Total Accessibility Matrix (T)

The total accessibility matrix (T) is obtained from the following procedure:

• Construct the connectivity matrix (C1). On the above network node C is having the highest degree (4).
• Construct the second order (two-linkages paths) connectivity matrix (C2). The total number of two-linkages paths (matrix C2) equals to C1*C1. Each cell in the C2 matrix is the result of the summation of the product of each corresponding row and column in the C1 matrix. For instance, cell A-B in matrix C2 (see above) is constructed from the following: 0*1 + 1*0 + 1*1 + 1*0 + 0*0. It indicates that there is only one possible two-paths link between node A and node B (A-C-B). Looking at the C2 matrix enable to find that there are two possible two-linkages paths between C and A (C-B-A and C-D-A).
• Repeat the construction of the Nth order connectivity matrices until the number of Nth-linkages paths is equivalent to the diameter (path between the farthest nodes) of the network. A 3rd order connectivity matrix (C3) would be equal to C1*C2. A network that has a diameter of 4 would require the construction of 4 matrices (C1 to C4). Since the above network has a diameter of 2, only two matrices, C1 (1st order connectivity) and C2 (2nd order connectivity), need to be constructed.
• Construct the total accessibility matrix (T). For the above network, matrix C2 (two-linkages paths) is added to matrix C1 (single paths; connectivity matrix). This summation represents for each node the total number of paths. For this network, there are thus 46 possible paths, with node C having the largest number (12); either originating from it or having it as a destination.