The Design of Rapid Transit Networks - Location Science - page 567

In small-world networks it is easy to travel both at the local and at the
global levels. Since such networks are tolerant against disruptions, they are robust.
However, metro networks have been shown not to be robust at the local level.
Nevertheless, networks of direct connections, where there exists an edge between
all pairs of stations for which passengers do not need to transfer to other line,
may be seen as small world networks (Sen et al. 2002 ; Seaton and Hackett 2004 ).
Other papers dealing with efficiency, robustness, vulnerability and small-world
phenomenon of metro networks are those of Latora and Marchiori ( 2002 ), Criado
el al. ( 2007 ), Derrible and Kennedy ( 2010 ), Barbadillo and Saldaña ( 2011 )and
Zhang et al. ( 2013 ). The paper by Roth et al. ( 2012 ) also deserves a mention. These
authors consider the dynamics of the largest metro networks and prove that they
converge to a unique network shape.
A new approach to the study of the connectivity of metro networks and thus
their robustness is grounded in the concept of hypergraphs and their associated
linear graphs. Given a collective transportation network made up of a set of lines
f L 1 ;:::;L l g ,whereL i Df s 1 ;:::;s l i g is the set of stations of line L i , the associated
hypergraph is the pair H D .V.H/;E.H//,whereV.H/is the set of all stations,
and the hyperedge set E.H/ Df L 1 ;:::;L l g consists of the station sets of the lines.
The associated linear graph is L.H/ D . f L 1 ;:::;L l g ;E.L.H///, where the edge
set E.L.H// is the set representing the transfer stations. In Barrena et al. ( 2013 )the
indices defined above are extended to collective transportation networks in order
to allow them to extract information on the easiness of transfer and to compare
different metro networks from this viewpoint. In that paper, the notions of clustering,
characteristic path length, local efficiency and global efficiency are extended to
hypergraphs and are applied to the comparison of several metro networks.

22.3 Location of Rapid Transit Networks: Models

and Algorithms

Construction projects for rapid transit networks can be classified into three groups:
those in which a single line is planned from scratch (Metro de Granada 2013 ), those
in which several lines are planned from scratch and simultaneously (for example,
Sociedad del Metro de Sevilla 2001 ), and those in which an existing network is to
be extended, which corresponds to a conditional network design problem (Metro de
Lisboa 2014 ).

22.3.1 Location of a Single Alignment

The problem of locating an alignment for a rapid transit system lies within the area
of location of dimensional structures either in a discrete or in a continuous space