IEEE Computational Intelligence Magazine - May 2023 - 51

B. Consensus Prior Information Extraction
As stated before, semi-supervised learning has an incomparable
advantage in terms of accuracy and robustness, but this method
relies on dataset labels. Large numbers of unlabeled datasets are
available in the real world since dataset labeling is too expensive.
According to the definition ofcommunity structure, highly similar
nodes should belong to the same community. Therefore, this
paper treats highly similar nodes as they have the same label to construct
the consensus prior information. More specifically, Node2vec
and the density-based aggregation strategy are adopted to
determine node similarity based on the input graph structure. In
addition, theJaccard coefficient is utilized to obtain the node similarity
based on the local node information.
Graph-level information is constructed based on network
embedding methods, and among them, Node2vec is one of the
most classic approaches. Node2vec modifies the random walk
process by applying the ideas of depth-first sampling (DFS) and
breadth-first sampling (BFS) to improve the flexibility ofthe walk
process and further improve the performance ofthe algorithm [35].
In view of its advantages, this paper selects Node2vec as the first
method for extracting the prior information.
In previous works, the representation of each layer was
obtained by applying Node2vec for each layer ofa network. As
mentioned above, a multilayer network comprises several layers
that contain different structures. To find a consensus low-dimensional
representation, a straightforward solution is to merge the
representation vectors of all layers by calculating their average.
However, in a multilayer network, each layer contains a different
amount ofinformation. Ifthe average-based aggregationmethod
is adopted, layers with different amounts of information will be
distributed with the same weight, resulting in information loss.
Therefore, to solve this problem, this paper proposes a densitybased
aggregation method that can quantize the information
content ofeach layer to dynamically set their weights. The proposed
strategy computes the weights by calculating the internal
density. However, a higher internal density tends to denote a
lower information quantity. Thus, calculating the weights
requires more effort. To minimize this inconvenience, the density
is redefined in (4).
fD ¼
Ml
NðN 1Þ=2
(4)
The internal density refers to the sparsity ofa cluster (i.e., community)
[36]. As described in (4), density is positively associated
with the number of edges and negatively associated with
the number of nodes. In a case where the number of nodes is
the same, the density value reflects the information quantity.
Therefore, to quantify the amount of information, each network
layer is regarded as a cluster to make the internal density
reflect the sparsity of the network layer. Ml is the number of
connections in the lth layer, and N represents the number of
nodes in a multilayer network.
The node-level information is obtained based on the Jaccard
similarity coefficient, which is an effective node similarity
measurement and is defined in (5). TheJaccard similarity can efficiently
measure the correspondence between the common features
of individuals [37]. It can be applied to CD problems and
thus has gainedmuch popularity among researchers [38], [39].
Jvi;vj
¼
NviðÞ \Nvj
NviðÞ [Nvj
(5)
where vi stands for the ith node and NðviÞ stands for the neighbor
of vi. In this paper, the neighbors of vi refer to all the
neighbors among the various layers ofvi. TheJaccard similarity
evaluates the similarity between nodes according to the number
ofcommon neighbors.
The consensus low-dimensional representation approach is
used to construct the prior layer and generate a high-quality
initial population. For the prior layer, MOEA-CPI first calculates
the cosine similarity between the nodes according to the
low-dimensional representation and then retains several of the
node pairs with the highest similarity to construct virtual connections
using a predefined threshold Cos, where Cos represents
the reservation ratio of the graph-level information and
the value range of Cos is [0,1]. The prior layer is constructed
by the virtual connections. During the initialization process,
the K-means method is applied to the consensus representation
to generate the initial clusters, and then the clustering process is
converted to the locus-based encoding scheme to generate the
initialization. Guided by the prior layer and initial population,
the algorithm is able to assign similar nodes to common communities.
Node-level information, which is described in
Section III-D, is used to guide the optimization process.
C. Encoding and Initialization Schemes
The encoding and decoding schemes play key roles throughout
the algorithm since a good encoding scheme can reduce the
required computational cost. The most widely used encoding
methods are the label-based and locus-based methods, as illustrated
in Figure 3. The label-based method encodes the number ofcommunities
as the encoding scheme, and some studies apply the
label-based method as the encoding scheme because this kind of
method is easy to use. However, this method has the drawback of
causing redundancy [30]. For example, the sequences
f1; 1; 2; 2; 3; 3; 3g and f3; 3; 2; 2; 1; 1; 1g are the same solution.
Therefore, this paper selects the locus-based method as the encoding
strategy. The locus-based encoding approach encodes [14],
[15] a chromosome (i.e., solution) as a sequence Chr ¼
fs1; s2; .. .; sNg with a length of N, where Chr represents the
chromosome, sN denotes the code ofthe Nth node, and N is the
number ofnodes.
Another important process is the initialization process, as
some studies have estimated that the initial population of any
optimization-based method can significantly affect its performance
[40], and a better initialization improves the quality of
the solutions acquired by evolutionary computation techniques
[31], [32]. As stated above, MOEA-CPI obtains graphlevel
information and node-level information. In simple terms,
MAY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 51
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