Graph Representations of Monotonic Boolean Model Pools 243consider monotonic functions without negative self-loops. This suggests that it
is reasonable to interpret the states of the quotient graph as signs of changes in
the continuous setting as well. This gives us another set of predictions Boolean
models are capable to make. We will discuss this aspect further in Sect. 5.
To explain the first point better, we state the following corollary.
Corollary 1.Consider a Boolean monotonic functionf :{ 0 , 1 }n →{ 0 , 1 }n
with an interaction graphIGglobal(f)=(σi,j)i,j∈{ 1 ,...,n}without negative self-
loops. Assume there is an edge(v, w)inGasync(f)
/
φfthen∀i∈diff(v, w)∃j∈comm(v, w):wi·vj=σi,j, (4)where we identified the equivalence classesv,wwith the corresponding elements
in the image ofφf.
Proof.Assume there is an edge (v, w)inGasync(f)
/
φf. According to the def-
inition of the quotient graph there exists an edge (s, t)∈Easync(f) such that
φf(s)=vandφf(t)=w. According to Theorem 1 and the assumption that the
interaction graph has no negative self loops Condition 3 is satisfied in the local
interaction graphIGf(s) and consequently also inIGglobal(f).
Example 6.Consider the statess=(1, 0 , 1 ,1),t=(1, 0 , 1 ,0) for our running
example. Their images underφfarev=(− 1 , 1 ,− 1 ,−1),w=(1, 1 ,− 1 ,−1). As
can be seen in Fig. 3 there is an edge (v, w)inGasync(f)
/
φfand Condition 4
should be satisfied for these nodes. Indeedw 1 ·v 4 =1·(−1) =σ 1 , 4.
4 Boolean Monotonic Ensembles
Motivated by the treatment of families of ODEs using QDEs, we now discuss
families of Boolean functions defined by an×nsign matrixΣ=(σi,j)i,j∈{ 1 ,...,n}
taking values in{− 1 , 0 , 1 }. Here,Σfixes the structure of Boolean model. In
addition to the constraints captured byΣwe assume throughout the section
that the considered functions do not have negative self loops.
Definition 6.For a givenn×nmatrix of signsΣ=(σi,j)i,j=1,...,nwe define
theBoolean monotonic ensemble
MB(Σ)={
f:{ 0 , 1 }n→{ 0 , 1 }n∣
∣IGglobal(f)=Σ}
.
As in the QDE setting, we are now looking for a compact representation of
possible trajectories for all models in the ensemble. The QDE graphGQDE(Σ)
and Theorem 1 motivate the following definition.
Definition 7.We call the graphGBooleanQDE (Σ)=(VQDEBoolean,EQDEBoolean(Σ))with
VQDEBoolean={− 1 , 1 }n,
EQDEBoolean(Σ)={(v, w)|∀i∈diff(v, w)∃j∈comm(v, w):wi·vj=σi,j}theEnsemble state transition graph (ESTG) ofΣ.