IEEE Computational Intelligence Magazine - August 2023 - 72

algorithm based on the framework of a
multi-objective evolutionary algorithm
with decomposition (MOEA/D) [42],for
which a problem-specificheuristicperturbation
operator was designed to improve
the convergence velocity and convergence
reliability ofthe algorithm. Experiments
verified the effectiveness of the
proposed community detection model
and designed heuristic operator. Recently,
a survey of community detection in [43]
indicated that some search strategies, such
as operators with domain knowledge, can
be exploited to further improve the detection
performance ofEA-based algorithms.
Based on the teaching-learning-based
optimization algorithm (TLBO) [44],
Chen et al. [19] proposed amulti-objective
discrete TLBO called MODTLBO/D for
complex network community detection.
In MODTLBO/D, two objectives based
on the negative ratio association (NRA)
and RC were adopted for optimization,
and the vector-based representation was
adopted for encoding each solution in the
population. Experiments on six real-world
networks demonstrated the effectiveness
of MODTLBO/D compared with the
existing EA-based community detection
algorithms (including single- and multiobjective
algorithms).
To solve the scalability of MOEAbased
methods on large-scale complex
networks, Zhang et al. [26] recently presented
a network reduction-based multiobjective
evolutionary algorithm called
RMOEA for large-scale complex networks
community detection, in which
the size of the networks was recursively
reduced as the evolution proceeds. In
RMOEA, the two objectives suggested
in [11] were adopted for optimization,
and each solution in the population was
encoded by the locus-based representation.
Experiments on synthetic and realworld
networks demonstrated the superiority
ofRMOEA compared with several
state-of-the-art community detection
algorithms for large-scale networks.
In addition to the aforementioned
MOEA-based algorithms focusing on
non-overlapping community detection
(each node in the network belonging to
one and only one community), some
interesting MOEA-based algorithms have
been proposed for overlapping community
detection (each node in the network
belongs to one ormore communities) [31],
[45], [46], [47], [48], [49], [50] or focused
on other types ofnetworks, such as signed
[51], [52],multilayer [53], [54], [55],
attributed [48], [56], [57], and dynamic
networks [58], [59], [60].
The empirical results of the aforementioned
algorithms have justified the
superiority of MOEAs for solving community
detection in complex networks
over single EAs and non-EAs. However,
to improve the performance, most existing
algorithms have employed different
search strategies for a single population,
which restricts the MOEAs in achieving
a better balance between exploration
and exploitation. Therefore, the macromicro
population-based co-evolutionary
multi-objective
algorithm called
MMCoMO is proposed, which can further
improve the quality of the detected
network partitions.
The proposed MMCoMO adopts the
same objective functions (KKM and RC)
as MODPSO [11] and RMOEA [26] to
model the community detection problem.
However, the proposed co-evolutionary
framework differs from the evolutionary
frameworks used in these algorithms. In
MMCoMO, the co-evolutionary framework
with macro-micro populations
combines exploration and exploitation to
achieve a good balance, and the micro
population adopts a local search strategy.
This is similar to somememetic algorithms
used for community detection [61],
[62]. The main difference between
MMCoMO and these memetic algorithms
is that, inmost memetic algorithms,
only one population is considered, and
designing the representation, operators,
and strategies is difficult in terms ofbalancing
the global evolution and local
refinement.
3)Co-Evolutionary Multi-Objective
Algorithms
To address large-scale problems
and achieve better performance, co-evolutionary
algorithms have been widely
adopted for solving many multi-objective
optimization problems [63]. Co-evolutionary
multi-objective algorithms can be
72 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2023
divided into cooperative and competitive
approaches. In cooperative approaches,
the fitnessofanindividualiscomputed by
collaborating with individuals of other
species [64]. For example, Gong et al. [65]
proposed a cooperative co-evolutionary
algorithm for hyperspectral sparse unmixing
problems, in which the decision variable
vector was divided into subsets and
assigned to different subpopulations
for optimization. Typically, cooperative
approaches adopt the divide-and-conquer
strategy, and subpopulations are designed
to solve the corresponding subproblems.
In competitive approaches, the fitness of
an individual is determined by encounters
with individuals of other species [66].In
this type of approach, individuals of subpopulations
typically compete for survival.
For example, Silva et al. [67] suggested a
predator-prey biogeography-based optimization(PPBBO)thatmodelsthe
predators
based on the individuals with the
worst values ofthe objective function, and
in whichtheypreyonindividuals with
the worst values.
Unlike these co-evolutionary multiobjective
algorithms, this study designed a
macro-micro population-based co-evolution
scheme. The proposed MMCoMO
employs two populations with different
representations that focus on different levelstobalance
theglobaland localsearches
(i.e., exploration and exploitation). The
two populations improve each other during
the co-evolutionary process, and no
competitive interaction exists between
them. Unlike CoCoMi [36],the two
populations in the proposed MMCoMO
were designed for the entire network and
have different representations. The following
section details the proposed
algorithm.
III. Proposed MMCoMO Algorithm
This section describes the proposed
MMCoMO algorithm for community
detection. First, the general framework of
the proposed algorithm is provided,
including the initialization, co-evolutionary,
and mergence phases. The proposed
macro-micro population-based co-evolution
scheme is then presented, which is a
crucial component ofMMCoMO. Subsequently,
two individual representations
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