IEEE Circuits and Systems Magazine - Q1 2023 - 30

If the pattern of malicious attacks can be well modeled using mathematics
and statistics tools, then analytical approximation methods are
recommended; but if there is no such a pattern, then analytical methods
are inapplicable while computational techniques are effective.
uniform weights for different attack strategies. In practice,
if the probabilities of a network suffering different
attacks are different, then it is meaningful to impose different
weights to them.
Possible realistic weighting methodologies include
decaying weights, importance-based weights, adaptive
weights, etc.
B. Termination Criteria
A realistic threshold of destruction is introduced in
Section III, which gives an alternative threshold to the
conventional settings, such as the Molloy-Reed criterion
[167] and the fixed-proportion threshold. However,
there still lacks a systematic investigation on the
determination of the time when a networked system is
deemed breakdown thereby the attack process can be
terminated. The destruction of networks can be investigated
from the perspectives of topological structures,
network functions, or both.
To determine proper termination criteria, analytical
and theoretical studies can be carried out, for example,
further development of the Molloy-Reed criterion [167],
percolation theory [178], and so on. Empirical studies
such as the realistic threshold introduced in Section III
can also be further investigated. Moreover, machine
learning techniques may be utilized for solving this
problem more effectively from a data-scientific perspective,
based on both real-world networks and synthetic
models. For example, given real-world data of network
destruction as training data, machine learning can be
used to estimate whether a given network is considered
breakdown, or when it would be breakdown, under
attacks.
C. Robustness Estimation
It is important to precisely and cost-efficiently approximate
various robustness of large-scale networks.
The existing analytical approximations are applicable
only to very limited specific issues of complex networks,
e.g., controllability robustness under random or
critical edge-attacks [51], [52], [53]. Considering Eq. (6)
as the controllability robustness measure, attacking
a single node (or edge) may either increase the number
of DN by 1 , or it does not change the number of
DN at all. Thus, the maximum damage to the network
controllability is limited.
In contrast, when Eq. (2) is
30
IEEE CIRCUITS AND SYSTEMS MAGAZINE
used to study the connectivity robustness, the range
of damages caused by each attack to LCC could vary
from 0 to N −1, namely with all possibilities. Therefore,
predicting the connectivity robustness is much more
uncertain and challenging than predicting the controllability
robustness, either analytically or computationally
[179].
In this direction, if the pattern of malicious attacks
can be well modeled using mathematics and statistics
tools, then analytical approximation methods are recommended;
but if there is no such a pattern (neither random
not specifically targeted), then analytical methods
are inapplicable while computational techniques are
effective.
A comprehensive investigation of analytical approximation
to robustness is needed, where some
potential research topics include: 1) modeling more
intrinsic attacks other than random or degree-based
attacks; 2) exploring the relationships between the topological
features and the robustness performance,
where if direct
relationships cannot be revealed
then indirected relationships may be explored, for
example some critical points (e.g.,
turning points)
of the robustness curve might be estimated using
topological features, so that a robustness curve can
be fitted based on these critical points. As for computational
approaches, not only the state-of-the-art
machine learning techniques can be developed and
applied, but also prior knowledge and theoretical
findings can be used to further improve the prediction
performances.
D. Robustness Optimization
Network robustness optimization via topological rewiring
is NP-hard [180]. The development of evolutionary
algorithms helps in effectively resolving this
difficult problem. Robustness optimization for largescale
complex networks is higher-dimensional and
computational expensive in general. In this regard, dimension
reduction can be archived by applying graph
embedding or using GNN [133], [134], [135], which not
only compress higher-dimensional network data into
lower-dimensional representations, but also extract
structural features for further processing. As for the
computational expenses in robustness evaluation,
surrogate models are advantageous for improving
FIRST QUARTER 2023
IEEE Circuits and Systems Magazine - Q1 2023

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