IEEE Circuits and Systems Magazine - Q4 2019 - 56

global significance. Centrality quantifies the significance
of a node (edge) based on various sources of information. Centrality measures may thus include degree centrality, Eigen-centrality, Katz-centrality, page rank centrality, closeness centrality, betweenness centrality, etc.
In PTN analysis, a few centrality measures have been
extensively studied, e.g., degree centrality, closeness
centrality, and betweenness centrality. The degree

centrality, as discussed in Section IV-A, rates a node's
significance according to its degree. Similarly, betweenness centrality emphasizes the capability of a node in
bridging multiple shortest paths in a network [52]. Specifically, the node betweenness centrality is defined as
C b (i ) =

/

i, j, k ! V

d jk (i )
,
d jk

(14)

10-4

10-3 10-2 10-1
100
Rewriting Probability (p)

C b (e im) =

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

(w)

1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Variation of w for Different Values of p

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

(w)

L(p)/L(0), C(p)/C(0)

and the edge betweenness centrality is defined as

i, j, k, m ! V

C(p)/C(0): Regular Structure
C(p)/C(0): Supernode Structure
L(p)/L(0): Regular Structure
L(p)/L(0): Supernode Structure
w: Regular Structure (-0.45)
w : Supernode Structure (-0.45)
(a)
Variation of w for Different Values of p

C(p)/C(0), L(p)/L(0)

1
0.8
0.6
0.4
0.2
10-4

10-3 10-2 10-1
100
Rewriting Probability (p)
C(p)/C(0): Regular Structure
C(p)/C(0): Supernode Structure
L(p)/L(0): Regular Structure
L(p)/L(0): Supernode Structure
w: Regular Structure (-0.71)
w: Supernode Structure (-0.8)
(b)

Figure 4. Small world network behavior for (a) Hong Kong;
and (b) London networks with and without supernodes, and
the value of ~ at p = 10 -4 is highlighted. (Note: L is used
instead of G d H to represent the average path length).
56

IEEE CIRCUITS AND SYSTEMS MAGAZINE

/

d jk (e im)
d jk

(15)

where djk is the total number of shortest paths between
nodes j and k, and d jk (i) or d jk (e im) is the shortest paths
between nodes j and k passing through node i or edge
eim. Appendix C summarizes the different perspectives
of betweenness centrality under various spaces of network representation.
For a given network, it is intuitive to assume that the
nodes having a higher degree have a higher probability
to serve as central nodes in the network, and thus, the
relationship between degree and betweenness centrality has been actively studied. The major observations
are as follows:
i) The dependency of betweenness upon degree is
found to follow a Poisson distribution in the L-space
representation [16], and a power-law distribution
in the L-space representation [22] and the C- space
representations [17].
ii) In the P-space representation, two variations of
power law distribution have been observed depending on the value of k. For small values of k,
the betweenness is almost zero leading to a steep
slope in the power-law distribution, whereas for
high values of k, a larger betweenness has been
observed, leading to a more regular power-law
distribution pattern [16], [17].
iii) In the B-space representation, the distribution pattern is found to be similar to that of the P-space
representation since, N proj nodes have low degree
and L proj nodes have high degree [17]. Furthermore, Bona et al. demonstrated that, the nodes
having a high betweenness centrality are mostly
situated in CBDs [25]. However, this observation
remains partially true because a node in the downtown/suburb which acts as an entry or exit point
for passengers traveling between the cities might
also contribute to a high betweenness centrality.
In an earlier work [53], we employed betweenness
centrality as a prime parameter for studying network behavior when the interaction between multiple transport
networks (bus and metro, for example) are ignored. Specifically, to demonstrate the unbalanced use or biasness
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