IEEE Circuits and Systems Magazine - Q1 2023 - 17

The expected robust control centrality (ERCC) [69],
[70] is a control centrality-based robustness measure
for node-attacks, defined as follows:
Ri EC i
()
4
j () =


()
j
c
()

,
(8)
where Cic
()j () represents the control centrality of node
j after a total number of i nodes have been attacked;
E ⋅[] is the statistical expectation. The generic robust
control centrality (GRCC) [69], [70] is a generalization of
ERCC, defined as follows:
Re EC e

ej,()()=
4
where Cec
()j {}
{}
c
 ()j {} ,
()


(9)
() represents the control centrality of
node j after a set of edges e{} have been removed, under
either node- or edge-attacks. Both ERCC and GRCC
measure the significance of a single node in controlling
part of the system, under random node- and edgeattacks,
respectively.
The reachability-based controllability robustness
[52], [71] is also a control centrality-based robustness
measure. Given a fixed number of H controllers that
can be pinned anywhere ( " free control " mode), the controllability
robustness is calculated by
R5 =
where ∑ 1cjc () represents the dimension of the conj=
H
trollable
subspace by the given H DNs. During the
attack, these DNs can be freely set in the remaining
network, as long as the control centrality is maximized.
Again,
NN ′=−i or N, depending on the specific situation
under consideration.
In the case that the given external controllers are
fixedly pinned at a set of given nodes ( " fixed control "
mode) [52], the controllability robustness is also measured
using Eq. (10), where however ∑
j=
H
1cjc () counts
the dimension of the controllable subspace by the given
H fixed controllers.
3) Communication Robustness
Different from the a priori measures of general connectivity
robustness, which are either spectral measures or
topological features, the a priori measures of communication
robustness are more comprehensive. For example,
the r-robustness [72], [73], [74] based on reachability, and
the comprehensive measure proposed in [75] consisting
of three indices, including edge betweenness centrality,
number of edge cut-sets, and node Wiener impact [76].
Nevertheless, the a posteriori measures for connectivity
FIRST QUARTER 2023
1
N−1 H
i=0
j=1
N∑∑ ,
c
cj
N
()
′
(10)
which ignores the non-dominant terms in Eq. (11) but
keeps only the dominant ones. The computation complexities
of both measures are the same, O(NM).
When the CNP values are plotted, a curve is obtained,
which is called the communication curve. Apparently,
higher values of R6 or R7 represent better communication
robustness against attacks.
B. Variants of Robustness Measures
Based on the fundamental a posteriori robustness measures
presented in Subsection II-A, several variants
have been developed with different concerns.
1) Rank-based Measure
Before being attacked, the initial proportions of LCC for
all connected networks are the same, namely nL 01
() = .
In contrast, the initially required proportion of DNs to
fully control a network varies from case to case. This inequality
of initial controllability may influence the measurement
of robustness. The rank-based controllability
measure offers an alternative to diminish this influence,
which is defined by
R8 =
1
N−1
N∑ ()
D
i=0
IEEE CIRCUITS AND SYSTEMS MAGAZINE
17
ri ,
(13)
R6 =
robustness remain useful for measuring communication
robustness [77].
The CNP-based robustness measure is a widely-used
a posteriori measure for communication robustness, defined
as follows [19]:
1
N−1 Γ()
i=0
i S
N∑∑ ,
j=1
( )
( )
N
j
2
2
the number of communicable node pairs, while ( )
( ) = N
Sj
2
N
2
j
2
2
The following simplified communication robustness
( ) < N
Sj
2
2
( ).
[20] provides a simpler CNP-based measure:
R
7 = ∑∑ ,
i=0 j=1
1
N
S
N
2
2
j
N−1 Γ()i
(12)
(11)
where Γ i() represents the number of connected components
in the remaining network after a total of i
nodes have been attacked; Sj represents the number of
nodes in the j th connected component;
is the number of all possible node pairs. When
S
( ) represents
( ), the network is fully connected, thus each
pair of nodes are communicable; while for the networks
that are not fully connected, the number of communicable
node pairs should be less than the all possible
node pairs, namely
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