IEEE Computational Intelligence Magazine - August 2023 - 73

that can effectively encode macro- and
micro-populations are suggested. Finally,
the two proposed interaction strategies
between the macro- and micro-populations
are described in detail.
Algorithm 1. General framework of
MMCoMO.
Input:A: adjacency matrix of a
network; gen: number of
generations; pop: size of a
population; pc: probability of
crossover for the micropopulation;
pm: probability of
mutation for the macropopulation;
gap: interval
between interactions;
Output: PF: final non-dominated
solutions;
Phase-1: Initialization
1: Pmi get the initial
micro-population;
2: Pma get the initial
macro-population;
3: SM get the initial similarity matrix
based onA and the diffusion kernel
similarity measure;
Phase-2: Co-evolution
4: for t ¼ 1to gen do
5: P0
ma get the offspring of
macro-population via uniform
crossover and bitwise mutation
with probabilitypm;
6: Evaluate P0
7: P0
ma with KKM and RC;
mi get the offspring of micropopulation
via one-way crossover
with probabilitypc and neighborbased
mutation;
8: Evaluate P0
9:
10:
11:
12:
mi with KKM and RC;
if t %gap ¼¼ 0 then
Pmi GuidanceðPma; Pmi; P0
Pmi LocalSearchðPmiÞ;
Pma InfluenceðPmi; Pma; P0
13: else
14:
15:
mi; SMÞ;
ma; SMÞ;
Pmi EnvironmentSelectionðPmi [P0
Pma EnvironmentSelectionðPma [P0
16: end if
17: end for
Phase-3: Mergence
18: PF NonDominatedSortðPmi [PmaÞ;
miÞ;
maÞ;
co-evolution scheme introduced in
Section III-B,and Algorithm 1 presents
the general framework of MMCoMO.
The inputs ofthe algorithm are defined as
follows: A is the adjacency matrix of the
given network, gen is the number ofgenerations
ofthe evolutionary process, pop is
the size ofthe population (i.e., the number
of individuals in a population), pc and pm
are the probabilities of crossover for the
macro-population and mutation for the
micro-population, respectively, and gap is
the number of interval generations
between interactions. NonDominated Sort
refers to non-dominated sorting, which is
used to sort the individuals in the population
and determine the solutions in the
Pareto set. EnvironmentSelection refers to
environment selection, which is used to
select individuals for the next generation
of the population. These two functions
were adopted from [13].MMCoMO
consists ofthree phases: initialization (Lines
1-3), co-evolution (Lines 4-17) and mergence
(line 18). In the first phase, the
macro-micro populations are initialized.
Specifically, for the macro-population,
MMCoMO adopts a similar initialization
strategy as that suggested in [46],with
which halfofthe macro-population is initialized
with the candidate central nodes of
larger degree and the remaining halfis initialized
randomly. Then, the initial macropopulation
is obtained and denoted as Pma.
For the micro-population, a simple strategy
is adopted for each population individual;
one neighbor of each node vi is
randomly selected as the label of vi. The
initial micro-population is then obtained
and denoted as Pmi. In addition, based on
the adjacency matrix A, the initial similarity
matrix (denoted by SM)can be
obtained with a size ofn n, where each
element SMi;j is the diffusion kernel similarity
value [68] between nodes vi and vj.
In the second phase, based on the suggested
co-evolution scheme, the macroandmicro-populations
evolve individually
andinteractwitheachother.Inparticular,
at each generation, the offspring of the
macro-population (denoted as P0
ma),
A. Overall MMCoMO Procedure
MMCoMO is implemented based on the
proposed macro-micro population-based
which adopts the medoid-based representation
[46], is generated via uniform crossover
and bitwise mutation with
probability pm [46] (Line 5). The offspring
ofthe micro-population (denoted as P0
mi),
which adopts a vector-based representation
[11], is generated via one-way crossover
[33] with probability pc and
neighbor-based mutation (i.e., the node's
label is changed to that of a neighboring
node) with probability 1=n (n is the number
ofnodes in the network) (Line 7). The
two offspring populations are evaluated
using the KKM and RC (Lines 6 and 8).
Then, the interactions between the
macro-micro populations perform every
gap generations (Lines 10-12). The proposed
guiding strategy is adopted in Algorithm
2 via the elite individuals of the
macro-population to guide the micropopulation.
Subsequently, the local search
suggested in [38] is used for the non-dominated
solutions in the micro-population
to improve the modularity. Finally, Algorithm
3 employs the proposed influencing
strategy using elite individuals of the
micro-population to positively influence
the macro-population. This process is
repeated in the second phase until the
maximum generation gen is reached. In
the last phase, the final macro-population
Pma
and micro-population Pmi can be
obtained and merged into a single population.
Subsequently, the non-dominated
sorting method [13] is adopted to obtain
the optimal solutions (i.e., PF)onthe first
front. Section II of the supplementary
material presents the complexity analysis
ofthe proposed algorithm. The following
introduces the proposed macro-micro
population-based co-evolution scheme,
which is a key component of
MMCoMO.
B. Macro-Micro Population-Based
Co-Evolution Scheme
1)Main Idea
To obtain balance better between exploration
and exploitation in MOEA-based
community detection algorithms, a
macro-micro population-based co-evolution
scheme is suggested for the proposed
MMCoMO, and Figure 1 shows the
overall framework, in which the macropopulation
at the macro level prefers
exploration and the micro-population at
the micro level prefers exploitation. The
elite solutions of the macro-population
AUGUST 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 73
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