Geoscience Reference
In-Depth Information
new indices based on the assessment of transportation networks from the angle
of complex network theory and robustness. In accordance with the glossary of
the IEEE (
1990
), robustness can be defined as the degree to which a system or
component can function correctly in the presence of invalid inputs or stressful
environmental conditions. In the case of rapid transit networks planning, future
ridership is an uncertainty input variable which also depends on the travel times
of alternative transportation modes.
Another issue affecting robustness lies in the disturbances of normal operations.
The paper of De-Los-Santos et al. (
2012
) considers robustness from the angle of
passengers in the presence of disruptions. The auxiliary function applied to define
robustness measures is the total transit time of passengers. Two cases are considered.
In the first case, passengers affected by the disruption have to wait for the failure
to be repaired or have to take an alternative route in the same network. In the
second case, the operator provides a bus-bridge service. An example for the Madrid
commuter system illustrates the applicability of the robustness indices developed by
the authors.
Over the past 15 years there has been an increased research interest in the
structural properties of the networks representing complex systems, which is
interesting for understanding the functioning of these systems. One of the most
cited examples in the scientific literature is that of transportation networks and,
in particular, metro networks. The concept of small-world phenomenon comes
from sociology. The corresponding networks are an intermediate class between
regular networks (with equal-degree nodes) and random networks (edge-generated
by a given probability). Small-world networks are highly clustered, like regular
networks, but they have a low average shortest path length between pairs of nodes
(Watts and Strogatz
1998
). Let G
D
.V;E/ be a graph and let d
ij
;v
i
;v
j
2
V be
the topological distance between v
i
and v
j
(the minimum number of edges in a
path between v
i
and v
j
). Then the characteristic path length L and the clustering
coefficient C are defined as
j
V
j
.
j
V
j
1/
X
i¤j
j
V
j
X
v
i
2V
1
d
ij
; and C
D
1
L
D
C
i
;
where C
i
is the number of edges in G
i
D
.V
i
;E
i
/, the subgraph of the neighbors of
v
i
, divided by the maximum possible number
j
V
i
j
.
j
V
i
j
1/=2.
In order to adapt these concepts to metric networks and to overcome some
indetermination, the average length of shortest paths and clustering coefficients were
substituted by global and local efficiency (Latora and Marchiori
2001
):
j
V
j
.
j
V
j
1/
X
i<j
j
V
j
X
v
i
2V
2
d
ij
; and E
loc
.G/
D
1
1
E
glob
.G/
D
E
glob
.G
i
/;
where G
i
is the subgraph of neighbors of v
i
.
