IEEE Circuits and Systems Magazine - Q1 2023 - 18

where riD () is the rank of the controllability matrix
after a total of i nodes have been attacked. Lower
ranks are assigned to the networks that possess better
controllability.
Figure 3 shows an illustrative example, where net1
requires a larger initial proportion of DNs than net2.
The controllability curve of net1 is flatter than that
of net2. Under two different measures, R3 returns
that net2 has better controllability robustness than
net1, but R8 returns that they have same performance.
Clearly,
initial states.
2) Combinatorial Measure
Although connectivity robustness has a certain positive
correlation with controllability robustness and communication
robustness, they actually have very different
measures and objectives. In general, good connectivity
is the prerequisite for good controllability and communication
ability, but the former does not guarantee the
latter in general.
Considering connectivity robustness and controllability
robustness together, adjustment is necessary
since better robustness means maximization Eq. (2)
but minimization Eq. (6). To unify them (e.g., both being
maximization), a combinatorial measure can be defined
using either the opposite of niD () [78], as follows:
R9=−1 ni ,
i=0
1
N∑()
D ()
or the reciprocal of niD () [79], as follows:
R
10 = ∑
i=0
1
N
N−1
ni
ni
L
D
()
()
.
N−1
(14)
Figure 3. Example of two different controllability robustness
measures. R3 and R8 are calculated using Eqs. (6) and (13),
respectively.
R8 diminishes the influence of the
robustness measure can be applied, which is defined
as follows:
R11 =
where Rpq,
PQ⋅ ∑∑Rpq,
,
1
p 11
==
q
represents the network robustness measured
under the p-th repeated simulation using the q-th
attack strategy; P is the number of repeated attack
simulations; Q is the number of different attack strategies.
After averaging, a robustness value will not be corresponding
to a specific attack strategy or sequence.
(15)
Maximizing R9 is equivalent to minimizing R3, while
maximizing R10 is equivalent to either maximizing R1,
or minimizing R3, or maximizing R1 and minimizing
R3 together.
3) Averaged Measure
All the above-mentioned a posteriori robustness measures,
except for ERCC and GRCC [69], [70], are calculated
based on a specific attack sequence, namely,
each robustness value is one-to-one corresponding
to a specific attack sequence. If network robustness
is required to be measured by a number of repeated
simulations, or several different attack sequences
are required to be considered, then the averaged
18
IEEE CIRCUITS AND SYSTEMS MAGAZINE
4) Other Measures
When cascading failure-based attacks are considered, the
robustness measure can be slightly modified, as follows:
H
R12 = ∑ ()
h=1
1
N
fh ,
(17)
where H is the required number of attacks to achieve
the attack task, for example, a significant destruction of
functionality [80], [81], [82], [83], [84]. Here, HN≤ implies
that it is not always necessary to attack all nodes
in order to destroy the network functions.
When the community structure is concerned, the
community robustness can also be calculated using Eq.
(1), where fi
() could be either the community integrity
that counts the number of remaining nodes in the community
[85], or the normalized mutual information [86].
It is noted that this survey article focuses on reviewing
the robustness measures of the networks with
FIRST QUARTER 2023
P Q
(16)
IEEE Circuits and Systems Magazine - Q1 2023

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