IEEE Circuits and Systems Magazine - Q4 2019 - 55

the geographic link length distribution has been
found to follow a power law, indicating that a substantial number of routes in the public transportation have a short geographical route length and
only a nominal number of routes have a long route
length. Furthermore, such an analysis sheds useful light on the city's demographics. (Note: Since
the latitude and longitude information of the stops
are given in a spherical coordinate system, the
great-circle distance is preferred over the Euclidean distance in evaluating the geographic distance
between two stops [47]).
v) In PTN analysis, the average shortest path length
between any two nodes in the network might not
always guarantee a minimum number of transfers.
Hence, combining the number of transfers with the
shortest path length offers a more realistic choice
for traveling between a chosen node pair. Zhang [18]
has demonstrated a way of measuring the shortest
path length in (10) taking into consideration the
number of transfers along the shortest path, i.e.,

/ / d ij (1 + trij)
G d tr H =

i

j

N (N - 1)

6i = j = 1, 2, ..N

(11)

where trij is the total number of transfers needed
to travel between nodes i and j.
E. Small-worldness in Public Transport Networks
First demonstrated by Watts and Strogatz [48], a class
of networks, called small-world networks, exhibit high
clustering and a low average path length. Empirically
the small-world property of a network can be verified by
C
c
C
v = rand =
d
m
d rand

(12)

where C rand and d rand are the clustering coefficient and
average path length values of the equivalent random
networks (degree conserved network of the same size)
[49]. If v 2 1, i.e., when C $ C rand and d . d rand, the network can be classified as a small-world network. Telesford et al. [50] pointed out that the comparison of average path length of a given network to its equivalent
random network is acceptable; however, the comparison of clustering of a network to that of its equivalent
random network does not fully capture the small-world
behavior since the clustering of a network is expected
to behave close to a lattice structure. It is also observed
in (12) that even a small change in C rand will affect the
value of the small-world parameter ^vh. Hence, a new
approach to capture the small-worldness of a network
can be adopted, as proposed by Telesford et al. [50], i.e.,
FOURTH QUARTER 2019

~=

d rand
- C
d
C latt

(13)

where C latt and d rand are the clustering coefficient and
average path length values of the equivalent lattice and
random network, respectively. In (13), when C . C latt
and d . d rand, we have ~ . 0 and such networks are considered small-world networks. By simulating the behavior of a small network, Telesford et al. [50] demonstrated
the variation of v and ~, where v 2 1 for all values of p
(except p = 1). This means that the network would show
the small-world property for all the rewiring probabilities (except p = 1), demonstrating that v 2 1 cannot fully capture the small-worldness. However, the variation
of ~ shows three major zones, viz. ~ 1 0, ~ .= 0 , and
~ 2 0 , capturing the random, small-world, and lattice
properties of the network [50]. Furthermore, interested
readers may refer to refs. [48], [49] for details on the basic rewiring approaches.
Some reported works have attempted to use (12) to
test the small-worldness of public transport networks
by verifying v 2 1, but such results have been found to
deliver misleading conclusions [11], [16], [24], [25], [37].
In our previous work [13], we adopted Telesford et al.'s
method to evaluate the small-world property of bus
transport networks, and the results of two networks
are shown here in Fig. 4. By observing the value of ~
in Fig. 4(a) we can see that the Hong Kong network becomes a small-world network if certain modifications
are made to the existing routes. However, from the value
of ~ shown in Fig. 4(b), we can also see that the modifications in the routes needed can be quite substantial
and hence difficult to implement.
Unlike Stanley Milgram's experiment conducted in
1967 for studying the small-world behavior of a social
network [51], finding a small value of average path length
in large public transport networks is much more difficult. In addition, it is widely know that G d H varies with
N [5]. Thus, a true measure of small-worldness should
consider the network size as one of the parameters
alongside with the clustering and average path length.
Small-worldness is undoubtedly an important network
behavior in public transport networks as it demonstrates the effectiveness of a transport network in terms
of both connectivity (clustering) and the travel distance
in hops (path length). However, existing measures of
small-worldness have merely been used to demonstrate
high clustering and low average path length, and a practical measure from the passenger's perspective would
be more desirable for public transport networks.
F. Bridges in Public Transport Networks
Centrality is a network parameter describing primarily
local information about nodes (edges), and yet having a
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