Flattening Based Cuckoo Search Optimization Algorithm for

Community Detection in Multiplex 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:

Community Detection, Flattening, Cuckoo Search Optimization, Multiplex Networks, Modularity.

Abstract:

Complex network analysis is a thriving research ﬁeld, with a particular focus on community detection. This

paper addresses the challenge of community detection in multiplex networks, which model multiple types of

relationships to reﬂect reality. Our approach consists of two key steps. First, we employ multiplex network

ﬂattening techniques to transform it into a one-dimensional network. Second, we introduce a cuckoo search-

based algorithm to maximize the modularity function and identify the best network partitions. Our algorithm

strategically combines the continuous aspects of the standard cuckoo search algorithm with the discrete nature

of community detection, to achieve better results. Experiments on both synthetic and real-world multiplex

networks demonstrate the efﬁciency and effectiveness of our approach.

1 INTRODUCTION

Real-life networks often involve various types of con-

nections between entities. For instance, human re-

lationships can encompass family ties, business as-

sociations, social interactions, and more. To model

such complex networks, researchers have introduced

the concept of multiplex networks (Boccaletti et al.,

2014). A multiplex network comprises several in-

dividual layers, with each layer representing a dis-

tinct type of connection or relationship (Kivel

a et al.,

2014). However, the challenge arises as each layer

in a multiplex network possesses its unique commu-

nity structures, potentially missing the comprehensive

composite community (Pizzuti, 2017). This complex-

ity contrasts with the relatively straightforward com-

munity structure in traditional networks, where sub-

graphs highlight nodes with stronger internal connec-

tions than with the rest of the network (Girvan and

Newman, 2002).

Despite the community detection problem being

one of the main topics of social network analysis for

over a decade, only recently it has been tackled in the

context of complex multidimensional systems (Gong

et al., 2013; Shen and Cheng, 2010). Several re-

cent surveys have suggested categorizing existing ap-

proaches that handle the presence of multiple layers

into three main classes (Huang et al., 2021; Roozba-

hani et al., 2021; Magnani et al., 2021): (1) Flatten-

ing techniques that combine all layers into one, fol-

lowed by the application of traditional network anal-

ysis methods. (Loe and Jensen, 2015); (2) Assem-

bly techniques that detect communities in each layer,

and then ﬁnd a consensus between them (Harakawa

et al., 2019); and (3) Direct detection techniques

that identify multiplex community structures without

layer simpliﬁcation (Mucha et al., 2010).

In this paper, we propose a two-phase community

detection approach in multiplex networks. In the ﬁrst

phase, we draw inspiration from the work of Berlin-

gerio et al. (2011) to apply ﬂattening techniques. In

the second phase, we apply an optimization algorithm

for community detection to the ﬂattened network. To

achieve robust results, we introduce a novel algo-

rithm called Cuckoo Search Algorithm (CSA) (Yang

and Deb, 2009). CSA is a relatively recent nature-

inspired optimization technique known for its effec-

tiveness in global optimization tasks (Yang and Deb,

2010). However, its direct application to community

detection is limited. To overcome this limitation, we

incorporate a random key representation that operates

within a continuous space. This allows for global and

local exploration within the search space, striking a

balance between exploration and exploitation. Addi-

tionally, we introduce a locus-based representation to

accommodate the discrete nature of network commu-

nity detection needs.

The paper is organized as follows: Section 2 de-

Boukabene, R., Benbouzid-Si Tayeb, F. and Dakiche, N.

Flattening Based Cuckoo Search Optimization Algorithm for Community Detection in Multiplex Networks.

DOI: 10.5220/0012363600003636

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2024) - Volume 3, pages 519-526

ISBN: 978-989-758-680-4; ISSN: 2184-433X

519

scribes the community detection problem in multiplex

networks. Section 3 outlines the proposed algorithm.

The experimental results on synthetic and real net-

works are presented in Section 4. Finally, Section 5

concludes the paper and explores potential future re-

search directions.

2 MULTIPLEX NETWORKS AND

COMMUNITIES

A multiplex network (Boccaletti et al., 2014) is a se-

ries of graphs G = {G

;α ∈ {1,...,d}} where G

(V,E

) represents the α

layer of G, and E

repre-

sents the α

type of connection among the same set

of nodes V . G can be represented by an adjacency ma-

trix A

[α]

= (a

i j

) ∈ R

N×N

for each layer G

, where:

i j



1 if(v

) ∈ E

0 else

(1)

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.

