IEEE Circuits and Systems Magazine - Q1 2023 - 21

can be approximated based on the uniformly-random
decreasing process of critical edges [51]. This analytical
method is applicable to approximating the controllability
robustness [51], as shown in Eq. (7), and the reachability-based
controllability robustness [52], as shown
in Eq. (10), under random edge-attacks.
For random-graph networks, based on the fact that
the generation mechanism is essentially the same as the
random edge-removal process from a fully-connected
network in a reverse manner, a precise approximation
can be designed. The given random-graph networks are
classified as either " dense " , " median " , or " sparse " . Then,
the hybrid approximation method uses different prior
knowledge to approximate the controllability curves
[53]. In comparison, the approach of [53] performs significantly
better in predicting the controllability curves
of random-graph networks under random edge-attacks;
while its disadvantage is clear that it is applicable only
to the abovementioned scenario.
2) Machine Learning-based Prediction
Machine learning algorithms, such as linear regression,
random forest, and neural networks, have been successfully
applied to predicting the number of DNs under random
or targeted edge-attack, such that the controllability
curves can be fitted [138].
During the network robustness optimization processes,
calculating the exact robustness values may not
be required for every generation. Therefore, fast estimation
can be used to improve the efficiency. For example,
in [49] and [50], three algorithms, including radial
basis function [139], inverse distance weighting [140]
and least-squares [141], form a surrogate ensemble for
estimating connectivity robustness values; attack simulations
are intermittently performed for obtaining real
robustness values, which are used for simultaneously
evaluation and updating the surrogates. The computational
time of optimization can be significantly reduced
by using such a surrogate ensemble [49], [50].
The CNN-based prediction approach treats complex
network data as gray-scale images [54], thereby fast approximating
the robustness performance against different
attacks in an end-to-end manner. Prior knowledge is
useful for preprocessing and filtering, which is utilized
to build an improved predictor [55], showing lower prediction
errors. A limitation of this straightforward approach

IEEE Circuits and Systems Magazine - Q1 2023

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