σ*
_{
1
}
*(t) , σ*
_{
2
}
*(t), … , σ*
_{
k
}
*(t)}*

the state of the nodes in the update

*t*, we have that

represents a set of possible states of the network for *t*
_{
m
} updates for a given initial condition and defines a possible trajectory of the network in its Space of States (SS)[10]. As we have a random boolean network the orbits are not necessarily equal for the same initial condition. The cardinality of *Σ*
_{
Total
} grows with increasing number of updates and has a maximum of *Ω*=*2*
^{
N
} different elements.

To characterize a trajectory we search for equal elements in (1) for each initial condition. The search for these states allows us to establish a relation with the frequency that the state is visited. To find equal states we use the Hamming distance *D*: If *Σ*
_{
t
}, *Σ*
_{
t’
} are elements of *Σ*
_{
Total
} in updates *t* and *t’* and *σ*
_{
i
}(*t*) = *σ*
_{
i
}(*t’*) for all i, then *D*(*Σ*
_{
t
}, *Σ*
_{
t’
}) =*0*. After identification of the states in the updates we count the number of times each one is visited.

Another important feature of the dynamics of a boolean network is the general behavior of its nodes. Often, a network breaks apart in two groups of nodes for all initial conditions: in the first group the nodes are always in a frozen state and in the second the nodes are always in a variable state [11]. This feature determines the cardinality of *Σ*
_{
Total
} because if the fraction of frozen states is high, the cardinality will be low and vice-versa. We identify all frozen and variable elements of the network for analysis.