A COMPARATIVE STUDY OF NEIGHBORHOOD TOPOLOGIES

FOR PARTICLE SWARM OPTIMIZERS

Angelina Jane Reyes Medina

, Gregorio Toscano Pulido

and Jos

e Gabriel Ram

ırez Torres

CINVESTAV-Tamaulipas. Km. 6 carretera Cd. Victoria-Monterrey, Cd. Victoria, Tamaulipas, 87261, Mexico

Keywords:

Particle swarm optimization, Neighborhood topologies, Parameter setting.

Abstract:

Particle swarm optimization (PSO) is a meta-heuristic that has been found to be very successful in a wide

variety of optimization tasks. The behavior of any meta-heuristic for a given problem is directed by both: the

variation operators, and the values selected for the parameters of the algorithm. Therefore, it is only natural

to expect that not only the parameters, but also the neighborhood topology play a key role in the behavior

of PSO. In this paper, we want to analyze whether the type of communication employed to interconnect the

swarm accelerates or affects the algorithm convergence. In order to perform a wide study, we selected six

different neighborhoods topologies: ring, fully connected, mesh, toroid, tree and star; and two clustering

algorithms: k-means and hierarchical. Such approaches were incorporated into three PSO versions: the basic

PSO, the Bare-bones PSO (BBPSO) and an extension of BBPSO called BBPSO(EXP). Our results indicate

that the convergence rate of a PSO-based approach has an strongly dependence of the topology used. However,

we also found that the topology most widely used is not necessarily the best topology for every PSO-based

algorithm.

1 INTRODUCTION

The behavior of any meta-heuristic for a given pro-

blem is directed by both: the variation operators, and

the values selected for the parameters of the algorithm

(parameter setting). The parameter setting issue plays

a key role on the performance of any meta-heuristic.

Tuning well these parameters is a hard problem, since

they can usually take several values, and therefore, the

number of possible combinations is usually high.

Kennedy & Eberhart (Kennedy and Eberhart,

2001) proposed an approach called “particle swarm

optimization” (PSO) which was inspired on the cho-

reography of a bird ﬂock. The approach can be seen

as a distributed behavioral algorithm that performs (in

its more general version) multidimensional search. In

the simulation, the behavior of each individual is a-

ffected by either the best local (i.e., within a certain

neighborhood) or the best global individual.

Several PSO proposals have been developed in or-

der to improve the performance of the original algo-

rithm (Kennedy, 2003; Kennedy and Eberhart, 2001;

Omran et al., 2008). Such approaches have shown

that the convergence behavior of PSO is strongly de-

pendent on the values of the inertia weight, the cog-

nitive coefﬁcient and the social coefﬁcient (Clerc and

Kennedy, 2002; van den Bergh, 2002). Other propo-

sals have sought to eliminate the dependence of such

parameters in order to avoid the parameter setting

problem. Investigations within the particle swarm

paradigm have found that the particles’ interconnec-

tion topology interact directly with the function be-

ing optimized (Kennedy, 1999). These studies have

shown theoretically that the neighborhood topology

affects (signiﬁcantly) the performance of a particle

swarm and that the effect depends on the function.

Thus, some types of interconnection topologies can

work well for some functions, while the same topolo-

gies can present problems with other test functions

(Kennedy, 1999). Despite the key role that the topo-

logy plays in PSO, it has been barely studied.

The remainder of the paper is organized as fo-

llows: Section 2 provides an overview of PSO and

its variants used in this paper. Neighborhood topolo-

gies and clustering algorithms are presented in Sec-

tion 3. Section 4 presents and discusses the results of

the performed experiments. Finally, Section 5 shows

the concluding remarks and future work.

152

Jane Reyes-Medina A., Toscano Pulido G. and Gabr iel Ramírez-Torres J. (2009).

