Graph Representations of Monotonic Boolean Model Pools 237This graph will be our focus in the subsequent sections. Notions and definitions
in this section are taken from [ 4 ].
We define an ensemble of ODE systems whose corresponding Jacobi matrices
share a sign structure. The usualsign operatoris denoted [·]:=sign(·),taking
values in{− 1 , 0 , 1 }and extended componentwise to vectors and matrices.
Definition 3 ([ 4 , p. 22]).Foragivenn×nmatrix of signsΣ=(σi,j)i,j=1,...,n,
σi,j∈{− 1 , 0 , 1 }and a state spaceX⊆Rnwe define themonotonic ensemble
M(Σ, X)={f∈C^1 (X,Rn)|∀x∈X:[J(f)(x)] =Σ},whereJ(f)denotes the Jacobian off.
We call a functionx∈C^1 ([0,T],Rn),T∈[0,∞],reasonable, if there is only
a finite set of pointstwithx ̇(t)=0in any bounded interval. We define thespace
of admissible trajectoriesby
E={x∈C^1 ([0,T],X)|xis reasonable}.Thesolution setSM(Σ,X)for an initial value problem withx(0) =x 0 ∈X
contains all reasonable solutions of corresponding ODE systems whose right hand
side function is contained in the monotonic ensembleM(Σ, X), i.e.,
SM(Σ,X):={x∈E|∃f∈M(Σ, X),x 0 ∈Xs.t.x ̇=f(x),x(0) =x 0 }.The restriction to reasonable solutions is a technical detail needed to allow for
a discretization which tracks the sign vectors [ ̇x(t)] for each solution and can be
deduced directly fromΣ[ 4 , p. 22]. For conciseness, we will identify an ODE with
its right hand side function, talking about ODEs as elements of the monotonic
ensemble
We illustrate the notions on our running example.
Example 2.Consider all solutions of ODE-systems ̇x = f(x) with f ∈
C^1 ([0,1]^4 ,R^4 ) having a Jacobi matrix with the sign structure
Σ=
⎛
⎜⎜
⎝
000 −
+000
0+0−
00 − 0
⎞
⎟⎟
⎠,
corresponding to the signed adjacency matrix of the interaction graph of the
Boolean function given in Example 1. They constitute a monotonic ensemble
denoted byM(Σ).
As an example for elements of this monotonic ensemble, we construct now
a functionf ∈M(Σ). To connect to our Boolean example, we construct an