3 PROPOSED SOLVING

APPROACH

The proposed process of community detection in mul-

tiplex networks using ﬂattening techniques involves

two distinct phases (Figure 2). Initially, the multiplex

network is consolidated into a mono-dimensional net-

work. Subsequently, we apply an optimization mono-

dimensional community detection algorithm.

In what follows, we will delve into a comprehen-

sive exploration of the proposed algorithm, providing

a detailed explanation of its two phases.

3.1 Flattening Techniques

Numerous approaches have been developed for com-

munity detection in multiplex networks, including the

ﬂattening technique, which transforms a multiplex

network G

into a mono-dimensional network (tra-

ditional network) G while preserving as much infor-

mation as possible. Berlingerio et al. (2011) propose

three ﬂattening techniques as follows:

• The ﬁrst technique, referred to as f

, assigns a

weight of 1 to the edge E

i j

in G if there exists

at least one dimension connecting nodes i and j in

(Figure 2a) (Eq2).

i j



1 if ∃d : (i, j,d) ∈ E

0 otherwise

(2)

• The second technique, denoted as f

, takes into

account the count of dimensions connecting any

pair of nodes, i and j, and uses this count as

the weight for the newly added mono-dimensional

edge (Berlingerio et al., 2011) (Figure 2b) (Eq3).

i j

= |d : (i, j, d) ∈ E| (3)

• The third technique, designated as f

, extends its

consideration to the neighbors of nodes i and j,

rather than solely focusing on the connection be-

tween these nodes. This modiﬁcation is grounded

in the assumption that common neighbors are

likely to belong to the same community as nodes i

and j (Berlingerio et al., 2011) (Figure 2c)(Eq4).

i j

| − 2

(4)

Where N

is the set of neighbors in dimension d

for a node.

3.2 Cuckoo Search Optimization

The Cuckoo Search Algorithm (CSA), introduced by

Yang and Deb (2009), is a nature-inspired metaheuris-

tic optimization algorithm. It draws inspiration from

the brood parasitism behavior observed in certain

bird species, notably the cuckoo (Yildiz, 2013). In

the proposed algorithm, called FCSA for ﬂattening-

based CSA, candidate solutions are represented us-

ing a novel structure that combines both the locus

and random key representations, using nests or eggs.

These solutions are created through a combination

of global and local walks. Subsequently, the algo-

rithm assesses their quality using the modularity func-

tion, leaning on the locus representation, and replaces

the existing solutions in the nests with any improved

ones. The random key representation serves as a vital

guide in steering the search and exploration processes,

maintaining the algorithm’s continuous nature. How-

ever, when it comes to replacing solutions within the

nests, the algorithm places a speciﬁc emphasis on the

locus representation to enhance community detection

results.

Next, we will delve into a comprehensive explo-

ration of the proposed algorithm.

3.2.1 Solution Representation and Objective

Function

FCSA starts with solution encoding to create a ran-

dom initial population. A candidate solution X is en-

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520

(a) Multiplex network. (b) Flattened network. (c) Graph partitions.

Figure 1: Proposed ﬂattening-based process of community detection in multiplex networks.

(a) Flattening function f

. (b) Flattening function f

. (c) Flattening function f

Figure 2: Multiplex network (1a) ﬂattening techniques.

coded and initialized using two-ﬁeld structure as il-

lustrated in Figure 3, where:

• The ﬁrst ﬁeld is the locus vector (Figure 3b)

that uses the locus-based representation (Pizzuti,

2008). In this representation, Each node u within

an individual is associated with a value v, repre-

senting one of its adjacent nodes. The decoding

step can be accomplished in linear time (Cormen

et al., 2022) (Pizzuti, 2008).

