IEEE Circuits and Systems Magazine - Q1 2023 - 27

Figure 7. Twelve four-node directed networks: (A) fully-connected (FUL); (B) weak fully-connected (WKF); (C) loop (LOP); (D)
ring-shaped non-loop (RIN); (E) cactus (CTS); (F) star-shaped out (SSO); (G) star-shaped in (SSI); (H) star-shaped random (SSR);
(I) directed chain (DCH); (L) chain-shaped (UCH); (K) disconnected (DIS); and (L) isolated (ISO).
Figure 8. Six four-node undirected networks: (A) fully-connected (FUL); (B) loop (LOP); (C) star-shaped (STR); (D) cactus (CTS);
(E) undirected chain (CHA); (F) isolated (ISO).
Spectral measures are widely used to measure network
connectivity robustness for undirected networks
[13]. Here, 3 adjacency matrix-based spectral measures,
namely spectral radius (AS-SR) [30], spectral gap
(AS-SG) [31] and natural connectivity (AS-NC) [32] measures,
and 3 Laplacian matrix-based spectral measures,
namely algebraic connectivity (LS-AC) [33], number of
spanning trees (LS-NS) [35], and effective resistance
(LS-ER) [34], are compared with a posteriori measures
for measuring undirected networks.
B. Attack Simulations
Tables 3 and 4 show the robustness ranks of different
4-node networks, where equal robustness share the
same ranks. Intensive simulations show the following
general results:
1) A posteriori measures return different values
when different attack strategies are applied, while
a priori measures return a unique value for the
network under different attacks. This suggests
that a priori measures are not capable of distinguishing
the network robustness under different
attacks; while a posteriori measures are capable
FIRST QUARTER 2023
of capturing even tiny robustness difference of a
network under different attacks.
2) Most measures can reflect the basic fact that fully-connected
networks possesses the best robustness,
followed by connected networks, then disconnected
networks, and lastly isolated networks,
which possess the worst robustness. This means
that both a posteriori and a priori measures can
generally reflect the robustness of networks, regardless
of the attack strategies.
3) For the 12 directed networks, no measure can
distinguish all of them with different robustness
values; whereas for the 6 undirected networks,
the measures described by Eq. (6) under EXA,
Eq. (12) under EXA and MBTA, as well as the 3
adjacency matrix-based spectral measures, all return
6 distinguished robustness values. This suggests
that the robustness measures for directed
networks remain a challenging research topic for
future investigation. It is also intuitively clear that
different directed networks may possess the same
robustness performance, which are impossible to
distinguish.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
27

IEEE Circuits and Systems Magazine - Q1 2023

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