Risk assessment and management of civil infrastructure networks: a systems approach - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 441

sparsity between clusters, then, low redundancy (i.e., few alternate paths)
can be expected between fi ctitious nodes; thus, connectivity relies on a few
elements of high betweenness. Figure 17.4 illustrates the fact that clustering
can be used to detect bottlenecks and other critical elements.
The hierarchical model is used to make analyses more effi cient by reduc-
ing the number of scenarios to be evaluated at the upper levels. Critical
elements for the network performance are detected as emergent properties
at different levels of abstraction, i.e., how individual elements participate
in the connections between fi ctitious nodes at different levels of abstraction.
Thus, in topological terms, links found in fi ctitious links at higher levels will
be considered to have higher criticality. Figure 17.5 shows how the concept
is applied throughout the hierarchy.
Two factors, F form ( e j ) and F strength ( e j ), are introduced to evaluate how the
form (i.e., topological importance according to connectivity) and the
strength (i.e., failure probability) contribute to the vulnerability of actual
links e j , respectively.
The impact of a damaging scenario is measured in terms of connectivity
loss (or increase in shortest-path). The Dijkstra algorithm (Brandes and
Erlebach, 2005) for fi nding shortest-paths within networks (before and after
link removal) can be used to evaluate a measure of the impact of damage
in the network. The factor F form ( e j ) describes the importance of the edge j
at level lv and is computed as:

( − ()
()

()

j

Dst Dst
Dst

, ,

() =

∑

Fe [17.3]

form j

,

st V ,

Bottleneck?
Critical point?

Failure prob.?

S T

Max flow/traffic path?
Min cost route?

17.4 Principle of maximum sparsity between clusters.