• The second ﬁeld corresponds to the key vector

(Figure 3c), where each node in the network is

associated with a random value ranging from 0 to

Modularity Q is by far the most used and best-

known quality function for measuring the quality of a

partition of a network (Newman and Girvan, 2004)

For each graph G, with n nodes and m edges, mod-

ularity is computed as follows:

Q =

∑

i j

−

]δ(c

) (5)

Where A

i j

represents the adjacency matrix, k

)

is the sum of edge weights of node i( j), c

) is the

community of node i( j), m is the sum of all the edge

weights in the graph. δ is the Kronecker delta function

δ(x,y) = 1 if x = y, 0 otherwise.

(a) Graph partitions.

(b) Locus vector.

Figure 3: Example of a solution representation (Boukabene.

et al., 2023).

3.2.2 Global Walk

In the proposed algorithm, the ﬁrst component,

known as L

evy ﬂights or global walk, is responsible

for generating new solutions near the best solution

found so far (Reda et al., 2022) (Boukabene. et al.,

Flattening Based Cuckoo Search Optimization Algorithm for Community Detection in Multiplex Networks

521

2023). To create a new cuckoo, both a new key vec-

tor and a new locus vector are generated. The new

key vector is produced using the L

evy ﬂights’ distri-

bution, as deﬁned in equation 6.

key

(t +1) = x

key

(t)+ αs(x

key

(t)− x

best

key

(t)) (6)

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.

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|

(7)

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

And

var(u) = [

Γ(1 + β)

βΓ((1 + β)/2)

sin(πβ/2)

(β−1)/2

]

1/β

var(v) = 1

(8)

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, we calculate the

difference between the old and new value of the

key vector |x

key

(t + 1) − x

key

(t)|. If the difference is

greater than the acceptance rate δ, we reinitialize the

corresponding node with the neighbor of the best so-

lution so far x

best

locus

(t). Otherwise, the node’s value re-

mains the same as the previous iteration.

The newly generated solution is evaluated using

the modularity measure (Eq. 5) on the locus vector. If

the new solution exhibits higher modularity than the

current solution, it replaces the current solution within

this nest.

3.2.3 Local Walk

The second component of the proposed algorithm is

the abandon operator which generates solutions far

from the best solution to prevent the search from get-

ting trapped in local optima and to encourage explo-

ration of different regions of the search space. By

introducing diversity into the search process, the al-

gorithm increases the chances of ﬁnding the global

optimum or better solutions (Reda et al., 2022).

The purpose of the abandon operator is to discover

foreign eggs from those of the host bird within a prob-

ability Pa. If foreign eggs are found, the host bird will

abandon them and replace them with new eggs. In

FCSA, this process is applied to the key vector, where

foreign solutions are replaced with new solutions gen-

erated using equation 9.

key

(t +1) = x

key

(t)+ S ∗ K (9)

With

K =



1 i f rand > Pa

0 Otherwise

(10)

And

S = (x

rand perm(n)

key

(t)− x

rand perm(n)

key

(t)) (11)

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 comparing the old and new

value of the key vector |x

key

(t +1)−x

key

(t +1)|. If the

difference exceeds the acceptance rate, a speciﬁc node

in the locus vector undergoes re-initialization through

a roulette selection process. The probability of halting

the roulette for this modiﬁcation is denoted as p and

is determined by the formula:

p =

i j

∑

k∈N(i)

Here, w

i j

represents the weight of the edge E

i j

be-

tween nodes i and j and N(i) denotes the set of neigh-

bor nodes connected to node i. If the calculated prob-

ability p exceeds a random number, a replacement is

made, and the current node is substituted with node

i. Otherwise, the node’s value remains the same as

in the previous step. If the newly generated solution

has greater modularity than the current solution, it re-

places the current solution for this nest.

4 EXPERIMENTAL RESULTS

AND DISCUSSION

In this section, we present the results of a series of

computational experiments conducted to assess the

effectiveness of our three proposed ﬂattening-based

CSA algorithms. We denote these proposed algo-

rithms as FCSA1, FCSA2, and FCSA3, each of which

combines our CS algorithm with one of the ﬂattening

functions, namely, f 1 (Eq. 2), f 2 (Eq. 3), and f 3 (Eq.

4), respectively. We implemented our algorithms in

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

522

Phyton and ran experiments on a personal computer

running Windows 10 Enterprise, equipped with 8 GB

of RAM and powered by an Intel(R) Core(TM) i7-

3537u CPU.

