IEEE Circuits and Systems Magazine - Q1 2023 - 24

The turning point of the NCC tendency curve can be used as the indicator
of severe network destruction, namely, when this turning point appears,
it means that there are only isolated nodes left in the
residual network.
is application dependent. For example, the robustness
of food webs is widely measured by R50, which is the
proportion of species (nodes) that has to be removed
to cause the extinction of 50% of the species in the food
web [64], [165], [166].
Different from the Molloy-Reed criterion [167],
which states that a network will lose its giant component
if kk2
2
network destruction is proposed, based on the change
of the number of connected components (NCC) [168].
Specifically,
/ > is reached, here a new measure of
for the a posteriori measure f ⋅() that
considers the network destruction, Eq. (1) can be
rewritten as
R14 =
where TT N()<
T ∑ ()wf i
1
+1
i=0
is the counted number of removed
objects before the threshold of " severe destruction " is
reached. Here, the threshold integer T separates the
attack process into two parts: before T is reached, the
network is considered as normal; after T is reached,
the network is deemed breakdown. Network robustness
will be measured only before this threshold is reached.
In the literature about node-attacks, T is set to be 0.5 N
in [127], 0.05 N in [169], or less than 0.2 N in [107]. All are
user-defined fixed integers.
In this article, instead of setting T to be a fixed value,
the network destruction is considered from the perspective
of NCC, which changes non-monotonically during
the attack process. In general, there is a clear turning
point in the curve of NCC. For a connected network, its
initial NCC is 1. The value of NCC increases as nodes and
edges are being attacked. During the targeted attack process,
the isolated nodes (generated due to attacks) will
never be removed until there are only isolated nodes
left in the residual network, since connected nodes are
always targets if they exist. Therefore, the turning point
of the NCC tendency curve can be used as the indicator
of severe network destruction, namely, when this turning
point appears, it means that there are only isolated
nodes left in the residual network. This indicator of destruction
is studied from a general perspective but not
for a specific application. Note that removing isolated
nodes is possible in any step of random attacks; hence,
24
IEEE CIRCUITS AND SYSTEMS MAGAZINE
i ⋅
T
,
(20)
TDi
i
= argmax
().
(21)
In attack simulations, T can be determined when
Di() < Di −()1 is successively detected for pN

times. Then, Ti pN=−, where p is a small decimal.
It is empirically observed that the determination
of T is insensitive to the change of p. Set p = 005.
in
the simulation, which means that when Di Di()<−()1
is successively detected for 005. N times, one has
Ti
=− N
005..
Equation (21) can also be applied to edge-attacks.
Since NCC will not decrease under edge-attacks,
Ti pN=− can be determined when Di Di()=−()1 is
successively detected for pN
 times.
Figures 5 and 6 show the attack simulation results under
node MDTA and edge MBTA, respectively.
Here, a total of 10 synthetic network models are
simulated, including the Erdös-Rényi (ER) randomgraph
[170], Newman-Watts small-world (SW-NW) [171],
Watts-Strogatz small-world (SW-WS) [29], random triangle
(RT) [98], random hexagon (RH) [172], extremely
homogeneous (EH) [164], Barabási-Albert (BA) scalefree
[173], [174], generic scalefree (SF) [175], onion-like
generic scale-free (OS) [15], q snapback (QS) [39], [176]
networks. In all simulations, the network size is set to
N =1000 with k = 10. Four network functions are measured,
that is, controllability robustness, connectivity
robustness, communication robustness, and the number
of connected components. The resultant values are
normalized, so they are all in [0, 1].
As can be seen from Fig. 5, all the vertical green
lines well match the turning points of the controllability
curves (blue dashed curves). In contrast, for the
connectivity and communication curves (brown and
black dashed curves), the vertical green lines appear
FIRST QUARTER 2023
it is not suitable to use this turning point as the indicator
of destruction.
The number of DN is non-decreasing and the numbers
of LCC and CNP are non-increasing. Moreover, stagnation
of DN, LCC, and CNP may occur frequently; therefore, it
is difficult if not impossible to determine the network destruction
using the changes of DN, LCC, or CNP.
Let Di() denote the NCC values after a total of i
() N∈[] and
nodes have been attacked, where Di
[]
iN
∈−
01
1,
,. The turning point of Di() is calculated by
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