IEEE Circuits and Systems Magazine - Q1 2023 - 14

Figure 1. General framework for a priori and a posteriori measures of network robustness. A priori measures perform one-time
calculations to evaluate the network robustness, where PM represents an a priori measure; while a posteriori measures require
an iterative process to gain the robustness.
networks. However, the predictive a priori measures
have limited scopes of applications [36]. Moreover, the
a posteriori measures are effective when the attack process
is terminated by a specific criterion, whereas the
a priori measures do not consider the stopping criteria.
Therefore, the time-consuming but precise a posteriori
measures remain to be the main approach for real-world
applications today.
The general framework of a priori and a posteriori
measures is shown in Fig. 1, which shows that a priori
measures evaluate the network robustness in a straightforward
one-time process; while a posteriori measures
require an iterative process until the stopping criterion
is met. It is clear that a posteriori measures could have
different options on the configuration of stopping criteria
and attack strategies, while this is invalid for a priori
measures.
With desirable robustness measure(s) chosen and
used as the objective(s) to optimize, network robustness
can be enhanced by model design [37], [38], [39],
[40], [41], edge addition [14], [42], or edge rewiring [13],
[43], [44], [45], [46], [47], [48], [49], [50].
In the literature, some general survey papers emphasizing
more on a priori measures of network robustness
are available [12], [14], but there does not seem to be
any that specifically emphasizes on the a posteriori
measures. To fill the gap, this article presents a survey
of the a posteriori measures of network robustness, including
definitions, computation, applications, and optimization.
The main contributions of this survey are
summarized as follows:
1) The a posteriori robustness measures are summarized
and compared, from the perspectives of
network functionality, attack strategies, robustness
performance prediction, and structural optimization.
2)
A threshold of network destruction is proposed,
which suggests a more practical robustness measure
of the functionality, especially when a network
has been severely destructed.
In so doing,
whether or not a modification of the network structure
can enhance the robustness has to be evaluated, usually
by using a posteriori robustness measures that usually
requires attack simulations.
Other than attack simulations, network robustness
can also be estimated using both analytical and computational
methods without iterative calculation. Analytical
approximations are applicable when the a priori
knowledge of the concerned network is available and
the attack strategy can be well modeled [51], [53], e.g.,
random attacks. In contrast, computational methods are
generally data-driven and thus applicable to any attack
methods with or without a specific pattern [54], [55], [56].
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IEEE CIRCUITS AND SYSTEMS MAGAZINE
3) Both a posteriori and a priori robustness measures
are experimentally compared on a series of
directed and undirected network examples. It is
found from simulations that a posteriori measures
have broader applicability.
4) Some possible future research directions with respect
to network robustness are suggested.
The remainder of this article is organized as follows.
Section II reviews the a posteriori measures of network
robustness, from the perspectives of network functionality,
malicious attacks, robustness performance prediction,
and optimization. Section III introduces a threshold
of network destruction. Section IV experimentally
compares a posteriori and a priori measures. Section V
presents some prospective research directions with respect
to a posteriori robustness measures of complex
networks. Section VI concludes the survey.
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