A COMPARATIVE STUDY OF NEIGHBORHOOD TOPOLOGIES FOR PARTICLE SWARM OPTIMIZERS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 152-159

DOI: 10.5220/0002324801520159

 SciTePress

2 PARTICLE SWARM

OPTIMIZATION

Particle swarm optimization (PSO) is an stochastic,

population-based optimization algorithm proposed by

Kennedy and Eberhart in 1995 (Kennedy and Eber-

hart, 1995), that simulates the social behavior of bird

ﬂocks or school ﬁsh. In PSO, a swarm of parti-

cles ﬂy through hyper-dimensional search space be-

ing attracted by both, their personal best position and

the best position found so far within a neighborhood.

Each particle represent a solution to the optimization

problem. The position of each particle is updated u-

sing equation (1) which is composed by the best po-

sition visited by itself (i.e. its own experience or y

in equation (2)) and the position of the best parti-

cle in its neighborhood determined by the commu-

nication topology used (Kennedy and Mendes, 2002;

Kennedy, 1999).

i j

(t + 1) = x

i j

(t) + v

i j

(t + 1), (1)

i j

(t + 1) = wv

i j

(t) + c

1 j

(t)(y

i j

(t) −x

i j

(t))

+ c

2 j

(t)(

i j

(t) −x

i j

(t)), (2)

for i = 1, ··· , s and j = 1, ··· , n

where w is the inertia weight (Shi and Eberhart,

1998), s is the total number of particles in the swarm,

n is the dimension of the problem (i.e. the number of

parameters of the function being optimized), c

and

are the acceleration coefﬁcients, r

1 j

, r

2 j

∼ U(0, 1),

(t) is the position of particle i at time step t, v

(t)

is the velocity of particle i at time step t, y

(t) is the

personal best position of particle i at time step t, and

is the neighborhood best position of particle i at

time step t.

Empirical and theoretical studies have shown that

the convergence behavior of PSO is strongly depen-

dent on the values of the inertia weight and the accel-

eration coefﬁcients (van den Bergh and Engelbrecht,

2006; Clerc and Kennedy, 2002). Wrong choices of

such parameters may produce divergent or cyclic par-

ticle trajectories. Several recommendations for values

of such parameters have been suggested in the spe-

cialized literature (Storn and Price, 1997), although

these values are not universally applicable to every

kind of problem.

A large number of PSO variations have been de-

veloped, mainly to improve the accuracy of solu-

tions, diversity, convergence or to eliminate the pa-

rameters dependency (Kennedy and Eberhart, 2001).

van den Bergh and Engelbrecht (van den Bergh and

Engelbrecht, 2006) and Clerc y Kennedy (Clerc and

Kennedy, 2002) proved that each particle converges

to a weighted average of its personal best and neigh-

borhood best position, that is,

lim

t→+∞

i j

(t) =

i j

+ c

i j

+ c

(3)

This theoretically derived behavior provides su-

pport for the Bare-bones PSO (BBPSO). BBPSO was

proposed by Kennedy in 2003 (Kennedy, 2003). The

BBPSO replaces the equation (1) and (2) with equa-

tion (4),

i j

(t + 1) = N



i j

(t) +

(t)

, |y

i j

(t) −

(t)|



(4)

Particle positions are therefore randomly selected

from N which is a Gaussian distribution with: mean,

equal to the average weighted of its personal best and

the global best positions (i.e. the swarm attractor)

and; deviation y

i j

(t) −

(t) which approximate zero

as t increases.

Kennedy also proposed an alternative version of

the BBPSO (EXP). He replaced the equations (1) and

(2) with equation (5),

i j

(t +1) =









i j

(t)+

(t)

, |y

i j

(t) −

(t)|



if U(0, 1)

> 0.5

i j

(t) otherwise

(5)

Based on the above equation, there is a 50%

chance that the j the dimension of the particle dimen-

sion changes to the corresponding personal best posi-

tion. This version of PSO biases towards exploiting

personal best positions.

