Improved Random Key Cuckoo Search Optimization Algorithm for

Community Detection in Social Networks

Randa Boukabene, Fatima Benbouzid-Si Tayeb and Narimene Dakiche

Laboratoire des Methodes de Conception de Syst

emes (LMCS),

Ecole Nationale Sup

erieure d’Informatique (ESI),

BP 68M - 16270 Oued Smar, Alger, Algeria

Keywords:

Social Networks, Community Detection, Cuckoo Search Optimization, Modularity, Random Key.

Abstract:

Social network analysis is a prominent and thriving research ﬁeld, with community detection being a partic-

ularly active area of study. In this study, we propose a cuckoo search-based approach for identifying the best

network partitions by maximizing the modularity function. The proposed algorithm combines wisely the con-

tinuous nature of the standard cuckoo search algorithm with the discrete nature of the community detection

problem to achieve the best results. Firstly, the algorithm incorporates the random key representation, which

operates in a continuous space. This representation enables the algorithm to perform global and local walks,

enabling both exploration and exploitation within the search space. Secondly, the algorithm utilizes the locus-

based representation to handle the discrete aspect of the community detection problem. Experiments on both

synthetic and real-world networks demonstrate the effectiveness and efﬁciency of our proposed algorithm.

1 INTRODUCTION

Recently, social networks, initially introduced by

Wasserman and Faust (1994), have gained consider-

able popularity in research and development, partly

due to more and more social media platforms such as

Facebook, Instagram, and Twitter, being built online

and the development of Web 2.0 applications. Social

networks model social relationships by graph struc-

tures using vertices and edges. Vertices model indi-

vidual social actors in a network, while edges model

relationships between social actors.

Social networks have the characteristic property to

exhibit a community structure, and highlighting the

hidden structures of many real-world networks has

drawn a wide range of researchers from many areas.

As a result of the huge interest in the topic, a large

number of surveys on community detection have been

published (Souravlas et al., 2021; Khan and Niazi,

2017; Cai et al., 2016). Communities are formally de-

ﬁned as sub-graphs in which nodes are more strongly

connected to each other than to the rest of the network

(Girvan and Newman, 2002).

Various approaches have been proposed to ad-

dress the community detection problem in social net-

works, as highlighted in the recent survey conducted

by Bara’a et al. (2021) wherein researchers formu-

lated the issue as a combinatorial optimization prob-

lem. This formulation enables the application of var-

ious optimization methods, including heuristics and

metaheuristics, and functions for measuring the qual-

ity of partitioning in communities of a network have

been proposed. Among them, modularity emerged

as one of the most well-known and widely employed

measures in this ﬁeld (Newman and Girvan, 2004).

Nowadays, nature-inspired metaheuristic algo-

rithms have gained signiﬁcant popularity in the op-

timization ﬁeld. The literature has been rich in regard

to different optimization strategies for detecting com-

munities by maximizing the aforementioned partition

quality indicators (Bedi and Sharma, 2016; Su et al.,

2022). In particular, metaheuristics, inspired by bio-

logical processes, have proven to be effective methods

for tackling community detection problems in com-

plex network instances (Osaba et al., 2020), with Evo-

lutionary Algorithms as the most resorted optimiza-

tion techniques in the last years (Pizzuti, 2017).

Cuckoo search algorithm (CSA) (Yang and Deb,

2009) is one of the recent nature-inspired algorithms

used extensively to solve optimization problems in

different ﬁelds of engineering (Yang and Deb, 2010).

It is very effective in solving global optimization be-

cause it is able to maintain a balance between local

and global random walks. Nonetheless, the appli-

Boukabene, R., Tayeb, F. and Dakiche, N.

Improved Random Key Cuckoo Search Optimization Algorithm for Community Detection in Social Networks.

DOI: 10.5220/0012202000003584

In Proceedings of the 19th International Conference on Web Information Systems and Technologies (WEBIST 2023), pages 113-120

ISBN: 978-989-758-672-9; ISSN: 2184-3252

113

cation of the CSA to solve the community detection

problem is very limited. Zhou et al. (2016) proposed

a multi-objective discrete CSA with local search tech-

niques. Their algorithm aimed at minimizing the ra-

tio association (RA) and the ratio cut (RC) to enhance

community quality. Shishavan and Gharehchopogh

(2022) used a hybrid approach by combining genetic

operators with the CSA to overcome local optima and

improve its applicability to the community detection

problem.

