IEEE Circuits and Systems Magazine - Q1 2023 - 26

TABLE 2.
Comparison of robustness performance under two measure schemes: Complete disconnection (CD) and thresholdbased
disconnection (TD). Numbers in parentheses represent the corresponding ranks of robustness.
Node MDTA
Controllability
Robustness
ER CD
TD
SWNW
SWWS
CD
TD
CD
TD
RT
CD
TD
RH CD
TD
EH CD
TD
BA CD
TD
SF
CD
TD
SO CD
TD
QS CD
TD
0.247 (2)
0.155 (3)
0.248 (3)
0.146 (2)
0.259 (4)
0.159 (4)
0.302 (7)
0.190 (6)
0.292 (6)
0.178 (5)
0.227 (1)
0.132 (1)
0.443 (8)
0.288 (8)
0.783 (10)
0.601 (10)
0.768 (9)
0.589 (9)
0.286 (5)
0.207 (7)
Connectivity
Robustness
0.476 (3)
0.534 (6.5)
0.474 (4)
0.538 (1.5)
0.473 (5)
0.537 (3.5)
0.459 (7)
0.533 (8)
0.464 (6)
0.538 (1.5)
0.478 (2)
0.536 (5)
0.418 (8)
0.534 (6.5)
0.205 (10)
0.376 (10)
0.215 (9)
0.380 (9)
0.484 (1)
0.537 (3.5)
Communication
Robustness
0.331 (3.5)
0.372 (6.5)
0.331 (3.5)
0.376 (4)
0.330 (5)
0.375 (5)
0.326 (7)
0.378 (3)
0.327 (6)
0.379 (2)
0.332 (2)
0.372 (6.5)
0.307 (8)
0.393 (1)
0.149 (10)
0.273 (10)
0.156 (9)
0.276 (9)
0.333 (1)
0.369 (8)
differently. The TD robustness measures are recommended
(or even necessary) to use for the following
reasons: 1) the resultant ranks are unique, such that the
robustness measures can be distinguished for different
networks; 2) the TD measures require much fewer
numbers of attacks to measure the robustness and thus
requires less computational time.
IV. Comparison Between A Priori and
A Posteriori Measures
Now, experimental results on a priori and a posteriori
measures are compared under 3 different node-attack
strategies, namely exhaustive attack (EXA) [164], MDTA,
and MBTA.
EXA averages the robustness values of a given
N-node network over all the N! possible node-attack sequences.
Note that any intentional attack (e.g., MDTA,
MBTA) is a particular case of the exhaustive attacks.
Since the sample size of N! becomes enormous as N increases,
only N = 4 is tested here, which well serves for
the purpose of demonstration. Figs. 7 and 8 show the
topologies of the 4-node directed and undirected networks,
respectively.
26
IEEE CIRCUITS AND SYSTEMS MAGAZINE
T Controllability
Robustness
891 0.390 (6)
0.286 (4)
881 0.361 (3)
0.269 (3)
881 0.372 (4)
0.234 (1.5)
861 0.376 (5)
0.302 (6)
861 0.344 (2)
0.294 (5)
891 0.327 (1)
0.234 (1.5)
782 0.424 (7)
0.372 (8)
545 0.632 (10)
0.610 (10)
564 0.615 (9)
0.588 (9)
901 0.448 (8)
0.361 (7)
Edge MBTA
Connectivity
Robustness
0.835 (4.5)
0.991 (2)
0.834 (6.5)
0.985 (5)
0.834 (6.5)
0.998 (1)
0.830 (8)
0.964 (6)
0.836 (2.5)
0.918 (8)
0.836 (2.5)
0.986 (4)
0.841 (1)
0.936 (7)
0.778 (10)
0.829 (10)
0.781 (9)
0.839 (9)
0.835 (4.5)
0.989 (3)
Communication
Robustness
0.806 (4.5)
0.983 (2)
0.808 (3)
0.971 (5)
0.806 (4.5)
0.995 (1)
0.801 (6.5)
0.936 (6)
0.812 (2)
0.892 (8)
0.813 (1)
0.975 (4)
0.800 (8)
0.896 (7)
0.669 (10)
0.718 (10)
0.676 (9)
0.733 (9)
0.801 (6.5)
0.978 (3)
T
381
406
347
426
455
411
446
465
460
381
Four a posteriori measures are simulated together,
namely the connectivity robustness measured by
Eq. (2), controllability robustness measured by Eq. (6),
communication robustness measured by Eq. (12), and
connectivity robustness measured by NCC, which is defined
as follows:
R15 =
where Ni
1
N−1
N∑Ni
i=0
NCC() represents the number of connected
components after a total of i nodes have been attacked.
A. A Priori Measures
A priori measures are quantified by specific network
features that can be calculated without performing attack
simulations. A priori measures require only onetime
calculation and they usually have lower time and
computational complexities compared to a posteriori
measures [13], [14].
Four topological a priori measures are compared
with a posteriori measures for both directed and undirected
networks, namely efficiency (EFF) [177], node betweenness
(NB) [28], edge betweenness (EB) [28], and
clustering coefficient (CC) [29].
FIRST QUARTER 2023
NCC()
(22)
IEEE Circuits and Systems Magazine - Q1 2023

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q1 2023
