UNCOVERING COMMUNICATION DENSITY IN PSO USING COMPLEX

NETWORK

Michal Pluhacek, Roman Senkerik, Adam Viktorin and Tomas Kadavy

Tomas Bata University in Zlin , Faculty of Applied Informatics

Nam T.G. Masaryka 5555, 760 01 Zlin, Czech Republic

{pluhacek, senkerik, aviktorin, kadavy}@fai.utb.cz

KEYWORDS

Swarm Intelligence, Particle Swarm Optimization,

Complex Network, Swarm communication.

ABSTRACT

In this study, we investigate the communication in

particle swarm optimization (PSO) by the means of

network visualization. We measure the communication

density of PSO optimizing four different benchmark

functions. It is presented that the communication density

varies over different fitness landscapes and in different

phases of the optimizing process. We analyze the results

in terms of use for future research.

INTRODUCTION

The Particle Swarm Optimization algorithm (PSO)

(Kennedy, Eberhart 1995, Shi, Eberhart, 1998, Kennedy

1997, Nickabadi et al., 2011) is known as one of the

leading metaheuristic optimizers. Heuristic methods are

widely used for solving industrial problems (Volná,

Kotyrba, 2016). In the past decades the inner dynamic of

the PSO algorithm has been studied in detail and many

modifications were proposed to tackle the known

weaknesses of the method (e.g. premature

convergence).

Recently the interconnection between metaheuristics

and complex networks (CNs) has been (Zelinka 2011a,

2011b, 2013, Senkerik et al., 2016) with interesting

results (Davendra, 2014a, 2014b).

We take inspiration in above mentioned examples of

interconnection of metaheuristics and CNs and use the

network-style visualization to uncover the density of

communication in the PSO. A network structure is

constructed from the inner communication of the swarm

and afterwards analyzed.

The rest of the paper is structured as follows: The PSO

is described in the next section, following is the

description of network construction process. The

experiment design is presented in the next section

followed by the results discussion. The paper concludes

with suggestion for future research.

PARTICLE SWARM OPTIMIZATION

The Particle Swarm Optimization algorithm (PSO) is an

evolutionary optimization algorithm based on the

natural behavior of birds. It was introduced by R.

Eberhart and J. Kennedy in 1995 (Kennedy, Eberhart

1995).

In the PSO algorithm the particles (representing

candidate solutions) fly in the multidimensional space

of possible solutions. The new position of the particle in

the next iteration is obtained as a sum of its actual

position and velocity. The velocity calculation follows

two natural tendencies of the particle: To move to the

best solution found so far by the particular particle

(personal best: pBest). And to move to the overall best

solution found in the swarm (global best: gBest).

In the original PSO the new position of particle is

altered by the velocity given by Eq. 1:

)(

)(

2

1

1

t

ijj

t

ijij

t

ij

t

ij

xgBestRandc

xpBestRandcvwv

(1)

Where: vi

t+1 - New velocity of the ith particle in iteration t+1.

w – Inertia weight value.

vit - Current velocity of the ith particle in iteration t.

c1, c2 - Priority factors.

pBesti – Local (personal) best solution found by the ith

particle.

gBest - Best solution found in a population.

xijt - Current position of the ith particle (component j of

the dimension D) in iteration t.

Rand1j, Rand2j – Pseudo random numbers, interval (0,

1).

The maximum velocity of particles in the PSO is typically limited to 0.2 times the range of the optimization problem and this pattern was followed in this study. The new position of a particle is then given by Eq. 2, where xi

t+1 is the new particle position:

11 t

i

t

i

t

i vxx(2)

Finally the linear decreasing inertia weight (Nickabadi

et al., 2011) is used in this study. Its purpose is to slow

the particles over time and improve the local search

capability in the later phase of the optimization. The

inertia weight has two control parameters wstart and wend.

A new w for each iteration is given by Eq. 3, where t

stands for current iteration number and n stands for the

total number of iterations.

Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD)

n

twwww endstartstart

(3)

NETWORK CONSTRUCTION

In this study we use the network structure as a tool to help use represent the communication in the swarm. The nodes in the network represent the particles in different time points (Particle ID with iteration code). This means that the theoretical maximal number of nodes in the network is the number of particles times the number of iterations. However a new node in the network is created only when a particle manages to find a new personal best solution (pBest). When a node is created, two links are also crated. First link is between the newly created node and previous node with the same particle ID (but different iteration code). This represents the information from pBest according to (1). Similarly the information from gBest represented by a link between the newly created node and a node that represents the last update of gBest.

THE EXPERIMENT

The following four well known test functions were used

in this study: Sphere function, Rosenbrock function,

Rastrigin function, Schwefel function with dimension

setting 10 and 100.

In the experiment the PSO was set in the following way:

Iterations: 1000;

Population size: 20;

c1,c2: 2;

wstart: 0.9;

wend: 0.4;

During the run of the algorithm the communication

network was constructed according to the rules

presented in the previous section.

Following is the visualization of the final networks. In the

network visualizations a color coding is used to differentiate

the phases of the run as percentage of the final number of cost

functions evaluations (CFE). (The first 20% of CFE are

represented by red color, magenta represents the 20-40% of

CFE, green is the 40-60% CFE., 60-80% CFE is represented

by yellow color and finally the 80-100% CFE is represented as

cyan).

The network visualizations for Sphere function are

presented in Fig. 1 (dim =10) and Fig. 2 (dim = 100)

alongside the gBest development in Fig. 3 and Fig. 4.

Similarly the network visualizations and gBest history

are presented in Fig. 5 – 8 for Rosenbrock function, Fig.

