Shimbel Distance Matrix (D-Matrix)

The Shimbel Distance Matrix (or D-Matrix) holds the shortest paths between the nodes of a network, which are always equal or lesser to the diameter. To construct this matrix, C matrices of Nth order are built until the diameter of the network is reached. Each C matrix is converted in a corresponding D matrix. In this case, two C matrices, C1(connectivity matrix) and C2 (two-linkages paths; C1*C1) are built since the diameter is 2.

• The first order Shimbel Matrix (D1) is a simple adaptation of C1, where all the direct links are kept. A value of 0 is assigned for all the cii cells since the shortest path between a node and itself is always 0. Cells that have a value of 0 in the C1 matrix (outside cii cells) remain unfilled on the D1 matrix.
• The second order Shimbel Matrix (D2) is built from the first order matrix D1 but only from its unfilled cells. A value of 2 is assigned for each cells on the D2 matrix that have a value greater than 0 on the C2 matrix, but if a value of 1 already exists (D1 matrix), this value is kept. This means that on the D2 matrix of the above figure, only the values of the yellow cells have been changed to 2. Since the diameter of this network is 2, the D2 matrix is the Shimbel distance matrix.
• Nth order Shimbel Matrix (DN). For a network having a diameter of 3, a D3 matrix would have to be built from a C3 matrix (C1*C2) because at least 1 cell would have remained empty in the D2 matrix. Repeat the construction of Nth order Shimbel matrices until the diameter is reached.
• The Shimbel Matrix (D). The order of the Shimbel distance matrix that corresponds to the diameter is the D matrix. The summation of rows or columns represents the Shimbel distance for each node. In the D matrix of the above example, node C is having the least summation of shortest paths (4) and is thus the most accessible, followed by node A (5), nodes B and D (6) and node E (7). The total summation of minimal paths is 28.