3 DESCRIPTION OF OUR

EXPERIMENT

In PSO, each particle inside of the swarm belongs to

an speciﬁc communication neighborhood. Therefore,

it was natural that several studies were performed in

order to determine whether the neighborhood topo-

logy could affect the convergence (Kennedy, 1999;

Kennedy and Mendes, 2002; Jian et al., 2004). These

studies relied on theoretical proposals and implemen-

tations of neighborhood topologies commonly used

by PSO. In such studies, some neighborhood topolo-

gies have performed better than others (Kennedy,

1999). However, only a few topologies and prob-

lems were tested at a time. Therefore, our hypothe-

sis to perform this study was that the topology used

in a particle swarm might affect the rate and degree

to which the swarm is attracted towards a particu-

lar region. Thus, the present study focused on seve-

ral swarm topologies, where connections were undi-

rected, unweighted, and they no vary over the course

A COMPARATIVE STUDY OF NEIGHBORHOOD TOPOLOGIES FOR PARTICLE SWARM OPTIMIZERS

153

of a trial. The neighborhoods topologies were con-

structed based on the index of each particle, then each

particle has a unique identiﬁer in the entire popula-

tion. We also decided to study how clustering al-

gorithms can improve the performance of the PSO.

In clustering algorithms, the euclidean distance were

used as a measure and the connections were updated

dynamically on each iteration of the trail.

Since this paper analyzes the ring, fully con-

nected, mesh, toroidal, tree, star topologies shown

in Figure 1 and two clustering algorithms: k-means

and hierarchical in order to determined whether the

type of communication employed to interconnect the

swarm accelerates or affects the algorithm conver-

gence. We described them below:

(a) Ring (b) Fully con-

nected

(d) Star (e) Toroidal (f) Tree

Figure 1: Neighborhood topologies used in this study.

3.1 Ring Topology

The ring topology is also known as the lbest version in

PSO (see Figure 1(a)). In this topology each particle

is affected by the best performance of its k immediate

neighbors in the topological population. In one com-

mon lbest case, k = 2, the individual is affected by

only its immediately adjacent neighbors.

In the ring topology, the neighbors are closely

connected and thus, they react when one particle has

a raise in its ﬁtness, this reaction dilutes proportion-

ally with respect to the distance. Thus, it is possible

that one segment of the population might converge on

a local optimum, while another segment of the pop-

ulation might converge to a different point or remain

searching. However, the optima will eventually pull

the swarm.

3.2 Fully Connected Topology

Fully connected topology is also known as the full

topology (see Figure 1(b)). All nodes in this topolo-

gy are directly connected among each other. In PSO

this topology is also known as the PSO’s gbest ver-

sion, in which all particles in the entire swarm direct

their ﬂight toward the best particle found in the whole

population (i.e. every particle is attracted to the best

solution found by any member of the swarm). That is,

(t) ∈ {y

(t), y

(t), . . . , y

(t)} = min{f (y

(t)), f (y

(t)),

. . . , f (y

(t))}, (6)

Kennedy et al. suggested (Kennedy and Eberhart,

2001; Kennedy and Mendes, 2002) that gbest popula-

tions tend to converge more rapidly to a optima than

lbest populations, but also, they are more susceptible

to converge to a local optima. However, this topology

is the most used by far.

3.3 Star Topology

In star topology, the information passes through only

one individual (see Figure 1(d)). One central node

inﬂuences and it is inﬂuenced by all other members

of the population.

In this article, the central particle of the star topol-

ogy is selected randomly. In each time step t, all par-

ticles of the entire swarm directs their ﬂight toward

one particle (the central particle), and the central par-

ticle directs its ﬂight toward the best particle of the

neighborhood. The star topology, effectively isolates

individuals from each other, since information has to

be communicated through the central node. This cen-

tral node compares the performance of every indivi-

dual in the population and adjusts its own trajectory

toward the best of them. Thus the central individ-

ual serves as a kind of buffer or ﬁlter, slowing the

speed of transmission of good solutions through the

population. The buffering effect of the central par-

ticle should prevent premature convergence on local

optima; this is a way to preserve diversity of poten-

tial problem solutions, though, it was expected that it

might destroy population’s collaboration ability.

