Table 1 Conceptual and numerical representations of a temporal-causal network model

Concept

Conceptual representation

Explanation

 States and connections

X, Y, X → Y

Describes the nodes and links of a network structure (e.g., in graphical or matrix format)

 Connection weight

ω _X,Y

The connection weight ω_X,Y ∈ [− 1, 1] represents the strength of the causal impact of state X on state Y through connection X → Y

 Aggregating multiple impacts on a state

c _Y (..)

For each state Y (a reference to) a combination function c_Y(..) is chosen to combine the causal impacts of other states on state Y

 Timing of the effect of causal impact

η _Y

For each state Y a speed factor η_Y ≥ 0 is used to represent how fast a state is changing upon causal impact

Concept

Numerical representation

Explanation

 State values over time t

Y(t)

At each time point t each state Y in the model has a real number value in [0, 1]

 Single causal impact

\( {\mathbf{impact}}_{X,Y}(t)={\boldsymbol{\upomega}}_{X,Y}X(t) \)

At t state X with a connection to state Y has an impact on Y, using connection weight ω_X,Y

 Aggregating multiple causal impacts

\( {\displaystyle \begin{array}{c}{\mathbf{aggimpact}}_Y(t)={\mathbf{c}}_Y\left({\mathbf{impact}}_{X_1,Y}(t),\dots, {\mathbf{impact}}_{X_k,Y}(t)\right)\\ {}={\mathbf{c}}_Y\left({\boldsymbol{\upomega}}_{X_1,Y}{X}_1(t),\dots, {\boldsymbol{\upomega}}_{X_k,Y}{X}_k(t)\right)\end{array}} \)

The aggregated causal impact of multiple states X_i on Y at t, is determined using combination function c_Y(..)

 Timing of the causal effect

\( {\displaystyle \begin{array}{c}Y\left(t+\Delta t\right)=Y(t)+{\boldsymbol{\upeta}}_Y\left[{\mathbf{aggimpact}}_Y(t)-Y(t)\right]\Delta \mathrm{t}\\ {}=Y(t)+{\boldsymbol{\upeta}}_Y\left[{\mathbf{c}}_Y\left({\boldsymbol{\upomega}}_{X_1,Y}{X}_1(t),\dots, {\boldsymbol{\upomega}}_{X_k,Y}{X}_k(t)\right)-Y(t)\right]\Delta t\end{array}} \)

The causal impact on Y is exerted over time gradually, using speed factor η_Y; here the X_i are all states with outgoing connections to state Y