IEEE Circuits and Systems Magazine - Q4 2019 - 43

be noted that choosing an extremely small value
of d th creates a lot of SSPs in a dense network,
whereas a large value of d th is meaningless, since
a long walking distance to reach another station
in the network is unreasonable. In either of the
cases, the chosen value of d th may bias the understanding of network behavior [13]. Hence, a
careful selection of the d th is important. SSPs are
more prevalently observed in bus transport networks as compared to metro transport networks.
Step 3: Generate the topology of a PTN from the data
extracted in Step 2. Initially, based on the graph type
and the space of representation, a square adjacency
matrix A with dimension N # N and elements a ij can be
derived to describe the connection between node pair
ni and nj. The element a ij = 1 if there exists a connection between nodes ni and nj, and 0 otherwise. A graph
can either be directed (digraph), undirected, weighted
or unweighted. The intent of choosing the graph type
solely depends on the necessity of the type of analysis
to be accomplished. For the analysis of transport structures, especially bus transport structures, a directed
graph is often chosen since the inbound and outbound
routes have different travel paths servicing different stations (except the round-trip journey routes). However,
an undirected graph is typically chosen in the analysis
of metro transport networks where the inbound and
outbound travel paths remain the same for a vast majority of routes. Furthermore, depending on the aim of the
network analysis, the graph can be represented in various spaces of representation as will be discussed in Section III. Thus, the type of graphs (directed, undirected,
weighted, unweighted) along with the space of representation (L-, P-, B- and C-space) defines the topology of a

E

A
C

E

A

D

B

C
F

Routes
Stops
1
A C D E
2
B C D F
3
C D F

(a)

A
B
C
D
E
F

B
0
0
1
0
0
0

C
1
1
0
1
0
0
(b)

D
0
0
1
0
1
1

1
C

F
A
0
0
1
0
0
0

III. Spaces of Network Representation
In this section, we describe different spaces of network
representation together with the adjacency matrix representation for analysis of public transport networks.
Our discussion will follow the basics introduced in Kurant and Thiran [15] and Ferber et al. [17] for representing a public transport network in different spaces of
network representation, as shown in Fig. 1. The various
topological representations are fundamentally related
to how the network and its parameters are being perceived. For instance, different aspects of interest may include information about the stations having more routes
traversing through them, the most significant station in
a network in terms of connectivity, the routes servicing more stations, edges with more overlapped routes,
the number of transfers needed to reach two different
stations in a network, and so on. Fig. 1 shows the most
commonly used representations of a PTN analysis along
with their adjacency matrix entries.
i) A graph in L-space, also called the space-of-stations, is shown in Fig. 1(b). In an L-space graph,
a public transport stop is treated as a node, and
a pair of nodes are connected by an edge if there
is at least one route servicing the two stops consecutively. The L-space representation is the most

A

D

B

PTN structure to be examined. Table I shows the graph
type and the space of representation chosen in various
PTN analysis in the literature.
Finally, for visualizing a network, there are many
open source network visualization tools, and the selection would depend on the need of the analysis. For a
comparison of different visualization tools, interested
readers are referred to ref. [14].

E F
0 0
0 0
0 0
1 1
0 0
0 0

2
A
1
0
0

B
0
1
0

(c)

F
C
1
1
0

E

A

3

D

B
Routes
1
2
3

E

D
1
1
1

E
1
0
1

F
0
1
1

C

3

D

B

A
B
C
D
E
F

F
A
0
0
1
1
1
0

B
0
0
1
1
0
1

1

C
1
1
0
1
1
1
(d)

D
1
1
1
0
1
1

E
1
0
1
1
0
1

F
0
1
1
1
1
0

2
Routes
1
2
3

1
0
1
1

2
1
0
1

3
1
1
0

(e)

Figure 1. (a) Simple public transport map with stations A-F being serviced by route no. 1 (shaded orange), no. 2 (black), and no. 3
(blue); (b) L-space graph; (c) B-space bipartite graph (route nodes are shown as squares); (d) P-space graph (complete sub-graph
corresponding to route no. 1 is highlighted in orange); (e) C-space graph of routes. The matrix of connectivity is shown below the
corresponding network representation.

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IEEE Circuits and Systems Magazine - Q4 2019

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