All experiments are conducted on both synthetic

and real-world networks. We generated synthetic

networks using the mLFR benchmark (Brodka and

Grecki, 2014), which extends the LFR benchmark

(Lancichinetti et al., 2008), with varied structures and

layers by adjusting both the mixing parameter (µ) and

the degree change chance (Dc). Increasing the values

of µ and Dc made the community detection task more

challenging by blurring the boundaries between com-

munities and increasing the differences in node de-

grees across network layers (Table 1). For real-world

networks, we employed a diverse set of 6 real-world

networks from various domains with diverse sizes and

characteristics (Table 2).

Table 1: The m-LFR128 Parameter Settings.

Parameter Value

Number of nodes 128

Node average degree 8

Number of layers [2, 3, 4]

Mixing parameter [0.1, 0.2, 0.3, 0.4, 0.5]

Degree change chance [0.2, 0.4, 0.6, 0.8]

Maximal community size 32

Minimal community size 8

Node maximal degree 16

Table 2: Real-world networks.

Network Nodes Edges Layers

ﬂorentine 16 35 2

Tailorshop 39 552 4

London Transport 369 441 3

CKM Physicians In-

novation

246 1551 3

EU-Air Transporta-

tion

450 3588 37

FAO Trade 183 16048 339

Plasmodium GPI 1203 2521 3

Celegans GPI 3879 8181 6

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. For real-world net-

works, where the ground truth community structure

is not available, we evaluated the quality of the de-

tected communities using Q

(Eq. 12) the average

modularity function (Eq. 5) across all layers.

∑

i=1

(12)

Where d is the number of layers and Q

is the mod-

ularity of layer i.

Furthermore, we conducted a sensitive analysis of

the FCSA algorithm to assess the impact of varying

its ﬁve parameters. Due to space constraints, we pro-

vide a summary of these parameters, Population size

n takes a value of 150, Maximum number of iterations

maxGen equal to 1000, Abandonment probability Pa

is 0.5, Step size scale α takes a value of 0.1, Accep-

tance rate δ is 0.1.

The proposed algorithms are compared against

the Louvain algorithm, which will be combined with

three ﬂattening techniques as proposed in (Berlinge-

rio et al., 2011), denoted Louvain1, Louvain2, and

Louvain3. Additionally, we will evaluate our al-

gorithms against ﬁve other community detection al-

gorithms in multiplex networks: Cinfomap (Rosvall

and Bergstrom, 2008), CBGLL (Lancichinetti and

Fortunato, 2012), GenLouvin (Mucha et al., 2010),

CLPAm (Barber and Clark, 2009), and MMCD (Ma

et al., 2018)). However, due to the unavailability of

the source code for these algorithms, the comparison

will only take place in real-world networks where re-

sult values are shared.

In what follows, we will ﬁrst present the per-

formance analysis of our newly proposed ﬂattening-

based CSAs on the synthetic multiplex networks, and

then on the real ones.

4.1 FCSA Performance Analysis on

Synthetic Networks

Figure 4 shows the results of our experiments on the

mLFR benchmarks. It serves as a visual representa-

tion of the performance of various algorithms, includ-

ing FCSA1, FCSA2 and FCSA3, compared with Lou-

vain1, Louvain2 and Louvain3. These algorithms are

evaluated in terms of their effectiveness in identify-

ing real communities, considering different values of

the mixing parameter (µ) and the degree change haz-

ard (Dc). Each column in this ﬁgure is devoted to

networks with a different number of layers, while the

rows represent networks with different degree change

chance (Dc). The graphs in the ﬁgure illustrate the

results in terms of NMI versus mixing parameter (µ).

When we consider a µ of 0.1, most of the al-

gorithms successfully identify the real communities.

However, as we increase µ to values beyond 0.1, the

algorithms encounter challenges in accurately detect-

Flattening Based Cuckoo Search Optimization Algorithm for Community Detection in Multiplex Networks

523

Figure 4: The NMI results for mLFR-128 networks with the increase of network layers (i.e., L = 2, 3, 4) and the change of

network structure (mixing parameter mu and degree change chance Dc).

ing the true communities. Notably, for µ values be-

tween 0.3 and 0.6, the Louvain algorithms consis-

tently outperform the other methods in community

detection. In more complex networks with higher val-

ues of the mixing parameter, we observe that FCSA

algorithms deliver superior results. A similar trend

can be observed regarding the degree change chance

(Dc). At a Dc value of 0.2, the algorithms perform

at their best, after which their performance starts to

decline as Dc increases. Furthermore, the number of

layers in the network signiﬁcantly inﬂuences commu-

nity detection, particularly for µ values between 0.3

and 0.6. It’s evident that the task is more manageable

for networks with fewer layers, and the complexity

increases when we move from two to three or four

layers.

