Risk assessment and management of civil infrastructure networks: a systems approach - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 443

the network, and may depend on different factors. The product in Equation
17.5 provides a qualitative metric that describes the relative importance of
network components based on the proportional contribution of the factors.
All factors in Equation 17.5 have been selected so that they are indepen-
dent and under the assumption that they fully describe the vulnerability
space under uncertain decisions. Three possible indices are proposed to
build a coeffi cient that assigns higher weights to higher levels:

()

n L

lv

+ 1

Llv
L

−+ 1 ⎛

⎜

⎞
⎟

1

⎛
⎜

⎞
⎟

⎛
⎜

⎞
⎟

c = [17.6]

lv

()

lv

n K

where L denotes the total number of hierarchy levels, n is the total number
of network nodes; |

( lv ) | is the cardinality of the set of fi ctitious nodes at
level lv ; and K ( lv ) is the k-connectivity of the fi ctitious network at level lv
(Brandes and Erlebach, 2005). The fi rst factor accounts for the fact that
more complex subsystems rely on fewer actual links as moving up in the
hierarchy; other weighting functions can be used depending upon the
problem. The second factor defi nes a degree of clustering at level lv ; i.e.,
the ratio of fi ctitious nodes with respect to the actual number nodes. Finally,
the third factor takes into consideration the k -connectivity of fi ctitious
networks at every level. Note that all three factors are defi ned within the
interval [0, 1] and decrease as the network description becomes closer to
the actual network.
The proposed index V ( lv ) ( e j ) can be used for ranking the edges according
to their relative contribution to the vulnerability at each hierarchical level.
The fi nal result is a ranking of links with decreasing contribution to vulner-
ability (either globally or by level). Thus a global index for a link can be
computed as: max lv V ( lv ) ( e j ). The proposed approach detects critical links as
those with high betweenness (which belong to many shortest-paths, with
low redundancy) and high failure probability. This value provides valuable
information for decision making and risk management, e.g., resource alloca-
tion for maintenance, replacement, etc.

Λ

17.4.3 Damage propagation analysis

Damage propagation is critical for the design, analysis and operation of
infrastructure systems. Networks are comprised of connected/related ele-
ments, increasing the likelihood that elements are affected by others; e.g.,
due to physical proximity to a source of damage (e.g., earthquake fault), or
to redistribution of fl ow after the removal of elements. Damage propaga-
tion depends on a certain level of correlation among network elements. The
proposed hierarchical model is constructed by grouping elements at differ-
ent levels of abstraction. The hierarchy itself can be understood as a measure