IEEE Circuits and Systems Magazine - Q1 2023 - 23

Table 1.
Summary of using heuristic algorithms to optimize network robustness.
Work
Measure
Gunasekara, et al.
[147]
Liu, et al. [163]
Liu, et al. [148]
Qiu, et al. [22]
Cai, et al. [25]
Eq. (2) and two
spectral measures
Eq. (2)
Eq. (2) and Eq. (4)
Eq. (2)
2 objectives:
1) maximize algebraic
connectivity
2) minimize the
number of removed
edges
Wang, et al. [49]
Wang, et al. [50]
Eq. (2)*
Eq. (2)*
Algorithm
multi-objective
evolutionary
algorithm
evolutionary
algorithm
multi-objective
evolutionary
algorithm
multi-population coevolution
NSGA-II,
NSGA-III,
and MODPSO
(Continued)
Constraint
N/A
N/A
degree preservation
for each node
degree preservation
for each node
N/A
Attack Object
node
node
node and edge
node
edge
surrogate-assisted
evolutionary
algorithm
surrogate-assisted
multi-objective
evolutionary
algorithm
Lou, et al. [164]
Lou, et al. [48]
Eq. (6)
Eq. (6)
random edge
rectification
random rewiring
degree preservation
for each node
degree preservation
for each node
average degree
preservation
average degree
preservation; and
underlying-topology
preservation
* with the assistance of surrogates and assortativity
node
node and edge
node and edge
node
the optimization process. Extensions of this common
scenario include considering different measures, using
advanced algorithms, imposing different constraints of
topology disturbances, and targeting different objects.
Since a single measure sometimes cannot fully reflect
the network robustness [20], [60], multiple robustness
measures are usually considered, which are simultaneously
optimized by using multi-objective optimization
algorithms [50], [147], [148].
III. Measuring Network Destruction
It is observed that a posteriori measures not only have
intuitively clear meanings for a network function, as
discussed in Section II-A, but also have clearer descriptions
about the network robustness, as introduced
in Section IV. One significant disadvantage of a posteriori
measures, however, is that their calculations are
FIRST QUARTER 2023
generally time-consuming. Nevertheless,
in some applications,
this can be (partially) solved by using analytical
and computational methods, as discussed in
Section II-D.
Here, another common shortcoming of a posteriori
measures is addressed. The calculation of most a posteriori
measures is based on the entire process from attacking
the first object to ending the attack at the last object.
Practically, if a network has been severely destructed or
malfunctioned, measuring its functionality will have no
meaning. Also, complete disconnection of networks may
not always be important in many applications. For example,
when cascading failures are concerned, as shown
in Eq. (17), complete disconnection is unnecessary to attempt.
Therefore, it is not always necessary to attack all
nodes or edges for measuring the network robustness.
Clearly, determining when to stop attacking or whether
a network is severely destructed or malfunctioned
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