Total Accessibility Matrix (T)

Simple Connectivity Matrix
Total Accessibility Matrix T

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

  • Construct the connectivity matrix (C1). It is a matrix where, for each cell, a value of 1 or 0 is used to denote that a connection exists between two node pairs. On the above network, node C has the highest degree, which is the sum of all the connections this node has (4).
  • Construct the second-order (two-linkages paths) connectivity matrix (C2). The total number of two-linkage paths (matrix C2) equals 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-path link between node A and node B (A-C-B). The C2 matrix indicates that there are two possible two-linkage 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-linkage paths is equivalent to the diameter (path between most distant nodes) of the network. A 3rd order connectivity matrix (C3) would be equal to C1*C2. A network with 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 the total number of paths for each node. 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.