IEEE Computational Intelligence Magazine - May 2023 - 54

First, MOEA-CPI extracts the graph-level information based
on Node2vec, which modifies the random walk procedure of
DeepWalk with a probability that can control the tendency ofthe
walk process (i.e., BFS and DFS). Suppose that the numbers of
nodes and edges in a network are N and M, respectively. Therefore,
the time complexity ofsampling with the random walk process
is OðMÞþ OðN T LÞ,where T is the walk length and
L represents the number of walks per node. For the word2vec
process, the time complexity is OðI NLTK logNÞ,where I is
the number of iterations and K is the window size. L, T,and K
are usually much smaller than N and M, so the time complexity
can be simplified as OðMÞþOðNÞþ OðI N logNÞ.The
node-level information is based on theJaccard similarity measure.
When calculating the Jaccard similarity between nodes, the first
and second neighbors need to be considered so that the time complexity
of the Jaccard similarity calculation is OðN logNÞ.
Regarding the multi-objective evolutionary optimization process,
this paper applies the NSGA-II framework, whose complexity
equals OðIjFj N2Þ,where F is the number ofobjective functions.
Therefore, according to the graph-level and node-level
information extraction process and the NSGA-II framework, a
total complexity of the proposed method is equal to OðMÞþ
OðNÞþ OðI N logNÞþOðN logNÞþ OðIjFj N2Þ.
For the sake ofconvenience, this complexity can be simplified to
an order ofmagnitude representation: OðN2Þ.
IV. Experiments
This section presents the experimental settings and experimental
analysis. More specifically, Section IV-A and Section IV-B introduce
the utilized datasets, baselines, and parameter settings. The
experimental results are shown in Section IV-C.Finally, an ablation
analysis, a parameter analysis, and a statistical analysis are introduced
in Section IV-D.
A. Datasets
This paper evaluates the accuracy and robustness of the proposed
MOEA-CPI algorithm on real-world networks as well
as synthetic networks. Brief descriptions of the attribute information
ofthese networks are shown in Table I.
1) Real-World Datasets
In this paper, four classic real-world datasets with different
scales are selected to evaluate the performance ofthe proposed
algorithm. This section introduces the information about the
real-world datasets in detail.
Social Network Dataset (SND): The SND is constructed
based on 71 employees of a law firm. The network contains
three layers, namely, cowork, friendship, and advice [43].
Word Trade Networks (WTN): WTN is constructed
according to different trade relationships, where its 183 nodes
represent different countries [44]. The network originally contained
339 layers, which represented different goods. However,
Gligorijevic et al. considered some layers to be too
sparse, so they preprocessed the network and retained 14 layers
to relieve its sparsity [4].
54 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2023
TABLE I The attributes of the utilized datasets.
NETWORK LAYERS NODES EDGES GROUND TRUTH
SND
3
WTN
CoRA
CiteSeer
Syn1
Syn2
Syn3
14
2
2
3
3
3
71
183
1662
3312
100
5000
1659
3334
9155
24790
1212
477257
10000 1919658
3
10
3
3
4
9
16
Bibliographic Dataset (CoRA): The CoRA network is constructed
from three categories of papers: those based on genetic
algorithms, neural networks, and probabilistic methods. The
CoRAnetwork consists of1662 machine learning papers [4].
Bibliographic Dataset (Citeseer): This network is constructed
according to the connections of 3312 papers, and
these papers can be divided into three categories, all of which
are regarded as ground truths. Citeseer contains two network
layers, namely, a citation connection layer and a layer representing
the similarity between papers [4].
2) Synthetic Datasets
To further evaluate the performance of the proposed algorithm,
this paper selects the mLFR benchmark to generate synthetic networks
with various scales and structures [45]. the mLFR benchmark
generates networks by controlling its mixing parameter (Mp)
and its degree change probability (Dc). Mp and Dc control the
number of connections between the communities in every layer
and the degree differences among nodes in different network
layers, respectively. Specifically, increases In Mp and Dc results in
increased multilayer network complexity [4].
To prove the performance ofMOEA-CPI more comprehensively,
this paper also generates some large-scale synthetic networks.
Syn1 consists ofthree layers, where each network layer has
100 nodes with an average degree of8. The Mp and Dc ofSyn1
are 0.4 and 0.4, respectively. Syn2 is composed of5000 nodes with
an average degree of128. In addition, the Mp and Dc ofSyn2 are
both 0.4. Furthermore, Syn3, which has more nodes and connections,
is constructed by setting N=10000 with an average degree
of256. The Mp and Dc ofSyn3 are 0.5 and 0.2, respectively.
Furthermore, 30 synthetic networks are generated with
different values ofMp, Dc and the number layers to verify the
robustness of MOEA-CPI. The former 16 networks have
three layers and 128 nodes with Mp ¼f0:4; 0:5g and
Dc ¼f0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8g. The latter 14 networks
still possess 128 nodes. To verify the effect of the number
of layers on the performance of MOEA-CPI, the setting
of the networks are Mp ¼ 0:5, Dc ¼f0:4; 0:6g, and
Layers ¼f3; 5; 7; 9; 11; 13; 15g.
In addition to datasets, metrics also play important roles in
experiments. In this paper, two widely used metrics are
adopted to assess the accuracy ofthe obtained community partition,
namely, the NMI [46] and ARI [46], which have been
introduced in Section II-A.
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