IEEE Computational Intelligence Magazine - May 2023 - 49

two categories [1]. The former merges a multilayer network
into a single-layer network and then applies the existing CD
algorithms to find its partition [12]. Since compressed networks
regard the connections among different relationships as
the same kind ofrelationship, this kind ofmethod cannot protect
the integrity of the community structure and may lead to
information loss [1]. The other category of methods first
applies a single-layer network CD algorithm to each network
layer and then extracts the uniform community partition via a
consensus aggregation strategy [13]. However, this kind of
algorithm does not address the complementary information
across various layers [1]. Notably, in a multilayer network, the
accuracy of community partitioning can be further improved
by taking all network layers into account simultaneously [1].
Given the shortcomings of extended single-layer network
methods, some researchers have focused on the structures of
multilayer networks and proposed a number ofmultilayer network-oriented
methods. The state-of-the-art algorithms are
divided into four categories, i.e., information diffusion-based
methods, spectral clustering methods, matrix and tensor factorization-based
methods, and modularity-based methods [4].In
particular, a modularity-based method can be regarded as an
optimization problem with respect to modularity [20].
Information diffusion-based methods are based on the
concept ofnetwork diffusion [21], [22]. One ofthe most classic
methods is similarity network fusion (SNF) [21], which first
calculates similarity matrices by applying the interchanging
process to all layers and then computes the fused matrix from
the determined similarity matrices. Finally, the community
partition is extracted by applying the spectral clustering
method. This kind of method can make full use of the
extracted network information through information diffusion.
Spectral clustering methods are based on eigen decomposition
[16], [17], [23]. One of the typical methods is SC-ML [17].
SC-ML firstcalculatesthe subspaces spanned by the principal
eigenvectors ofthe corresponding Laplacian matrix for each layer.
Subsequently, all subspaces are aggregated into a consensus subspace
by regarding each subspace as a point on the Grassmann
manifold, and the community structure is extracted using Kmeans.
SC-ML achieves high accuracy and robustness since it can
preserve the integrity ofmultilayer networks.
Matrix and tensor factorization-based methods extract the
consensus community partition by jointly factorizing the adjacency
matrices of a multilayer network [4], [5], [24].Gligorijevic
et al. proposed the NFCCEmethod, which consists offour different
kinds of algorithms [4]. The algorithm calculates the lowdimensional
representation of each layer, and then a common
low-dimensional representation is computed by fusing the representations
of different layers. Finally, the consensus community
partition can be extracted from the consensus representation.
NFCCE can preserve the information contained in each network
layer via collective factorization, enabling it to achieve good accuracy
and robustness.
Modularity-based methods can be formulated as modularity
optimization processes [20]. Optimization-based methods find the
optimal solution by optimizing the objective (or fitness) function
of the population. Among them, multi-objective evolutionary
methods have been proven to be some ofthe most efficient methods
for solving CD problems and have drawn much attention in
recent years [6], [14], [15], [25], [26], [27]. MOEAs can maintain
relatively high accuracy for networks with various structures due
to the remarkable optimal solution search capabilities ofevolutionary
algorithms. In addition, MOEAs can effectively equilibrate the
information of each network layer by flexibly selecting various
objective functions to minimize the effect of noise information.
Moreover, a multi-objective optimization method (MLMaOP)
regards each network layer as an objective and applies the nondominated
sorting genetic algorithm (NSGA)-II to optimize each
layer simultaneously [15]. Different from MLMaOP, MOEAMultiNet
takes the similarity between network layers into
account [6]. This method sets two objective functions, which correspond
to the community structure ofeach layer and the similarity
between two layers, namely, the average modularity and NMI
between network layers. Different from traditional MOEAs, a
semi-supervised MOEA method (SS-MOML) combines semisupervised
learning and an MOEA. Guided by prior information,
SS-MOML can find high-quality consensus community partitions
[28]. Previous studies have proven that an MOEA can effectively
extract the community structures of multilayer
networks [6], [14], [15].
As mentioned above, different kinds ofalgorithms have been
proposed to detect the community structures of multilayer networks,
and all of them have achieved good performance [4].
However, each type of method has drawbacks. For example,
information diffusion-based methods obtain solutions through the
information diffusion process. Thus, their clustering performance
is susceptible to the restriction ofthe network structure. In particular,
in a network with very sparse or even many isolated nodes, the
performance ofinformation diffusion is seriously affected. Spectral
clustering-based methods and matrix (tensor) factorization-based
methods suffer from similar problems. They are both vulnerable
to the noise informationina network [29]. Optimization-based
methods also have some disadvantages, such as the tendency to
find local optima. However, it is undeniable that optimizationbasedmethods
are able to find approximate local optimal solutions
in acceptable time through heuristic information guidance and
population iteration [30]. Experiments have shown that such
methods are particularly suitable for solving NP-hard problems,
e.g., CD [31], [32]. Given this situation, this paper addresses the
local optimality issue ofMOEAs, designs a novel algorithm based
on prior information, and experimentally verifies the significance
ofprior information for improving the accuracy and robustness of
the developed algorithm.
C. Semi-Supervised Learning-Enhanced CD Methods
Prior information is ofgreat significance for algorithms since it can
be utilized to guide them to further improve their accuracy and
robustness for CD problems [1], [18]. On the basis of the four
kinds of algorithms mentioned above, some researchers have
incorporated prior information into traditional CD algorithms by
MAY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 49
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