3.4 Mesh Topology

In this type of topology (see Figure 1(c)), one node is

connected to several nodes, commonly each node is

connected to four neighbors (in this case, we connect

each node to the ones which are in the north, south,

east and west of the particle’s location).

In the mesh topology, the particles in the corners

are connected with its two adjacent neighbors. The

particles on the mesh’s boundaries will have tree ad-

jacent neighbors and the particles on the mesh’s cen-

ter will have four adjacent neighbors. Thus, there

exists overlapping neighbors in each particle, allow-

ing redundancy in the search process. The particles

IJCCI 2009 - International Joint Conference on Computational Intelligence

154

will be assigned to each node of the mesh from left to

right and top-down. The mesh remains the same form

throughout the algorithm’s execution.

3.5 Tree Topology

It is also known as a hierarchical topology (see Figure

1(f)). This topology has a central root node (the top

level of the hierarchy) which is connected to one or

more individuals that are one level lower in the hier-

archy (i.e., the second level), while each of the second

level individuals that are connected to the top level

central root individual will also have one or more in-

dividuals which are one level lower in the hierarchy

(i.e., the third level) connected to it (the hierarchy of

the tree is symmetrical).

The tree topology is constructed as a binary tree

(using the particle’s index as nodes), the root node

is selected randomly among swarm and the remain-

ing particles are distributed in the tree branches. The

nodes (particles) in the tree must be, as possible, ba-

lanced in the tree branches. The root node searches

for the best ﬁtness obtained by their children (i.e. the

second level) to redirected its ﬂight. The second level

nodes search for the best ﬁtness found by both, chil-

dren and father.

3.6 Toroidal Topology

Topologically, a torus is a closed surface deﬁned as

the product of two circles (see Figure 1(e)). This topo-

logical torus is often called as Clifford torus.

The toroidal topology is similar to the mesh topo-

logy, except that all particles in the swarm have four

adjacent neighbors. As is shown in Figure 1(e), the

toroidal topology connects every corner particle with

its symmetrical neighbor. The same occurs with the

toroid boundaries. The assignment from particles to

nodes will be similar to the mesh topology assign-

ment.

3.7 Clustering Algorithms

Clustering is deﬁned by the average number of neigh-

bors that any two connected nodes have in common

(Kennedy, 1999). In PSO, the particles naturally clus-

ter in more than one region of the search space usu-

ally indicate the presence of local optima. It seems

reasonable to investigate whether information about

the distribution of particles in the search space could

be exploited to improve particle trajectories (Kennedy

and Eberhart, 2001).

We have implemented two types of clustering al-

gorithms: k-means and hierarchical clustering.

• K-means clustering algorithm:

Given a data set through a certain number of clus-

ters (assume k clusters) ﬁxed a priori. The main

idea is to deﬁne k centroids, one for each clus-

ter. These centroids should be placed in a cunning

way because of different location causes a differ-

ent result. The next step is to take each point be-

longing to a given data set and associate it to the

nearest centroid. In PSO, we assume to have 4

clusters. We used the euclidean distance, in order

to associate a particle to its nearest centroid.

• Hierarchical clustering algorithm:

Hierarchical clustering considers the distance be-

tween one cluster and another cluster to be equal

to the shortest distance from any member of one

cluster to any member of the other cluster. If

the data consists of similarities, then hierarchical

clustering considers the similarity between one

cluster and another cluster to be equal to the great-

est similarity from any member of one cluster to

any member of the other cluster.

In order to implement the hierarchical clustering

algorithm into PSO, we deﬁned 4 clusters to be

searched for. The shortest distance between parti-

cles were calculated using the euclidean distance.

