IEEE Circuits and Systems Magazine - Q1 2023 - 28

Table 3.
Robustness ranks of the 12 directed 4-node networks, using 4 a posteriori measures and 4 a priori measures.
Directed 4-Node
Networks
R1 (Eq.(2))
EXA
R15 (Eq.(22))
R3 (Eq.(6))
R7 (Eq.(12))
R1 (Eq.(2))
MDTA
R15 (Eq.(22))
R3 (Eq.(6))
R7 (Eq.(12))
R1 (Eq.(2))
MBTA
R15 (Eq.(22))
R3 (Eq.(6))
R7 (Eq.(12))
A Priori
Measures
EFF
NB
EB
CC
1.5
1.5
1.5
1.5
3.5
3.5
2.5
3.5
2.5
2.5
1.5
2.5
1
2.5
2.5
1.5
1.5
1.5
1.5
1.5
3.5
3.5
2.5
3.5
2.5
2.5
1.5
2.5
2
10
8
1.5
3.5
3.5
3
3.5
7
7
6
7
6.5
6.5
5
6
3
11
11
8.5
3.5
3.5
4.5
3.5
3.5
3.5
2.5
3.5
2.5
2.5
3.5
2.5
5
5.5
5
8.5
5
5
4.5
5
3.5
3.5
5
3.5
6.5
6.5
3.5
8
4
9
10
4
8
8
8
8
10.5 10.5
7
7
10
10
11
9.5
6.5
6.5
10
6
10
2.5
2.5
8.5
3.5
3.5
7.5
3.5
2.5
2.5
7
2.5
10
2.5
2.5
8.5
8
8
8
7
10
10
10
9.5
10.5
10.5
11
10
8
5.5
6
8.5
8
8
6
9.5
3.5
3.5
2.5
3.5
9
9
7
9
7
8
9
8.5
8
8
9
9.5
8
8
9
8
6.5
6.5
9
6
10
2.5
2.5
8.5
FUL WKF LOP RIN CTS SSO SSI SSR DCH UCH DIS ISO
11
11
7
11
10
10
7.5
11
10.5
10.5
7
11
6
7
7
3
12
12
12
12
12
12
12
12
12
12
12
12
12
NA
NA
8.5
4) The two connectivity robustness measures, R1
in Eq. (2) and R15 in Eq. (22), return identical robustness
values (ranks) in all the cases for both
directed and undirected networks, indicating that
these two measures are highly-correlated. It is
worth mentioning that R1 has been widely used
for robustness measure, while R15 is rarely used.
As can be seen from Section III, more useful information
can be dug out from NCC, which may be
underestimated as compared to LCC.
5) CC cannot provide distinguished robustness values
for different networks in many cases; while
NB, EB, and LS-ER cannot measure the isolated
networks. In contrast, the a posteriori measures
are able to return different robustness values for
all networks. This suggests a wider applicability
of the a posteriori measures.
The advantages of a posteriori measures, which are
summarized from the experimental comparisons shown
in Tables 3 and 4, are as follows:
1) A posteriori measures have intuitively clear meanings
for each network function and each attack
strategy, while a priori measures always return a
unique robustness value for a given network.
2) A posteriori measures can provide more robustness
information about the given networks. For
example, although the 3 adjacency-spectral mea28
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CIRCUITS AND SYSTEMS MAGAZINE
sures can return distinguished robustness values
for different undirected networks, as shown in
Table 4, they disagree with the assertion that LOP
is more robustness than CTS. In contrast, a posteriori
measures tell much more useful information:
LOP is more robustness than CTS with respect to
RR
115,, and R7, but CTS is more robustness than
LOP with respect to R3 under EXA and MDTA;
two networks are equally robust under MBTA
with respect to R1 and R15.
3) It is clear that the spectral measures provide
better indicators than the topological measures,
but the former can only be applied to
undirected networks. Moreover, NB, EB, and
LS-ER cannot even be calculated for isolated
networks. In contrast, a posteriori measures can
be applied to any network, and provide better
indicators to network robustness, thus have a
wider applicability.
V. Prospective Research Directions
Some prospective research directions are summarized
from four aspects: 1) exploring better weighting
methods and termination criteria for Eq. (1); 2) designing
more efficient and precise analytical and computational
estimation methods; 3) performing more efficient
FIRST QUARTER 2023
A Posteriori Measures
IEEE Circuits and Systems Magazine - Q1 2023

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