IEEE Circuits and Systems Magazine - Q4 2019 - 54

accomplish a journey. The different perspectives of average path length are given in Appendix C. A detailed
comparison of average path length in different spaces
has been given in Tables IV to VI. From the values of G d H
given in Tables IV to VI, it is evident that the average

× 105

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2
1
0

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200
Path Length
(a)

300

400

× 106

Count

1.5
1
0.5
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300
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Path Length
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40
Path Length
(c)

Regular Structure
Supernode Structure
Figure 3. Average path length distribution for (a) Hong Kong;
(b) London; and (c) Bengaluru networks with and without
considering supernodes.
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IEEE CIRCUITS AND SYSTEMS MAGAZINE

path length in the L-space representation is significantly longer than that in the P-space representation. Thus,
the average number of links traversed by a user is much
larger than the number of transfers made to reach the
destination. A few other notable observations concerning the average path length are
i) An inhomogeneous distribution of stops within a
city leads to Gaussian or asymmetric unimodal distribution (with longer tail ends) in the L-space and
P-space representations. Thus, a fewer number of
stops in the suburbs/downtown in a city leads to
long travel distances. This accounts for the long
tail ends in the distribution. This phenomenon is
consistent with the plethora of stops observed at
city centers leading to short travel distances [12],
[16], [17]. A rather unique feature can be observed
in the distribution pattern in ref. [17], where a secondary peak in the tail end of the distribution along
with the major peak has been observed, indicating
that in addition to a major central business district
(CBD), a supporting minor CBD exists in the city.
ii) As studied in ref. [15], the average path length of a
network is significantly affected in L-space by the
existence of shortcut paths. Despite the absence
of physical connectivity between a few nodes in
the PTN (e.g., between a bus stop and a metro station which are geographically close, or stops on
either sides of a road segment), they can be virtually connected by a short walking distance and
such nodes can be represented as short distance
station pairs (SSPs) or supernodes. Thus, merely
representing the physical connectivity of two
different transportation networks does not justify the true measure of the average path length
[10-13]. However, a slight reorganization of the
network topology using supernodes provides a
better and more practical insight on the average
path length estimation in PTN analysis [10], [13].
iii) Fig. 3 shows the path length distribution of bus
stops for the three cities analyzed in our previous work with and without considering supernodes in the network [13]. For all the three cities in
Fig. 3, it has been observed that the path length
values are comparatively small when the supernode representation is used which conveys more
clear information on the actual path length to be
traversed in practice. Thus, in a PTN analysis, the
supernode representation offers a more realistic
path length estimation.
iv) The link length distribution (the distribution of
geographical distance between the stops) conveys captivating information on the route length
adopted by public transport networks. In ref. [10],
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