Lecture 12 Network Analysis (3) 12-1 Network Connectivity – C1Matrix To measure the connectivity of a network, we can use a Matrix set – called C matrices. Connectivity matrices are useful in evaluating network accessibility. The first order connectivity matrix, defined as the C1matrix, is based on the direct connection between nodes. 2018/11/24 Jun Liang, Geography @ UNC

12-1 Network Connectivity – C1 Matrix (Cont.) 3 4 5 6 7 8 Σ 1 3 2 5 6 7 8 4 2018/11/24 Jun Liang, Geography @ UNC

12-1 Network Connectivity – C1 Matrix (Cont.) About C1 Matrix: Cell value 1 represents a direct connection between two nodes (P and Q). Cell value 0 represents no direct connection between two nodes. The last column shows the sum of values for each row. The row sum indicates the number of links directly connected to the corresponding node. 2018/11/24 Jun Liang, Geography @ UNC

12-2 Network Connectivity – C2 Matrix C1 matrix is based on direct connection. Transportation systems always consist of both direct and indirect connections. (for example, airline passengers can choose their preferences – no connection, one connection, or more.). So we will need high-order C matrix: C2= C1 X C1 Each cell in C2 matrix is the sum of the products of all elements in the corresponding row and column. 2018/11/24 Jun Liang, Geography @ UNC

12-2 Network Connectivity – C2 Matrix (Cont.) Cell value in C2: The row P, column Q cell in the C2 matrix is the sum of the products of elements in row P and the corresponding elements of column Q in the C1 matrix. For example: 2018/11/24 Jun Liang, Geography @ UNC

12-2 Network Connectivity – C2 Matrix (Cont.) 3 4 5 6 7 8 Σ 10 16 9 11 12 14 1 3 2 5 6 7 8 4 2018/11/24 Jun Liang, Geography @ UNC

12-2 Network Connectivity – C2 Matrix (Cont.) About C2 Matrix: Every element in the C2 matrix signifies the number of unique ways that one could move from one node to another through an indirect path of exactly two links. The row sum in the C2 matrix indicates the number of different ways one could move from the corresponding node to all other nodes in exactly two steps in the system. 2018/11/24 Jun Liang, Geography @ UNC

12-3 Network Connectivity – C3 Matrix Indirect connectivity can be extended beyond the second order: C3= C1 X C2 About C3 matrix: It is the connectivity matrix for indirect connection of exactly three steps. Every element represents the number of ways to move from one node to the other in exactly three steps. The row sum in the C3 matrix indicates the number of different ways one could move from the corresponding node to all other nodes in exactly three steps in the system. Matrix order larger than diameter value will contain redundant information. 2018/11/24 Jun Liang, Geography @ UNC

12-3 Network Connectivity – C3 Matrix (Cont.) 4 5 6 7 8 Σ 33 19 9 10 59 32 38 39 41 49 1 3 2 5 6 7 8 4 2018/11/24 Jun Liang, Geography @ UNC

12-4 Network Accessibility Accessibility matrix (T matrix): Sum of all meaningful C matrices from the first order to the order equal to the network diameter. In the above example, T= C1+C2+ C3 A cell in the T matrix is the sum of values of the corresponding row and column in the C1,C2 , C3 , ….matrices. A node with a higher row sum has greater accessibility to the rest of the network. 2018/11/24 Jun Liang, Geography @ UNC

12-4 Network Accessibility (Cont.) 3 4 5 6 7 8 Σ 10 46 27 13 11 12 80 9 44 52 54 56 67 In general, the larger the network accessibility value, the more route choices are available in the system and the better connected the nodes. 2018/11/24 Jun Liang, Geography @ UNC