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

From: Mathematical analysis of the emergence of communities based on coevolution of social contagion and bonding by homophily

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 ωX,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 cY(..) 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 ηY ≥ 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 ωX,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 Xi on Y at t, is determined using combination function cY(..)
 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 ηY; here the Xi are all states with outgoing connections to state Y