IEEE Circuits and Systems Magazine - Q4 2019 - 57

^ w i a hZ = c

t Pa
m + k ia
t Na Z

(16)
(17)

w i = w ia + C b (i )

(18)

where all symbols are listed in Table 3. Similar to equations
16-18, the node weight is evaluated on layer b. Fig. 5 shows
the influential nodes ^w i $ 0.8 h in the network according to
the node weight assigned with and without considering the
interaction between the layers. We can see that the assigned
node weights differ significantly between the individual
mono-layer analysis and the integrated multi-layer analysis. This indicates that ignoring the inter-connectedness
between the transport layers leads to a unrealistic conclusions. Betweenness centrality has been employed as the
prime parameter for assigning node weight for the multilayer analysis since passengers may prefer using multiple
transport networks (bus and metro) to complete their trips.
One of the main advantages of using betweenness
centrality as a measure of significance of a node is that
the removal of high betweenness nodes can adversely
affect the average path length of the entire network as
these nodes essentially control the traffic movement in
the network by bridging various routes and nodes. Consideration of betweenness of nodes has recently been
incorporated under robustness analysis and is attracting a significant research attention [28], [34], [54]-[57].
G. How Close are the Stops in
a Public Transport Network?
Closeness centrality is yet another parameter giving
node level information, and in particular indicates how
close a node i is to the rest of the network. Normally,
closeness is evaluated in terms of hop count, i.e., total
number of hops required to reach all other nodes in a
network from a given node, i.e., we have
C c (i) = 1
/ d ij

(19)

H. Social Behavior in Public Transport Networks
Observing the social behavior at public transport stops
and routes in a PTN is interesting. Specifically, the polarization of connectivity of the stops and routes towards
other stops and routes is practically useful. Such social
behavior can be studied in terms of assortativity. While
degree, as discussed in Section VI-A, captures the connectivity of a node in the network, assortativity captures
the connectivity among similar kind of nodes in the network. In other words, assortativity reflects the bias of
nodes to connect with nodes of similar kind. Thus, assortativity is also a local parameter providing node level
51.7
51.6
Latitude

^ Pa = a 1 ) PT hZ and ^ Pb = a 2 ) PT hZ

key perspectives on closeness centrality under various
spaces of representation. Due to the limited available
results on closeness centrality related to PTNs and the
rather restricted analysis in the L-space representation,
the practical significance of evaluating closeness centrality of PTNs is still not widely recognized. In addition,
in a PTN under the L-space representation, a particular
stop is seldom expected to be close to all other remaining nodes in the network as it is typically connected to
a portion of the network. However, closeness centrality
in other spaces might offer insightful information, and
should therefore deserve further investigation.

51.5
51.4
51.3
51.2
-0.6

FOURTH QUARTER 2019

-0.4

-0.2
0
Longitude
(a)

0.2

0.4

0.2

0.4

51.6
51.5
51.4
51.3

The smaller the value of dij, the closer node i is to
all other nodes. Prior works [12], [22] have considered
the closeness centrality values for weighted and unweighted network structures, respectively, and the corresponding distributions have been found to follow an
exponential distribution. Appendix C summarizes the

Influential Nodes (MTN Layer)
Influential Nodes (BTN Layer)

51.7

Latitude

of PTNs, a node weight was assigned considering the
bus (layer a) and metro (layer b) transport layers as
individual mono-layers where the layer interaction is ignored. Later, a method of spatial amalgamation was applied to integrate the two layers, and accordingly, a new
node weight was assigned to the nodes in the integrated
multi-layer, i.e.,

51.2
-0.6

Influential Nodes on
Integrated Multilayer
-0.4

-0.2
0
Longitude
(b)

Figure 5. Influential nodes in the London PTN with (a) monolayer analysis; (b) multi-layer analysis.

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