Valued Graph Matrix (L-Matrix)
The construction of the valued graph matrix (L-matrix) follows the
- The distances in the network are transcribed in matrix L1
(direct connectivity distance) for each pair directly connected.
An infinite value is given for pairs not directly connected.
- Calculation of the Nth order L matrix. The operation
is similar to the creation of the Shimbel Matrix. What differs in
this case is that we are not working with the minimum number of
paths, but with the minimal distance, which could give different
results. The shortest path between node A and B is obviously the
A-B link. However, there is also a A-C-B link, which summation of
distances could be smaller (actually it is not). The calculation
of the L2 matrix requires the cross-summation of the L1 matrix where
each cell in a column is added with each cell in a row. The B-A
cell on matrix L2 is thus calculated by the cross summation of column
B and row A. Only the smallest value of the five operations is kept,
which is 10 in this case. Since the above network has a diameter
of 2, only two steps are necessary and the L2 matrix becomes the
L-Matrix. The summation of each row on the L2 matrix represents
the minimal distance required to reach all the other nodes in the
network. For node B, it is 43.