IEEE Computational Intelligence Magazine - May 2023 - 48

the MOEA-CPI algorithm has distinct accuracy and robustness
advantages over some state-of-the-art methods.
The contributions of this paper can be summarized as
follows.
❏ This paper applies Node2vec and the density-based aggregation
strategy to extract graph-level information based on the
input graph structure for constructing the prior layer and generating
a high-quality initialization. The walk-based sampling
strategy ofNode2vec can lead the acquired prior information
(i.e., the graph-level information) to possess high-dimensional
structural information, whichmakes the prior layer and initialization
possess global information.
❏ This paper applies the Jaccard similarity coefficient to extract
node-level information for guiding the mutation strategy. The
Jaccard similarity only considers the node-level information,
and the mutation strategy is essentially a node-to-node operation.
Because ofthe natural similarity between the node-level
information and the mutation strategy, the node-level information
is used in a guidance role to further improve the performance
ofthemutation strategy.
❏ After carefully exploring the structures ofmultilayer networks,
great differences are observed between different layers in terms
of the clarity of their community structures. Therefore, a
weighted strategy is proposed to dynamically assign the weight
for each layer to make the most ofthe valuable extracted information
and eliminate invalid information, thus further
improving the accuracy and robustness ofthe algorithm.
The rest ofthe paper is organized as follows. The formulation
ofCD for multilayer networks and the existing algorithms
are introduced in Section II. Section III introduces the proposed
MOEA-CPI algorithm in detail. The specific experimental
settings, experimental results, and algorithm analysis
are revealed in Section IV. Section V concludes this paper.
II. Related Work
This section first defines the problems discussed in Section IIA.
Then, Section II-B introduces the existing CD algorithms
for multilayer networks. Finally, semi-supervised learningenhanced
CD methods are introduced in Section II-C.
A. Problem Definition
In the past few years, researchers have conducted studies on different
kinds ofmulti-relationship networks, such as multiplex networks,
interdependent networks, and interconnected
networks [19].Thispaper focusesonmultilayernetworks(multiplex
networks), which can be represented as G ¼fV; Elgðl ¼
f1; 2; ...LgÞ. L denotes the total number of network layers, V
stands for the node set, and El represents the connections between
the nodes in the lth layer. Amultilayer network includes a series of
network layers, where each layer has the same node set VðjVj¼
NÞ and a different edge set El. The existing MCD algorithms
regard each network layer as a weighted and undirected graph (or
matrix), and these algorithms code amultilayernetwork as aseries
of adjacency matrices AðlÞ 2 RNNðl ¼f1; 2; ...LgÞ.In addition,
complementarity remains across the various layers in
48 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2023
multilayer networks, which can reflect the multiple relations
between objects with higher accuracy. For example, Figure 1(a)
shows a two-layer network with the same nodes in each layer. In
the first layer, only the community ofnode set {1, 2, 3} is clear,
while in the second layer, only the community ofnode set {4, 5,
6} is clear. In other words, the information of each layer is
incomplete and complementary [4]. Hence, the community
structure produced by a single layer is not precise for defining a
multilayer network. To address this issue, the core purposes of
MCD are to combine the information contained across the
layers, obtain a unified partition, and ensure that each node is in
the same community in different layers, i.e., produce common
community partitions.
In this paper, two authoritative metrics, named the normalized
mutual information (NMI) [4] and adjusted Rand
index (ARI) [4], are selected to assess the performance of the
proposed algorithm. More specifically, the NMI,defined in
(1), is one of the most widely used metrics for assessing CD
algorithms by measuring the similarity between the produced
partition V and the ground-truth partition C.
NMIðV; CÞ¼
IðV; CÞ
jHðVÞþHðCÞj=2
(1)
where V and C denote the detected partition and the ground
truth, respectively. I stands for mutual information. HðVÞ and
HðCÞ denote the entropies of V and C, respectively. The
range of NMI is from 0 to 1, and the larger the value is, the
better the clustering results are.
The ARI stands for the proportion of correct decisions of
two types. One validly classifies the node pairs from the same
class as they belong to the same cluster. The other correctly distinguishes
the node pairs in different clusters, as shown in (2).
ARI ¼
where
RIðV; CÞ¼
TP þTN
TP þ FP þ FNþTN
(3)
where FP indicates false positive decisions and FN denotes
false negative decisions. In contrast, TP and TN represent true
positives and true negatives, respectively. EðRI) is the expectation
value of RI. The more the value of ARI approaches 1,
the better the algorithm performs.
B.MCD
Unlike the success of single-layer network CD, the research on
CD for multilayer networks has not formed a uniform framework
nor reached a consensus definition. The existing algorithms can be
divided into two categories, i.e., extended single-layer network
methods and multilayer network-oriented methods.
The extended single-layer networks are directly extrapolated
from the traditional network CD algorithms based on
some strategies. Generally, these methods can be divided into
RI EðRIÞ
maxðRIÞ EðRIÞ
(2)
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