Using the cuckoo search algorithm directly for

community detection can potentially diminish its ef-

fectiveness because the algorithm was primarily de-

signed for continuous problems. This presents a chal-

lenge because community detection problems inher-

ently possess a discrete nature. In light of this, the

aim of this paper is to develop and assess the perfor-

mance of Cuckoo search algorithm to discover com-

munities in social networks. Firstly, the proposed al-

gorithm incorporates the random key representation,

which operates in a continuous space. This repre-

sentation allows performing global and local walks,

enabling both exploration and exploitation within the

search space. Secondly, the algorithm utilizes the

locus-based representation to meet the requirements

of the discrete nature of the network community de-

tection problem.

The structure of the paper is as follows: Section 2

gives the community detection problem description.

Section 3 describes the proposed cuckoo search algo-

rithm. The results of experiments conducted on syn-

thetic and real networks are presented in Section 4.

Finally, Section 5 concludes the paper and discusses

possible future work.

2 PROBLEM FORMULATION

A social network can be modeled by a graph, de-

noted G = (V, E), which includes N =

nodes and

m =

edges. V represents the set of nodes and E

represents the set of edges connecting two nodes in

G. A is an N × N adjacency matrix of the network,

where A

i j

represents the element at the i

row and j

column of A. If there is an edge connecting nodes i

and j, A

i j

= 1; otherwise A

i j

= 0.

The problem of community detection requires

the partitioning of the network into sub-clusters or

communities. This partitioning is denoted as C =

,...C

}, where each element C

l = 1,2,...,k is a

proper subset of V , and k is the total number of com-

munities. Moreover, for any two communities C

and

∈ C and i ̸= j, V

∩V

= φ. The quality of the com-

munities obtained is often measured by the so-called

modularity function Q (Newman and Girvan, 2004).

Q is a value between -1 and 1 that measures the den-

sity of the links within the communities compared to

the links between the communities; it can also be de-

ﬁned as the difference between the edges of the net-

work connecting vertices of the same type and the ex-

pected value of the same quantity in a network with

the same community divisions but random links be-

tween vertices. It is computed according to equation

Q =

∑

i j

−

]δ(c

) (1)

Where A

i j

represents the adjacency matrix, k

)

is the degree of node i( j), c

) is the community of

node i( j), m is the sum of all the links in the graph. δ

is the Kronecker delta function δ(x, y) = 1 if x = y, 0

otherwise.

3 PROPOSED CUCKOO SEARCH

BASED SOLVING APPROACH

Cuckoo search algorithm (CSA) (Yang and Deb,

2009) is a metaheuristic optimization algorithm, in-

spired by the brood parasitism behavior of some of

a bird family called cuckoo (Yildiz, 2013). Cuckoo

birds lay their eggs in another nest, called the host

nest where they imitate the color and patterns of the

host eggs to prevent them from being discovered. if

the host bird discovers them, the cuckoo eggs can be

thrown or the host bird leaves its nest. To make the

algorithm suitable for solving optimization problems,

the eggs of the nest are associated with a set of can-

didate solutions for a given optimization problem. In

particular, CSA is based on three idealized rules:

• Each cuckoo lays one egg at a time, and deposits

it in a randomly selected nest.

• The best nests containing high-quality eggs will

be passed on to subsequent generations.

• The number of available host nests is ﬁxed, and

the egg laid by a cuckoo is discovered by the host

bird with probability Pa. In this case, the host bird

can either discard the egg or simply abandon the

nest and build a brand-new one.

This algorithm follows a simple approach where

each nest is associated with a single egg, and each

cuckoo can lay only one egg (thus representing one

solution). We can safely make no difference between

an egg, a nest, or a cuckoo. The aim is to use the new

and potentially better solution (cuckoo egg) to replace

existing solutions in the nests.

