ISM - Interpretive Structural Modeling

1. 1. Binary relation matrix D = [π_ij] (1 if e_i affects e_j).

   $$D=[{\pi }_{ij}{]}_{n\times n},{\pi }_{ij}\in \{0,1\}$$

2. 2. Reachability matrix M* via Boolean powers until stable.

   $$M=D+I,M^{\ast }=M^k\text{ s.t. }{M}^k=M^{k+1}$$

3. 3. Reachability set R(t_i) and antecedent set A(t_i).

   $$R(t_i)=\{e_j\mid m_{ji}^{\ast }=1\},A(t_i)=\{e_j\mid m_{ij}^{\ast }=1\}$$

4. 4. Level partition: e_i at current level iff R(t_i) ∩ A(t_i) = R(t_i).

   $$R(t_i)\cap A(t_i)=R(t_i)⟹e_i\in \text{level }l$$

5. 5. Build hierarchical digraph from level partition.

   $$\mathcal{G}=(V,E),V=\{e_1,\dots ,e_n\}$$