240 R. Schwieger and H. Siebert
(+, +, +, +)(+, +, +, -)(+, +, -, +)(-, +, +, +)(-, +, -, +)(+, +, -, -)(+, -, +, +)(+, -, +, -)(+, -, -, +)(-, -, +, +)(-, -, -, +)(+, -, -, -)(-, +, +, -)
(-, -, +, -)(-, +, -, -)(-, -, -, -)(+, +, +, +)(-, +, -, +) (+, +, +, -)(+, +, -, +)(+, +, -, -)(+, -, +, +)(+, -, +, -)(+, -, -, +)(-, -, -, +)(+, -, -, -)(-, +, -, -)(-, -, -, -)Fig. 3.Left: State transition graph of the monotonic ensemble in the running example.
Right: The quotient graphGasync(f)
/
φffrom Example 5. Self-loops are not shown.Proposition 1 ([ 4 ,p.25]).Letv, w∈{− 1 , 1 }n=VQDE(Σ),v=w.Then,
(v, w)∈EQDE(Σ)iff
∀i∈diff(v, w)∃j∈comm(v, w):wi·vj=σi,j (1)The proof of Proposition 1 is given in [ 4 , p. 25]. In the following section, we
will transfer the statement into the Boolean setting.
To conclude this section, we note that there is no one-to-one correspondence
between the qualitative state transition graph and the corresponding sign matrix
Σ. It is possible to change elements on the diagonal ofΣwithout changing the
graphGQDE(Σ). This is due to the fact that the sets diff(v, w) and comm(v, w)
are disjoint and thus the diagonal elements do not play a role in Eq. ( 1 ). Conse-
quently, the edge set does not change when changing the diagonal ofΣ.
Remark 1.LetΣbe a sign matrix. Then
GQDE(Σ)=GQDE(Σ−D),whereDis any diagonal matrix with entries in{− 1 , 0 , 1 }.
In the following we will examine analogues to the graphGQDE(Σ) and the
setM(Σ) in the Boolean setting.
3 Quotient graphs for Boolean networks
In the Boolean setting, paths in the asynchronous state transition graph can
be seen as trajectories of the system. To transfer the results from the previous