WEBIST 2023 - 19th International Conference on Web Information Systems and Technologies

114

In the proposed algorithm, hereinafter RKCSA for

random key CSA, the nests represent the current so-

lutions or candidate solutions represented with a new

structure combining both the locus representation and

a random key representation. The eggs or nests corre-

spond to the new solutions generated by both global

and local walks. The algorithm evaluates these new

solutions using the modularity function based on the

locus representation and replaces the existing solu-

tions in the nests with the improved ones. By con-

tinuously generating and evaluating new solutions for

the locus vector, RKCSA aims to enhance the qual-

ity of the solutions over time. The replacement pro-

cess ensures that only better solutions are retained in

the nests, leading to the continual improvement of

the overall solution quality. It is worth noting that

the random key representation, depicting the continu-

ous nature of the algorithm, also plays a crucial role

in guiding the search and exploration process. How-

ever, the replacement of solutions in the nests specif-

ically focuses on the locus representation to improve

the community detection results. By combining the

exploration capabilities of the random key represen-

tation and the reﬁnement process of the locus repre-

sentation, the proposed algorithm aims to achieve bet-

ter performance in optimizing the modularity function

and identifying the best network partitions. The pop-

ulation of nests is subjected to repeated cycles until

the stop criterion is satisﬁed.

Next, we will delve into a comprehensive explo-

ration of the proposed algorithm, providing a detailed

explanation of its step-by-step process in algorithm 1.

3.1 Solution Encoding and Decoding

The implementation of RKCSA at hand begins with

egg encoding to generate a random initial population.

A candidate solution X, i.e. an egg, (Figure 1a) is

coded by a two-ﬁeld structure where:

• The ﬁrst ﬁeld is the locus vector (Figure 1b)

that uses the locus-based representation (Pizzuti,

2008). Each individual in the population consists

of N genes {g

,....,g

} equivalent to the number

of nodes in the network. Each gene g

in an in-

dividual is assigned a value v, which represents

one of the adjacent nodes of node u. In this rep-

resentation, it is necessary to perform a decoding

step to identify all the components of the graph,

so that the nodes participating in the same com-

ponent are assigned to the same community (Piz-

zuti, 2008). This decoding step can be performed

in linear time using the method described in (Cor-

men et al., 2022).

• The second ﬁeld is the key vector (Figure 1c) that

takes a random value between 0 and 1 associated

with each node in the network. This random key

representation enables the algorithm to operate in

a continuous space, facilitating smooth transitions

between solutions and allowing for efﬁcient ex-

ploration of the search landscape. This contin-

uous aspect is crucial in navigating the complex

and high-dimensional solution space of commu-

nity detection problems.

To generate the initial solutions, we need to initial-

ize both the locus-based vector and the random key

vector. For the locus vector initialization, we assign

to a node one of its neighbors. On the other hand, to

initialize the key vector, we assign a random number

between 0 and 1 to each node in the network. Ac-

cording to this representation, prior knowledge of the

number of communities is not needed.

(a) Graph partitions.

(b) Locus vector.

Figure 1: Example of a solution representation.

3.2 Global Walk

The ﬁrst component of the cuckoo search algorithm

known as the L

evy ﬂights or global walk generates

new solutions in proximity to the best solution found

so far (Reda et al., 2022). In the proposed algorithm,

we need to create both a new key vector and a new lo-

cus vector to generate a new cuckoo. The generation

of a new key vector is done using the L

evy ﬂights’

distribution according to equation 2.

key

(t + 1) = x

key

(t) + αs(x

key

(t) − x

best

key

(t)) (2)

Where x

key

(t), x

key

(t +1) and x

best

key

are the current,

the new, and the best key solution, respectively. s is

the step size while α is the scale of the step size.

