IEEE Circuits and Systems Magazine - Q1 2023 - 19

static connection, whereas the networks with dynamic
and temporal connections are not discussed. This is
because the robustness measures of dynamic and temporal
networks have very different characteristics and
applications. For example, the robustness of a temporal
network is measured by calculating the relative loss of
efficiency caused by attacks [87], as follows:
R13 =−1
∆ε
ε
where ε0 represents the global efficiency of the temporal
network within a given time window, and ∆ε represents
the efficiency loss caused by attacks. Although it
may be regarded as an a posteriori measure, it has a different
form from Eq. (1) that performs iterative attackand-evaluation
operations.
C. Attack Strategies
From the attacker's perspective, searching for the most
destructive attacking sequence is a desirable task,
which can also help the defender in considering how to
design a best possible network topology with the strongest
robustness. Therefore, attack strategy is also a focal
topic in the study of network robustness.
For a given network, a priori measures return a single
deterministic value about the network robustness,
which will not change when different attack strategies
or different numbers (rounds) of attacks are applied. In
contrast, a posteriori measures are able to reflect different
robustness performances when attack strategies
(or attack sequences) vary. The issue of network robustness
within different contexts has been extensively investigated,
with many edge- and node-attack strategies
proposed to destruct the network functions, regarding
the connectivity, controllability, communication ability,
and so on.
Random attacks remove or malfunction randomlyselected
objects (nodes or edges), while targeted attacks
aim at attacking deliberately-selected objects,
for example, the highest-degree node or the largestbetweenness
edge. Given an importance measure g
for either nodes or edges, targeted attacks perform
sequential attacks to object j, with arg max g, meaning
that object j is the most important according
to measure g.
1) Degree- and Betweenness-based Attack Strategies
For targeted attacks, it is assumed that the targeted object
is more important than the others in maintaining
the network functionality. The most frequently-used
measures of importance are the degree centrality and
betweenness centrality, for both nodes and edges.
FIRST QUARTER 2023
,
(18)
In fact, the maximum degree-based targeted attack
(MDTA) and maximum betweenness-based targeted attack
(MBTA) are the most widely-used strategies.
To integrate multiple importance measures into one,
weights and probabilities may be considered:
p
j =× K
i
∑αi
gij
,
,
∑ gij
j=1
,
where pj is the probability of attacking object j i;α is
the weight for importance measure gg
ii j
;
12 represents a combination of deK
kj
×+ ×
∑∑
kj
bj
gree and betweenness, where kj and bj are the degree
and the betweenness of node j; weights α1 and
α2 adjust the distributions of different features, which
can be set manually [88], or with α2 being replaced by
1 1
j11bj
==
K
j
−α [89].
Similarly, three parameters can be used [90] to
control the weights of degree, betweenness and harmonic
closeness, respectively. Attacking the highestbetweenness
node inside the LCC makes MBTA more
destructive in the later stages of the attack process [91].
These measures have also been used in some strategies
to attack interdependent networks [89], [90], [92],
[93], [94], networks of networks [95], [96], and weighted
networks [97].
Both MDTA and MBTA are not only destructive to
connectivity robustness, but also effectively degrade
other network functions such as controllability and
communication ability [19], [80], [81], [98].
2) Topology-based Attack Strategies
Beside degree and betweenness, commonly-used measures
of importance include closeness [99], Katz centrality
[100], neighborhood similarity [101], branch
weighting [102], structural holes [103], and so on. However,
ranking the importance of nodes or edges is practically
intractable for large-scale networks, since most
measures cannot guarantee that removing the targeted
object will globally and consistently cause the greatest
damage to the network.
The hierarchical structure of a directed network enables
the random upstream (or downstream) attack to
the network controllability, which results in a more destructive
attack strategy than random attacks [68]. The
module-based attack strategy [104], [105] aims at attacking
the nodes with inter-community edges that are crucial
to maintain the connectivity among communities.
Practically, the removal costs for different nodes are not
the same, so attack strategies could also be designed to
minimize the total costs [106].
IEEE CIRCUITS AND SYSTEMS MAGAZINE
19
,
portance measure gi for object j. For example, pj =
αα
(19)
is the im
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