IEEE Circuits and Systems Magazine - Q1 2023 - 16

where wN i
i =−
i =−
1/() means that the i th attacked nodes
have been removed from the network. Compared to
Eq. (2), wN i
attack stages as i
1/() assigns higher weights to the later
increases. Different weighting parameters
wi also change the range of robustness measure,
where R1 00 5∈[]
,.
but R2 01∈[]
,.
The measure shown in Eq. (2) for network robustness
under node-attacks can be extended to edge-attacks
[44], as follows:
ND =
R ∑∑ ,
e
1
1 =
M 1
+
ni
L
e
() =
1
M 1
+
i00
==
i
with the superscript e indicating edge-attacks, where
the denominator remains N under
the
tion that the number of nodes is unchanged during
edge-attacks.
When the values niL () or niL
12
,, or Re
connectivity robustness against attacks.
2) Controllability Robustness
Controllability robustness reflects how well a networked
system is in maintaining its controllable
state. Consider a general linear time-invariant (LTI)
networked system,
xA  =+ , where x∈RN
xBu
u∈Rb
and
are the state vector and control input, respectively,
and A∈R ×NN and B∈R ×Nb are constant matrices
of compatible dimensions. Conceptually, this
LTI system is state controllable if and only if there exists
a control input u that can drive the system state
x from any initial state to any target state in the state
space within finite time. A commonly-used criterion
is that the LTI system is state controllable if and only
if the controllability matrix C = [B AB A2B ... AN-1B]
has a full row-rank [66]. The concept of structural
controllability is a slight generalization of the state
controllability, to deal with two parameterized matrices
A and B, in which the parameters characterize
the structure of the underlying system in the sense
that if there are specific parameter values ensuring
the system to be state controllable then the system is
structurally controllable.
When considering a network of many LTI systems,
the node system with control input is called a driver
node (DN). Network controllability is investigated from
two aspects: 1) to gain the full control of the entire dynamical
system, one aims to determine how many and
which nodes to control [16], [17]; 2) for each single node,
the aim is to determine the dimension of its controllable
subspace [67], [68].
16
IEEE CIRCUITS AND SYSTEMS MAGAZINE
e () are plotted, a curve
is obtained, which is called the connectivity curve. A
higher RR
1 value indicates an overall better
where niD () and NiD () represent the density and
number of DNs needed to retain the network controllability
after a total of i nodes have been attacked;
N ′
can be set to either NN ′=−i or NN ′ =
, depending
on specific preference, namely whether or not an
attacked node still belongs to the network depends on
the situation under consideration. Usually, attacked
nodes are assumed to be malfunctioned (but still in
the system) in connectivity robustness measures, but
will be removed from the network in controllability
robustness measures.
Similarly, controllability robustness under edge-attacks
is measured by
R
3 = ∑∑ ,
e
1
M 1
+
ni
D
e
() =
1
M 1
+
i00
==
i
where M is the number of edges in the network. When
the values niD () or niD
Ni
N
D
e
()
M
M
(7)
Ni
N
L
e
()
M
M
(4)
where E*
assumprepresents
the number of edges in the maximum
matching E*, which is a basic concept in classical
graph theory [16]. Under node-attacks, the controllability
robustness is measured by
R ∑∑ ,
i=0
3 =
11 D
D () =
N−1
ni
N
N
i=0
N−1
Ni
N
()
′
(6)
Define the density of DNs by nD = , where ND
ND
N
is the minimum number of DNs needed to retain a full
control of the network, which can be calculated using
either the minimum inputs theorem (MIT) [16] for directed
networks or the exact controllability theorem
(ECT) [17] for both directed and undirected networks,
defined as follows:




max,NA usingECT[17],
{}
−
*
max,NE ,
{}
1
1
−
rank() ,
usingMIT [16],
(5)
e () are plotted, a curve is obtained,
which is called the controllability curve. A lower
R3 or Re
against attacks.
Different from considering the density of DNs, the
control centrality measures the control ability of a
single node in a directed network [68], defined by
cC N
()
c
j = /, where Cc
j
()
c
()
j = rank ()C
j
()
g
is the generic
dimension of the controllable subspace of node j that
can be calculated according to the Hosoe theorem [67];
C represents the controllability matrix. Under this measure,
the greater the cc
the node j is as a DN.
()j
value is, the more " powerful "
FIRST QUARTER 2023
3 value represents more robust controllability
IEEE Circuits and Systems Magazine - Q1 2023

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