Improved Random Key Cuckoo Search Optimization Algorithm for Community Detection in Social Networks

115

In Mantegna’s algorithm (Mantegna, 1994), the

step size s can be calculated using two Gaussian dis-

tributions u and v via the transformation of equation

s =

|v|

(3)

With

u ∼ N(0, var(u)) et v ∼ N(0, var(v)) (4)

And

var(u) = [

Γ(1 + β)

βΓ((1 + β)/2)

sin(πβ/2)

(β−1)/2

]

1/β

var(v) = 1

(5)

Where u and v are random numbers from a

normal distribution, N(Γ) is the normal(gamma)

distribution. var(u(v))is the variance of u(v) is the

gamma distribution, and β = 1.5.

To generate a new locus vector as shown in ﬁgure

2, we perform the following steps:

1. Calculate the difference between the old and new

key vectors |x

key

(t + 1) − x

key

(t)|.

2. If the difference is greater than the acceptance rate

key

(t + 1) − x

key

(t)| > δ, we reinitialize the cor-

responding node with the neighbor of the best so-

lution so far x

best

locus

(t). Otherwise, the node’s value

remains the same as the previous iteration.

3. The newly generated solution is evaluated using

the modularity measure (Eq.1) on the locus vector,

and if it has greater modularity than the current

solution, then it replaces the current solution for

this nest.

3.3 Local Walk

The second component of the cuckoo search algo-

rithm is the abandon operator. Its purpose is to gen-

erate solutions that are far from the best solution ob-

tained. This is done to prevent the search from getting

trapped in local optima and to encourage exploration

of different regions of the search space. By introduc-

ing diversity into the search process, the algorithm in-

creases the chances of ﬁnding the global optimum or

better solutions (Reda et al., 2022).

In the abandon operator, the host bird has a proba-

bility Pa of discovering foreign eggs in its nest. If for-

eign eggs are found, the host bird will abandon them

and replace them with new eggs. In RKCSA, this pro-

cess is applied to the key vector, where foreign solu-

tions are replaced with new solutions generated using

equation 6.

key

(t + 1) = x

key

(t) + S ∗ K (6)

With

K =



1 i f rand > Pa

0 Otherwise

(7)

And

S = (x

rand perm(n)

key

(t) − x

rand perm(n)

key

(t)) (8)

Where rand is a random number within [0,1], Pa

is the discovery probability and rand perm(n) is the

permutation function that chooses a random number

in the range [0,n].

After generating a new key vector, the locus vector

needs to be updated by following these steps:

1. Calculate the difference between the old and new

key vectors |x

key

(t + 1) − x

key

(t + 1)|;

2. If the difference is greater than the acceptance rate

(i.e, |x

key

(t + 1) − x

key

(t + 1)| > δ), then the corre-

sponding node in the locus vector is reinitialized

by one of its neighbors, unlike the global walk,

whose value is reinitialized by the neighbor of the

best solution. Otherwise, the node’s value remains

the same as in the previous step;

3. The newly generated solution is evaluated using

the modularity measure (Eq.1) on the locus vector,

and if it has greater modularity than the current

solution, then it replaces the current solution for

this nest.

4 EXPERIMENTAL RESULTS

AND DISCUSSION

To assess the efﬁciency and performance of our pro-

posed algorithm, RKCSA, we conducted a compre-

hensive set of experiments on both synthetic and real-

world networks using a personal computer running

Windows 10 Enterprise, equipped with 8 GB of RAM

and powered by an Intel(R) Core(TM) i7-3537u CPU

with a clock speed of 2.50 GHz.

For the synthetic networks, we used the well-

known LFR benchmark (Lancichinetti et al., 2008),

which allows us to generate networks with various pa-

rameters. In our experiments, we focused on varying

the mixing parameter µ from 0 to 0.8, with an interval

of 0.05. This deliberate variation was intended to cre-

ate a scenario where the boundaries between commu-

nities became increasingly blurred, thereby making

the task of community detection signiﬁcantly more

WEBIST 2023 - 19th International Conference on Web Information Systems and Technologies

116

Figure 2: An example of the locus vector generation.

challenging. Then, we created two groups of net-

works with different community sizes by specifying

the minimum and maximum number of communities.

For real-world networks, we also employed a di-

verse set of 11 real-world networks from various do-

mains with diverse sizes and characteristics. Details

are presented in Table 1.

Table 1: Real-world networks.