3.8 Test Functions

Nine test functions were selected from the specialized

literature. Such test functions are described below:

A. Sphere function, deﬁned as

f (x) =

∑

i=1

where x

∗

= 0 and f (x

∗

) = 0 for −100 ≤x

≤ 100

B. Schwefel’s problem , deﬁned as

f (x) =

∑

i=1

∏

i=1

where x

∗

= 0 and f (x

∗

) = 0 for −10 ≤x

≤ 10

C. Step function, deﬁned as:

f (x) =

∑

i=1

(bx

+ 0.5c)

where x

∗

= 0 and f (x

∗

) = 0 for −100 ≤x

≤ 100

D. Rosenbrock function, deﬁned as

f (x) =

∑

−1

i=1

(100(x

−x

i−1

)

+ (x

i−1

−1)

where x

∗

= (1, 1, . . . , 1) and f (x

∗

) = 0 for −30 ≤

≤ 30

E. Rotated hyper-ellipsoid function, deﬁned as

f (x) =

∑

i=1

(

∑

j=1

)

where x

∗

= 0 and f (x

∗

) = 0 for −100 ≤x

≤ 100

F. Generalized Schwefel Problem 2.26, deﬁned as:

f (x) = −

∑

i=1

sin(

|)),

where x

∗

= (420.9687, . . . , 420.9687) and

f (x

∗

) = −4426.407721 for −500 ≤x

≤ 500

A COMPARATIVE STUDY OF NEIGHBORHOOD TOPOLOGIES FOR PARTICLE SWARM OPTIMIZERS

155

G. Rastrigin function, deﬁned as:

f (x) = −

∑

i=1

−10cos(2πx

) + 10),

where x

∗

= 0 and f (x

∗

) = 0 for −5.12 ≤x

≤5.12

H. Ackley’s function, deﬁned as

f (x) = −20exp



− 0.2

∑

i=1



−

exp



∑

i=1

cos(2πx

)



+ 20 + e,

where x

∗

= 0 and f (x

∗

) = 0 for −32 ≤x

≤ 32

I. Griewank function, deﬁned as:

f (x) =

4000

∑

i=1

−

∏

i=1

cos



√



+ 1,

where x

∗

= 0 and f (x

∗

) = 0 for −600 ≤x

≤ 600

4 DISCUSSION OF RESULTS

This section compares the performance of the ba-

sic PSO, BBPSO and BBPSO (EXP) algorithms dis-

cussed in Section 2. We implemented: ring, full,

star, mesh, toroidal and tree neighborhood topologies

and; k-means and hierarchical clustering algorithms

on each PSO algorithm. It is important to note that

neighborhood topologies were determined using par-

ticle indexes and were not based on any spatial infor-

mation. For both clustering algorithms, the euclidean

distance (spatial information) has been used to deter-

mined the distance among particles.

For the basic PSO algorithm, we used w = 0.72

and c

= c

= 1.49. These values have been shown

to provide good results (Clerc and Kennedy, 2002;

van den Bergh, 2002; van den Bergh and Engelbrecht,

2006).

For all the algorithms used in this section, the

swarm size was s = 50. 200 iterations were perform

by each algorithm (24 algorithms, since there were

implemented 6 topologies + 2 clustering techniques

in 3 PSO variants). Every resulting approach were ex-

ecuted 30 independent runs. These values were used

as defaults for all experiments which use static con-

trol parameters. Also, the distribution of the particles

were 10 ×5 when the mesh and toroidal topologies

were used. For hierarchical and k-means clustering

algorithms, 4 groups were asked for.

Since there are 3 algorithms, 8 different neighbor-

hood topologies, and 9 test functions, it was difﬁcult

to show numerical results. In order to present such

results in a friendly-comparison way, we selected to

show the numerical results as box-plot graphics. In

Figures 2 and 3 shows such results.

From Figures 2 and 3, it is easily to see that the

BBPSO (EXP) version is highly dependent of the in-

terconnection topology, since it only presented a good

behavior when the fully connected, star and hierarchi-

cal approaches were used. However, when the remain

topologies were used, it presented the worst results

among the three algorithms. Therefore, we can say

that PSO and BBPSO algorithms showed a more ro-

bust behavior.

