# 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. It only considers the shortest path and does not account for alternative routes. To construct this matrix, C matrices of nth order are built until the network’s diameter (d) is reached. So, if a network has a diameter of 4, four C matrices must be constructed. Each C matrix is converted into 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 (blue cells). A value of 0 is assigned for all the cii cells since the shortest path between a node, and itself is always 0 (green cells). Cells with 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 cell on the D2 matrix with 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 blue 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 has 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 between all nodes is 28.