Network analysis - Spatial Data Analysis: An Introduction for GIS Users - page 76

N1

A1

From
node

To
node

Node Vertex

Arc Length

N2

N3

1
2
3
4
5
6
7
8
9
10
11
12
13

1
3
2
5
4
4
6
7
7
8
8
10
10

3
4
4
7
7
8
8
9
10
10
11
12
13

1.358
0.529
1.230
0.665
2.335
0.991
1.204
0.674
0.797
2.588
0.728
0.687
0.598

A3

A2

N4 N5

N6

A5

A4

A6

A7

N7

N8

N9

A8

A9

A10

A11

N10 N12

A12

N11

A13

N13

Figure 6.1 Synthetic road network. Arc numbers are prefi xed by 'A' and node numbers by 'N'.

h e initial focus of this section is on simple summaries of networks, in terms of how
connected places are or how complex are the networks which connect these places.

Network connectivity

6.3

h e connectivity of a network can be represented using the connectivity matrix (Taafe
et al. , 1996). h e connectivity matrix is a square matrix that contains the arc labels as
its column and row headings. h e matrix indicates those nodes that are connected by
an arc (assigned a value of 1) and those that are not (given 0). h e connectivity matrix
for the network shown in Figure 6.1 is given in Table 6.1. In this case, nodes 1 and 3,
for example, are connected by an arc—that is, the value for column 3 and row 1, or for
column 1 and row 3, is 1, indicating a connection.
h e matrix in Table 6.1 includes a i nal column that is the sum of all the row ele-
ments and it indicates the number of ways in which a node can be reached in one step
from other nodes. h e maximum number of possible ways (4) is for nodes 4, 7, 8,
and 10. h e connectivity matrix, indicated by C 1 , in Table 6.1 refers to nodes that are
directly connected by arcs—it is termed 'i rst order. h is idea can be extended to the
case of nodes that are connected by two arcs with a further node in between—this is
termed 'second order' (and the matrix is indicated by C 2 ). h e matrix in the second-
order case is obtained by multiplying C 1 with itself (given by C 1 C 1 ). Matrix multiplica-
tion is illustrated below, but further supporting material is given in Appendix A if
clarii cation is required. To multiply a matrix by itself the matrix is multiplied by
the transpose (as dei ned in Appendix A) of itself. Note that, in this case, the result of