IEEE Computational Intelligence Magazine - May 2023 - 52

FIGURE 3. Diagram of label-based and locus-based encoding methods.
The figure shows the encoding (red arrows) and decoding (black
arrows) processes of both methods.
the optimal solution is obtained by a genetic operation (i.e.,
selection, crossover or mutation) based on the initial population.
Although the performance of algorithms does not rely
only on the initial population, it is undeniable that initialization
plays a vital role in optimization-based algorithms. Therefore,
this paper generates the initial population according to
the extracted graph-level information to improve the quality
of the prior information from the global perspective (as introduced
in Section 3.2). After that, the population is converted
with the locus-based encoding scheme, which is used for further
optimization.
D. Optimization Process
The optimization process is one ofthe most important components
in an MOEA. MOEA-CPI applies the NSGA-II as the
multi-objective optimization framework. In this section, two
important components are introduced, namely, the weighting
strategy and genetic operation.
An MOEA finds optimal solutions by optimizing different
objective functions, which play a vital role in such algorithms. In
this paper, the kernel K-means (KKM) and ratio cut (RC) strategies
are selected as two objective functions. KKM andRC are two
widely used functions, where KKM is used to optimize the internal
connections and RC is applied to optimize the external connections
[41]. TheKKM andRC functions are shown in (6).
8
min
where Vh
<
:
KKM ¼ 2ðn kÞ
Pk LVi;Vi
RC ¼
i¼1
ðÞ
jjVi
¼VVh, LðVh; VhÞ¼
P
connection strength and LðVh;VhÞ¼
Pk
i¼1
LVi;ViðÞ
jjVi
(6)
ðvi;vj2VhÞ Aij is the internal
P
external connection strength.
As mentioned above, a multilayer network consists ofseveral
network layers with different structures, which causes the clarity of
the community structure in each layer to be different. As shown in
Figure 4, this paper projects each layer of an example multilayer
network into a two-dimensional surface. the results show that the
community structures (i.e., the network clusters) are clear in some
network layers but fuzzy in others. For example, the nodes in the
52 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2023
vi2Vh;vj2Vh Aij is the
FIGURE 4. An example of a multilayer network. Upon portraying the
network structure, the community structure of the first layer is fuzzy, but
the community structures of the second and third layers are clearly
visible. In addition, different communities are colored differently.
third network (L3) have three clusters, but it is difficult to observe
the cluster distributions in the first and second network layers (L1
and L2). Ifa network layer that has an obvious community structure
can be extruded and the importance ofa network layer with a
chaotic community structure that can be reduced, the accuracy of
the algorithm can be further improved. Therefore, to dynamically
accommodate multilayer network structures, this paper proposes a
weighting strategy that first computes the community structure of
each layer based on K-means due to its outstanding time complexity
(i.e., OðNÞ). Then, the community structure clarity of each
layer can function as a weight by applying the modularity (Q):
1
Q ¼
2Ml
XN
i;j
Aij
di dj
2Ml
d Ci; Cj
(7)
where Aij represents the corresponding adjacency matrix ofnodes
i and j, N represents the number of nodes, and Ml denotes the
number ofedges in the lth layer. di denotes the degree ofnode i,
and Ci stands for node i belonging to community C. When Ci ¼
Cj, dðCi; CjÞ¼ 1, anditis0 otherwise. Q reflects the community
structure ofa network. The closer Q is to 1, the clearer the community
structure is.
During the optimization process, another vital component
is the genetic operation, which helps the algorithm escape
from local optima and further find better solutions. MOEACPI
is equipped with two genetic operators, namely, the
crossover and mutation strategies.
In recent decades, studies on genetic operators have produced
great achievements. This paper selects the uniform crossover strategy.
First, the algorithm generates a random binary mask with a
length ofN, and offspring are generated according to the binary
mask. Specifically, when the binary mask equals 0, the corresponding
offspring gene selects the value ofthe first parent; otherwise, it
selects the value ofthe second parent. Compared with other crossover
strategies, such as single-point crossover, uniform crossover
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