IEEE Computational Intelligence Magazine - May 2023 - 57

TABLE III The ablation analysis results of the proposed
MOEA-CPI algorithm. the Syn dataset is constructed by the
mLFR benchmark with Mp ¼ 0:5 and Dc ¼ 0:2.
TABLE IV The rankings of the algorithms based on Table II.
SND WTN CORA CiteSeer SYN1 SYN2 SYN3 RANKING ðRÞ
NETWORK
Basic
Basic+weighted
Basic+weighted+Jaccad
MOEA-CPI
SND
0.487
0.551
0.766
0.808
SYN
0.237
0.285
0.297
0.544
MOEA-CPI 1 1 2
SS-MOML 2 2 1
COMCLUS 7 6 7
S2-jNMF 6 7 3
CSNMF 3.5 3 5
SC-ML
3.5 4 6
MIMOSA 8 8 8
GMC 5 5 4
1 1.5 2.5 2 1.5714
3 5 5 2.8571
2
5 1.5 6 7 5.6429
7 8 3 5.7143
8 2.5 6 4.4286
6 2.5 1 3.8571
5 7 8 7.4286
4 2.5 4
4.5
6
3
4
8
7
FIGURE 8. Parameter analysis of Cos andJac. The datasets of (a) and (b)
are Syn (a synthetic network) and the SND. The figure shows that the 2
kinds of prior information actually play guiding roles and the MOEA-CPI
performs better when Cos ¼½0:1; 0:2 andJac ¼½0:1; 0:3.
adding the weighting strategy, the performance is improved
because the upgraded method can dynamically adjust the
weight ofeach layer, but its NMI value is still not competitive
with those of the state-of-the-art methods. 2) The performance
of the Basic+weighted+Jaccard method is improved,
especially on the SND, because the Jaccard similarity between
the nodes of the SND is clear and the node-level information
can guide the mutation strategy to distribute similar nodes into
the same community. In addition, the node-level information
does not perform well enough on the Syn network because
the similarity structure is disordered. 3) Finally, graph-level
information is applied to the algorithm, and the performance
is further improved. This is because graph-level information
can be used to reconstruct the topological structure and generate
high-quality initialization from the global perspective.
Under the impacts of such factors, our proposed MOEA-CPI
algorithm can maintain high performance even in situations of
insufficient iteration and complicated network structures.
In summary, the proposed algorithm achieves substantial
NMI increases on both SND networks and artificial networks
after adding two formulations ofprior information. The experimental
results demonstrate that the MOEA-CPI method proposed
in this paper can effectively use the consensus prior
information to guide the MOEA to obtain more accurate community
structures for multilayer networks.
2) Parameter Analysis
This section demonstrates the analysis of the Cos and Jac
parameters via evaluating the NMIs produced by the algorithm
with different parameter settings on two networks (as shown in
Figure 8).
FIGURE 9. The result of the Bonferroni-Dunn test. The red line indicates
the ranking level when a ¼ 0:1.
To further evaluate the effect ofthe prior information, this section
conducts an analysis experiment aimed at the effects ofthe Cos
and Jac parameters. A real-world social network and a synthetic
network are selected for this experiment. Figure 8 shows that the
NMI is at its lowest value when Cos ¼ 0andJac ¼ 0, indicating
that no prior information is applied to the algorithm. With the
increase In Cos andJac, the NMI improves since the graph-level
and node-level information affect the algorithm from the global
and local perspectives. On the Syn network (as shown in Figure 8
(a)), MOEA-CPI performs the best when Cos ¼½0:1; 0:2 and
Jac ¼½0:1; 0:3. However, as the values of these two parameters
continue to increase, the NMI drops to a low value. This is
because as the parameter representing the reservation ratio of the
prior information increases, the accuracy ofthe prior information
decreases, which leads to performance degradation. Figure 8(b)
demonstrates the parameter analysis results of the SND network,
showing that MOEA-CPI can achieve better accuracy when
Cos ¼½0:1; 0:3. Generally, the proposed MOEA-CPI method
performs better in terms ofaccuracy and stability when Cos andJac
range from 0.1 to 0.2 and 0.1 to 0.3, respectively, on both realworld
networks and synthetic networks.
3) Statistical Analysis
In this section, to further prove the superiority ofMOEA-CPI,
some statistical analyses are conducted [49], [50]. Table IV
MAY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 57
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