Civil Engineering Reference
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L
), both
fi ctitious and real networks are the same, and so are nodes and links
(Gómez
et al.
, 2011a).
between each pair of fi ctitious nodes. At the bottom level (
lv
=
Fictitious networks:
A fi ctitious network is defi ned in terms of fi ctitious
nodes and links at each level. Let
(
lv
)
be the set of fi ctitious nodes at
Λ
V
(1)
{
V
(2)
,
V
(1)
, . . . ,
V
(2)
}.
Similarly,
E
(
lv
)
is the set of fi ctitious links
E
i
(
lv
)
at level
lv
. The fi ctitious
network at level
lv
describes a system at such level of abstraction and
has an associated graph
G
(
lv
)
(
level
lv
. Then,
Λ
(1)
=
=
{
v
1
,
v
2
, . . . ,
v
n
} and
Λ
(2)
=
(
lv
)
,
E
(
lv
)
). The adjacency matrix
A
(
lv
)
denotes whether fi ctitious nodes
V
i
(
lv
)
and
V
(
lv
)
are connected through a
fi ctitious link in
E
(
lv
)
.
Λ
Each of the
L
levels in a hierarchy represents a complete description of the
network. Each level is comprised of
k
lv
>
k
lv
−1
fi ctitious nodes and is a refi ne-
ment of its immediately superior level (i.e., it is a partitioning of higher
resolution). Note that
k
1
n
; thus, the associated partitionings
are trivial clusterings. The hierarchical representation of a network consists
of a set of graphs
G
(
lv
)
(
=
1 and
k
L
=
(
lv
)
,
E
(
lv
)
) with
lv
Λ
=
1, 2, . . . ,
L
, representing fi ctitious
networks.
In addition to the defi nition of fi ctitious nodes and links, three types of
relationships can be observed within the hierarchy. Figure 17.2 summarizes
the hierarchical representation of infrastructure networks.
Inter-cluster relationships:
These are relationships among fi ctitious
nodes, i.e., fi ctitious links.
Inter-level relationships
lv =
1
Fictitious links
(inter-cluster relationships)
Fictitious node
(
lv
)
V
i
(
lv
)
E
ij
lv =
2
V
1
V
2
V
n
Y
Intra-cluster relationships
X
17.2
Elements in the hierarchical model.
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