IEEE Computational Intelligence Magazine - May 2023 - 58

TABLE V The p-values yielded by comparison algorithms on
different datasets. " - " represents a value is less than
0.00003. The results demonstrate that our proposed MOEACPI
method performs better than the other comparison
algorithms.
TABLE VI The time cost (unit: seconds) comparison. Among
them, " - " represents a method that is unable to detect the
community structure within the specified time (i.e., 1e4
seconds).
MOEA-CPI vs. zUdp
MIMOSA
4.47
S2-jNMF
COMCLUS
GMC
SC-ML
CSNMF
SS-MOML
-
Bdp
-
Hlp
-
Hbp
-
3.16 0.0016 0.0112 0.0096 0.0095
3.11 0.0019 0.0133 0.0096 0.0095
0.025 0.175
2.24
2.18 0.0292 0.2044
0.98 0.3222
0.1
0.1
0.0876
0.0876
1.75 0.0802 0.5614 0.1604 0.1604
1
0.3222 0.3222
shows the rankings of eight different algorithms (i.e., MOEACPI,
SS-MOML, COMCLUS, S2-jNMF, CSNMF, SC-ML,
MIMOSA and GMC) presented in Table II.
This section first performs the Friedman test (x2
F ¼ 26:8936. As the chi-square table
(10), and the results are listed in reverse order. Moreover, the
unadjusted p, Bonferroni-Dunn p, Holm p, and Hochberg p
are named Udp, Bdp, Hlp, and Hbp, respectively. Among them,
Udp is obtained by searching the standard distribution table,
while Bdp, Hlp, and Hbp can be calculated based on (11).
According to the comparison mentioned above, the superiority
ofour proposed MOEA-CPI approach is strongly proven.
F) according
to (8),where A denotes the number of algorithms and D represents
the number of datasets. According to the rankings
shown in Table IV, x2
shows, x2
0:05 ¼ 14:067 when the number of degrees of freedom
is A 1 ¼ 8. Since the acquired value of 26.8936 far
exceeds 14.067, various results produced by different algorithms
possess confidence values of95%, as shown in Table II.
x2
12D
F ¼
AðAþ 1Þ
" #
X
j
R2
j
AðAþ 1Þ2
4
(8)
In addition, the Bonferroni-Dunn test is conducted to further
prove the differences among the tested algorithms. The
critical difference (named CDa) can be acquired by (9).
CDa ¼ qa
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
AðAþ 1Þ
6D
(9)
where qa can be acquired by searching the Z table according to a
desired confidence level a. In this paper, the P value is
AðA 1Þ=2=28, so q0:1=2.913. Then, CD0:1=3.814 based on
the q value and (9). The ranking difference between two algorithms
that is larger thanCDa represents a severe difference performance
between the two algorithms. Therefore, as shown in
Figure 9, the results prove that the proposedMOEA-CPI method
performs better than COMCLUS, S2-jNMF, and MIMOSA
with 90% confidence. Finally, the rankings ofthe other three algorithms
(i.e., CSNMF, SC-ML, and GMC) are close, which also
proves that our proposed algorithm has superior performance over
that ofthese comparison algorithms.
Finally, two powerful analysis methods (i.e., Holm's and
Hochberg's methods) are applied to further compare the performance
between each pair of algorithms. More specifically,
aiming at algorithm i and algorithm j, the statistic between
these two algorithms (i.e., the z value) can be calculated by
58 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2023
4) Time Complexity Experiment
This section further analyzes the computational costs of different
algorithms (i.e., MOEA-CPI, MOEA-MultiNet, S2jNMF,
and SC-ML) running on the same computer. The
results are shown in Table VI.
As showninTable VI, the temporal cost of MOEA-CPI is
lower than that ofMOEA-MultiNet but higher than those ofS2jNMF
and SC-ML. The reason for this finding is that EA-based
algorithms have generally higher time complexity levels than
NMF-based methods and spectral clustering methods. Furthermore,
MOEA-CPI needs to extract graph-level and node-level
information as guidance to accurately detect community structures,
and this step costs extra time. In different application scenarios, specific
algorithms can be chosen based on the type of encountered
tasks, e.g., based on the accuracy priority or the time priority.
V. Conclusion
This paper proposes a novel MOEA-CPI approach by combining
a weighting strategy and prior information guidance (i.e.,
z ¼ q
8
<>
Ri Rjffiffiffiffiffiffiffiffiffiffiffiffi
AðAþ1Þ
6D
Bdp ¼ min vi; 1fg; vi ¼ðA 1ÞUdp
Hlp ¼ min vi; 1fg; vi ¼ max ðAjÞUdpj
:>
Hbp ¼ min ðAjÞUdpi
; ðA 1Þj i
The statistical results ofthe z, Udp, Bdp, Hlp and Hbp values are
described in Table V.Based on Hlp and Hbp, the results show that
the proposed MOEA-CPI method performs better in accuracy
and robustness than the MIMOSA, S2-jNMF, COMCLUS,
GMC and CSNMF (90% confidence) methods, and the SC-ML
method (80% confidence). Moreover, the results ofSS-MOML
are also close to 70% confidence. Therefore, our proposedmethod
has superior performance in terms ofsolving theMCD problem.
(10Þ
SND
Syn1
WTN
Citeseer
MOEA-CPI MOEA-MultiNet S2-jNMF SC-ML
0.91
0.46
2.81
43.02
33.46
632.27
5707.54
37.83
587.44
-
-
111.82
0.43
0.66
1.87
83.12
no
; 1j i
(11)
IEEE Computational Intelligence Magazine - May 2023

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