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 |