Network Nodes Edges

Karate 34 78

Dolphin 62 159

Polbooks 105 441

Football 115 613

Lesmis 77 254

Jazz 198 2742

Metabolic 453 2040

08blocks 300 584

Neural 297 2148

Polblogs 1490 16718

Netscience 1589 2742

In order to evaluate the performance of RKCSA

algorithm, we compared it with six other algo-

rithms namely: GN (Girvan and Newman, 2002),

CNM (Blondel et al., 2008), Louvain (Blondel et al.,

2008), BCD (Binary Swarm Optimization) (Beldi and

Bessedik, 2019), ABC (Artiﬁcial Bee Colony) (Da-

kiche et al., 2022), and CSA (The standard Cuckoo

Search Algorithm).

4.1 Evaluation Metrics

Given the fact that we have two types of networks,

with and without the ground truth community struc-

ture, we adopt two widely used criteria to evaluate

the accuracy of community detection algorithms. For

the synthetic networks where the ground truth com-

munity structure is known, we used the Normalized

Mutual Information (NMI) (Danon et al., 2005) met-

ric to assess the similarity between the detected com-

munities and the ground truth. The NMI is calculated

according to equation 9.

NMI(A,B) =

−2

∑

i=1

∑

j=1

i j

log(C

i j

N/C

. j

)

∑

i=1

log(C

/N) +

∑

j=1

. j

log(C

. j

/N)

(9)

Where A and B are two communities in the net-

work. C

i j

is the number of nodes in community A

∈ A

that are also in community B

∈ B. c

) is the num-

ber of groups in A(B). C

and C

. j

are the sum of the

elements of C in row i and column j respectively. N

is the number of nodes.

For real-world networks, where the ground truth

community structure is not available, we used the

Modularity function (Eq 1) to measure the quality of

the detected communities.

4.2 RKCSA’s Parameters Tuning

The most important cuckoo search algorithm includes

three parameters: Pa (abandonment probability for

Improved Random Key Cuckoo Search Optimization Algorithm for Community Detection in Social Networks

117

worse nests), step size scale (α), and acceptance rate

(δ). Among these parameters, Pa and α play a cru-

cial role in achieving improved solutions. Therefore,

in this paper, we have undertaken a sensitivity anal-

ysis of performance for RKCSA to examine the ef-

fects of varying the values of ﬁve parameters: pop-

ulation size (n), the maximum number of iterations

(maxGen), and the CSA parameters Pa, α, and δ. Af-

ter a deep analysis of the results, we observed that

the performance of the algorithm was sensitive to the

choice of parameter values. For the population size

(n), we found that values between 150 and 200 led

to the best outcomes. Regarding the maximum num-

ber of iterations (maxGen), we observed that values

between 800 and 1000 were optimal. The step size

scale (α) was found to be most effective when set to

0.1. For the acceptance rate (δ), we obtained the best

results when it was set to 0.1. The complete details

are not reported for the sake of a concise presentation.

Algorithm 1: Random key cuckoo search algorithm

for community detection.

Data: A network G = (V,E)

Result: Community structure of the network

C = {C

,...,C

}

Initialize the algorithm parameters n

(population size), maxGen (maximum

number of iterations), Pa (discovery

probability), α (step size scale), δ

(acceptance rate).

Generate an initial population of n hosts with

both key vector and locus vector x

key

, x

locus

respectively.

while t < maxGen do

Generate cuckoos by global walk.

Evaluate the quality of the cuckoo

solution Q(x

locus

Choose a nest j among n nests at random.

if (Q(x

locus

) > Q(x

locus

)) then

Replace cuckoo i with the new

cuckoo j.

end

A fraction Pa of the nests are abandoned.

New nests are generated by local walk.

Keep the best solutions (nests with

quality solutions).

Rank the solutions and ﬁnd the best

current solutions.