In summary, comparing various ﬂattening tech-

niques proves to be challenging due to performance

variations across diverse network structures and pa-

rameter settings. It’s worth noting that FCSA consis-

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

524

Table 3: Comparison of modularity results on real-world networks.

Network Florentine Tailorshop London

Transport

CKM EU-Air FAO

Trade

Plasmo-

dium

Celegans

FCSA1 0.2781 0.2370 0.7259 0.6973 0.0107 0.0301 0.1563 0.4498

FCSA2 0.3250 0.2377 0.7239 0.7030 0.0131 0.1740 0.1601 0.5320

FCSA3 0.2725 0.2332 0.7325 0.6960 0.0110 0.0215 0.1559 0.4324

Louvain1 0.2780 0.2335 0.7227 0.6938 -0.0399 0.0555 0.1765 0.5480

Louvain2 0.3250 0.2374 0.6529 0.6918 -0.0173 0.1938 0.1766 0.5480

Louvain3 0.2385 0.1696 0.6224 0.6715 -0.0230 0.0449 0 0.4250

Cinfomap - - 0.7456 0.6579 0.0120 0.3113 0.4601 0.4905

CBGLL - - 0.7352 0.6833 0.0479 0.2378 0.4734 0.4732

Gen-

Louvain

0.2936 0.2238 0.7892 0.6944 0.1154 0.3174 0.5183 0.5445

CLPAm - - 0.5352 0.5702 0.1033 0.3124 0.3406 0.3998

MMCD 0.3060 0.2107 0.7932 0.7056 0.1219 0.3174 0.5237 0.5524

tently delivers optimal results for complex networks

characterized by high values of mu and DC.

4.2 FCSA Performance Analysis on

Real Networks

Table 3 presents the experimental modularity results

for real-world networks, with values highlighted in

bold indicating the highest modularity values in their

respective columns based on problem resolution type.

As the modularity values for real-world networks are

gathered from the literature, the use of dashes indi-

cates that the speciﬁc result values for the correspond-

ing algorithm and dataset were not available. Our

analysis reveals noteworthy trends, particularly the

superior performance of the second ﬂattening func-

tion, outperforming other techniques in 6 out of 8

real-world networks when both FCSA and Louvain

are employed. Compared to Louvain, FCSA demon-

strates superiority in 4 out of 7 real-world networks

and achieves comparable results for the Florentine

network.

In a comparison between FCSA and direct meth-

ods, FCSA exhibits superiority in multiple instances,

surpassing CLPAm and Cinfomap in 3 out of 6 cases,

CBGLL in 2 out of 6 cases, GenLouvain in 3 out of 8

cases, and MMCD in 2 out of 8 cases. Upon anal-

ysis, it becomes apparent that a signiﬁcant amount

of information is lost when ﬂattening multiplex net-

works with a high number of layers and high density.

This information loss could explain why direct meth-

ods outperform our proposed algorithm. However, for

small networks with a limited number of layers, such

as Florentine and Tailorshop, our approach yields bet-

ter results due to the minimal loss of information dur-

ing ﬂattening.

5 CONCLUSIONS

In this paper, we introduced FCSA, a novel commu-

nity detection algorithm utilizing three distinct strate-

gies to ﬂatten multiplex networks. The incorporation

of locus and random key representations in FCSA

enhances network encoding and search capabilities.

Experimental results on synthetic networks demon-

strate the algorithm’s superior performance for com-

plex networks. However, when applied to real-world

networks, its effectiveness is more pronounced for

small networks. It is crucial to note that the trans-

formation of the multiplex network into a mono-

dimensional network introduces some information

loss, which stands as a current limitation of our algo-

rithm. Our future objective is to extend the algorithm

to operate directly on the multiplex network, aiming

to overcome this speciﬁc limitation.

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