Network analysis - Spatial Data Analysis: An Introduction for GIS Users - page 79

connectivity of a road network. Tools like the T matrix are potentially very useful aids
to transport network development. h e following section details some single-value
summaries of network characteristics.

Summaries of network characteristics

6.4

Several simple summaries of network characteristics exist. h ese include the g (gamma)
index and the a (alpha) index (Taafe et al. , 1996), both of which are introduced below.
A better connected network has larger values of g and a (Chou, 1997) and together
they provide a useful summary of network complexity and connectivity.
For a given number of nodes, more arcs indicate greater connectedness. h e
minimum number of arcs (or links), l , needed to connect n nodes is given by:

l =- 1

(6.1)

min

With a minimally connected network, removal of any one arc will result in two
unconnected networks—that is, there are no loops or circuits in the network. For the
network in Figure 6.1, the number of nodes is the same as the number of arcs—there
is a circuit comprising arcs 5, 6, 10, and 9. It is therefore not minimally connected, but
it would be if arc 5 or arc 10 was removed from the circuit.
h ere is a range of measures of network characteristics, such as complexity of
a network or the degree to which it is connected. h e g index is one summary of
network complexity. It is the ratio of the number of links (arcs) in a network to the
maximum number of links possible. For a planar graph with n nodes, the maximum
number of links is given by 3( n - 2). A graph is a set of nodes connected by arcs; a planar
graph has no intersecting arcs—the intersections in Figure 6.1 are represented by
nodes which connect separate arcs and so it is a planar graph. In non-planar graphs,
such as three-dimensional air transport networks, the maximum number of links is
n ( n - 1) / 2 (Chou, 1997). h e g index for a planar graph is given by (Chou, 1997):

l l

g == -

(6.2)

l 3( n 2)

max

where l is the number of links in the network and l max is the maximum number of pos-
sible links (i.e. 3( n - 2)). h e g index can take values from 0 to 1 where small values
indicate simpler networks with few links and larger values indicate more links, and
therefore a better connected network (Chou, 1997).
h e network illustrated in Figure 6.1 has 13 nodes and 13 lines (arcs), so n = 13 and
l = 13. In this case, g is calculated by:

13 13

g = = = 0.394

3(13 - 2) 33