Update t ← t + 1.

end

4.3 RKCSA Performance Analysis on

Synthetic Networks

Figure 3 presents the experimental results on LFR

benchmarks. It shows the performance of different

algorithms in detecting the real communities for var-

ious values of the mixing parameter, µ. Plots (a)

and (b) in Figure 3 correspond to the same network

size with two different community sizes (small 20-

50 and big 30-100). For µ = 0.05, most of the algo-

rithms are able to accurately detect the real commu-

nities. However, as the value of µ increases beyond

0.05, the algorithms start to struggle in identifying the

real communities, except for the Louvain algorithm

which maintains its effectiveness until µ = 0.2. From

µ = 0.25 to µ = 0.5, the Louvain algorithm consis-

tently outperforms the other algorithms in detecting

real communities. In terms of the metaheuristic al-

gorithms, RKCSA demonstrates better performance

than the other metaheuristics for µ values less than

0.5. It is able to maintain its ability to detect real com-

munities effectively. However, for µ values greater

than 0.5, we observe that the CSA, BCD, and RKCSA

algorithms converge towards similar results and out-

perform the other algorithms (CNM, GN, and Lou-

vain) in terms of community detection accuracy.

These results highlight the varying performance of

the algorithms based on the value of the mixing pa-

rameter, with some algorithms performing better in

speciﬁc ranges of µ. It also indicates the effective-

ness of RKCSA and other metaheuristic algorithms

in community detection tasks, particularly for certain

parameter settings.

4.4 RKCSA Performance Analysis on

Real Networks

Table 2 presents the experimental results of modu-

larity on real-world networks. The numbers in bold

indicate the highest modularity values in each corre-

sponding row. It is observed that the RKCSA algo-

rithm achieves optimal modularity results for 7 out

of 11 real-world networks, accounting for 63.64% of

the networks. On the other hand, the Louvain algo-

rithm obtains the best results for 5 networks, which

corresponds to 45.45% of the networks. Furthermore,

RKCSA outperforms GN, CNM, ABC, BCD, and

CSA algorithms on 10 out of the 11 networks, making

it a competitive method for the community detection

problem. These results highlight the effectiveness of

RKCSA in achieving high modularity values and its

capability to accurately identify community structures

in various real-world networks.

WEBIST 2023 - 19th International Conference on Web Information Systems and Technologies

118

(a) 20-50

(b) 30-100

Figure 3: NMI results on LFR benchmarks with n = 500. (a) Small size communities. (b) Big-size communities.

Table 2: Comparison of modularity results on real-world networks.

Network GN CNM Louvain BCD ABC CSA RKCSA

Karate 0.4013 0.3807 0.4198 0.4198 0.4198 0.4188 0.4198

Dolphin 0.5194 0.4955 0.5185 0.5253 0.5285 0.4572 0.5285

Polbooks 0.5168 0.5020 0.5270 0.5189 0.5116 0.4915 0.5285

Football 0.5996 0.5497 0.6043 0.5146 0.6009 0.3671 0.6044

Lesmis 0.5381 0.5006 0.5548 0.5430 0.5594 0.4879 0.5600

Jazz 0.4051 0.4389 0.4438 0.3120 0.4384 0.2392 0.4426

Metabolic 0.4048 0.4172 0.4405 0.3128 0.3664 0.2569 0.4055

08blocks 0.8599 0.8750 0.8750 0.8104 0.8750 0.8749 0.8750

Neural 0.3010 0.3728 0.3926 0.2319 0.3137 0.1901 0.4254

Polblogs 0.4180 0.4270 0.4269 0.3642 0.3702 0.2234 0.4234

Netscience 0.9579 0.9551 0.9592 0.9006 0.9086 0.8909 0.9500

Improved Random Key Cuckoo Search Optimization Algorithm for Community Detection in Social Networks

119

5 CONCLUSIONS

In this paper, we presented a novel cuckoo search al-

gorithm, RKCSA, for community detection in social

networks. Where we proposed a new solution rep-

resentation that combines the locus and random key

representations of the network to enhance its search

ability. Experiments on both synthetic and real-world

networks show that RKCSA can accurately and ef-

fectively uncover the community structure. We also

demonstrated the superior performance of RKCSA

compared to the standard CSA algorithm. However,

in real-life networks, we can ﬁnd multiple relation-

ships between a couple of nodes. Therefore, we

aim to extend our algorithm to handle multilayer net-

works.

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