When PSO was used, the best topologies which

showed better results were toroidal, ring and mesh

(see Figures 2(a), 2(b), 2(c), 2(e), 3(b), 3(c), 3(d),

which corresponds to the sphere, schwefel, step, ro-

tated hyper-ellipsoid, ratrigin, ackley and griewank

test functions, respectively). For the generalized

schwefel test function (see Figure 3(a)) the k-means

clustering algorithm was the approach which per-

form the best. For the Rosenbrock test function all

topologies and clustering algorithms performed well.

In summary, when toroidal, mesh and ring topolo-

gies were used, PSO presented an good performance.

When BBPSO was used, the topologies which per-

formed better were full, star and hierarchical cluster-

ing algorithms in all test functions (see Figures 2 and

3).

In our opinion, BBPSO presented the best perfor-

mance, since it outperformed the other two PSO ap-

proaches in six out of nine test functions (see results

shown in Figures 2(a), 2(b), 2(c), 3(b), 3(c) and 3(d)

which corresponds to sphere, schwefel, step, ratrigin,

ackley and griewank test functions). BBPSO obtained

similar results with respect to the results obtained by

PSO in Rosenbrock test function (see results shown in

Figure 2(d)). The original PSO algorithm outperform

the others two PSO approaches in two out of nine test

functions (see results shown in Figures 2(e) and 3(a)

which corresponds to the rotated hyper-ellipsoid and

the generalized schwefel test functions).

From our results, we can conclude that, the topo-

logy plays a key role in PSO. The original PSO ap-

proach should be used with the toroidal, mesh and

ring topologies, whilst BBPSO should be used with

the fully connected, star or hierarchical clustering

methods.

5 CONCLUSIONS AND FUTURE

WORK

Our main conclusions are the following:

• We found that the use of mesh, toroidal and ring

topologies promote better convergence rates in the

PSO algorithm.

• The use of the fully connected, star and hierarchi-

cal clustering approaches promote better conver-

gence rates in the BBPSO algorithm.

• The topology most widely used (fully connected

topology) did not perform well in PSO algorithm

IJCCI 2009 - International Joint Conference on Computational Intelligence

156

whilst presented a good performance in BBPSO.

• Ring topology (which it is the another topology

widely used) presented a good convergence rate.

• The good selection of a topology can increase the

performance of a PSO-based algorithm.

Some possible paths to extend this work are the fol-

lowing:

• Experiment with other PSO’s models and diffe-

rential evolution algorithms.

• To include the parameter’s values w, c

and c

a similar study, in order to identify the relation

among parameters (including the topology).

ACKNOWLEDGEMENTS

The ﬁrst author acknowledges support from CONA-

CyT through a scholarship to pursue graduate stu-

dies at the Information Technology Laboratory at

CINVESTAV-IPN. The second author gratefully ac-

knowledges support from CONACyT through project

90548. Also, This research was partially funded by

project number 51623 from “Fondo Mixto Conacyt-

Gobierno del Estado de Tamaulipas”

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0 5000 10000 15000

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(a) Sphere test function

0 20 40 60 80 100 120

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(b) Schwefel test function

0 5000 10000 15000

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

0e+00 2e+07 4e+07 6e+07 8e+07

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(d) Rosenbrock

0 10000 20000 30000 40000 50000

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(e) Rotated hyper-ellipsoid function

Figure 2: Box-plots produced from the results of 30 independent runs for the ring, mesh, toroidal fully connected, star, and

tree topologies; and k-means and hierarchical clustering algorithms.

IJCCI 2009 - International Joint Conference on Computational Intelligence

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−40000 −30000 −20000 −10000

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(a) Generalized Schwefel test function

50 100 150 200 250

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(b) Ratrigin test function

0 5 10 15 20

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

0 50 100 150

f(t)

PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP PSO BB EXP

Ring Mesh Toroidal Full Star Tree K−means Hierarchical

(d) Griewank test function

Figure 3: Box-plots produced from the results of 30 independent runs for the ring, mesh, toroidal fully connected, star, and

tree topologies; and k-means and hierarchical clustering algorithms.

A COMPARATIVE STUDY OF NEIGHBORHOOD TOPOLOGIES FOR PARTICLE SWARM OPTIMIZERS

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