From Reachability Matrix to Hierarchical Matrix

The Hierarchical Reachability Matrix is constructed in order to partition the element set into levels.

Process

1. Two subsets of the element set are constructed for each element e_i:
   • The reachability set R_i which involves all of those elements that are reachable from element e_i
   • An antecedent set A_i which involves all of those elements that reach element e_i.
     In other words for each element e_i the set R_i can be determined as the set of elements whose columns have an entry of 1 in row i of the Reachability Matrix.
2. After constructing the sets Ri and Ai their intersection Ri Ai is found next, that is all of those elements which are common to both Ri and Ai. Those elements ei for which Ri Ai = Ri are not reachable from any of the remaining element and thus constitute top/first level elements of the hierarchy (Table I).
3. During the next step these top level elements – the rows and columns corresponding to them – are striking from the Reachability Matrix and the procedure is repeated. The resulting elements are also top level elements of the hierarchy; hence they constitute second/top level elements (Table II). By repeating the above process – until there are no more elements to take part in – it is feasible to determine all the levels of the element system. In Tables I – IV you can see step-by-step the calculation of reachability and antecedent sets based upon the Reachability Matrix of the above example.