9 – 12 for Schwefel function in Fig 13 – 16 for Rastrigin

functions.

It is clear from the visualizations that the number of

newly created links in different phases of the algorithm

varies. The numerical representation of newly created

links is presented in Table 1 – 4.

Figure 1: Network visualization with highlighted phases - Sphere function; dim = 10

Figure 2: Network visualization with highlighted phases - Sphere function; dim = 100

Figure 3: gBest history - Sphere function; dim = 10

Figure 4: gBest history - Sphere function; dim = 100

Figure 5: Network visualization with highlighted phases - Rosenbrock function; dim = 10

Figure 6: Network visualization with highlighted phases - Rosenbrock function; dim = 100

Figure 7: gBest history - Rosebrock function; dim = 10

Figure 8: gBest history - Rosebrock function; dim = 100

Figure 9: Network visualization with highlighted phases - Schwefel function; dim = 10

Figure 10: Network visualization with highlighted phases - Schwefel function; dim = 100

Figure 11: gBest history - Sschwefel function; dim = 10

Figure 12: gBest history - Sschwefel function; dim = 100

Figure 13: Network visualization with highlighted phases – Rastrigin function; dim = 10

Figure 14: Network visualization with highlighted phases – Rastrigin function; dim = 100

Figure 15: gBest history - Rastrigin function; dim = 10

Figure 16: gBest history - Rastrigin function; dim = 100

Table 1: Newly created link overview – Sphere function

Newly created links by CFE %

dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100

10 448 334 730 1520 2396

100 584 604 550 678 1280

Table 2: Newly created link overview – Rosenbrock

function

Newly created links by CFE %

dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100

10 424 298 684 1030 1030

100 440 404 452 816 1156

Table 3: Newly created link overview – Schwefel

function

Newly created links by CFE %

dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100

10 330 126 350 1408 1810

100 268 128 466 754 1378

Table 4: Newly created link overview – Rastrigin

function

Newly created links by CFE %

dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100

10 248 294 942 1666 22

100 230 222 374 938 1472

RESULTS DISSCUSION

Firstly, according to the results presented in the

previous section it is clear that the number of newly

created nodes in different phases of the algorithm

varies. However when put into context with the history

of gBest, often the smallest part of newly created nodes

represents the most dramatic improvement of gBest

value and vice versa.

Secondly, the shape of the network and number of

newly created nodes seems to be affected by the fitness

landscapes in terms of modality and complexity.

Further, there seems to be a tendency for some particles

to improve after a very long time window without

improvement (possibly escaping local optima), this

trend can be observed namely in Figs. 5, 10 and 13.

In most cases the majority of the newly created nodes is

crated in the last phases of the optimization. This is

most likely due to the decreasing inertia weight and

small velocities of the particles.

CONCLUSION

In this study we have presented the possible use of

network visualization to highlight the trends in

communication density in the particle swarm

optimization. We have concluded that the

communication density varies significantly in different

phases of the optimization and also varies based on the

fitness landscape.

There are two main directions for our future research.

First is employment of more advanced network analysis

for classification of various fitness landscapes and

second is a feedback-loop style control of the swarm

based on the number of newly created links in a

specified time window.

ACKNOWLEDGEMENT

This work was supported by Grant Agency of the Czech

Republic – GACR P103/15/06700S, further by the

Ministry of Education, Youth and Sports of the Czech

Republic within the National Sustainability Programme

Project no. LO1303 (MSMT-7778/2014. Also by the

European Regional Development Fund under the

Project CEBIA-Tech no. CZ.1.05/2.1.00/03.0089 and

by Internal Grant Agency of Tomas Bata University

under the Projects no. IGA/CebiaTech/2017/004

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Davendra, D., Zelinka, I., Senkerik, R. and Pluhacek, M. Complex Network Analysis of Discrete Self-organising Migrating Algorithm, in: Zelinka, I. and Suganthan, P. and Chen, G. and Snasel, V. and Abraham, A. and Rossler, O. (Eds.) Nostradamus 2014: Prediction, Modeling and Analysis of Complex Systems, Advances in Intelligent Systems and Com-puting, Springer Berlin Heidelberg, pp. 161–174 (2014).

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Kennedy J., “The particle swarm: social adaptation of knowledge,” in Proceedings of the IEEE International Conference on Evolutionary Computation, 1997, pp. 303–308.¨

Nickabadi A., Ebadzadeh M. M., Safabakhsh R., A novel particle swarm optimization algorithm with adaptive inertia weight, Applied Soft Computing, Volume 11, Issue 4, June 2011, Pages 3658-3670, ISSN 1568-4946

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AUTHOR BIOGRAPHIES

MICHAL PLUHACEK was born in the Czech

Republic, and went to the Faculty of

Applied Informatics at Tomas Bata

University in Zlín, where he studied

Information Technologies and obtained

his MSc degree in 2011 and Ph.D. in 2016

with the dissertation topic: Modern

method of development and modifications of

evolutionary computational techniques. He now works

as a researcher at the same university. His email address

is: pluhacek@fai.utb.cz

ROMAN SENKERIK was born in the Czech Republic,

and went to the Tomas Bata University in

Zlin, where he studied Technical

Cybernetics and obtained his MSc degree

in 2004, Ph.D. degree in Technical

Cybernetics in 2008 and Assoc. prof. in

2013 (Informatics). He is now an Assoc.

prof. at the same university (research and courses in:

Evolutionary Computation, Applied Informatics,

Cryptology, Artificial Intelligence, Mathematical

Informatics). His email address is: senkerik@fai.utb.cz

ADAM VIKTORIN was born in the Czech Republic,

and went to the Faculty of Applied

Informatics at Tomas Bata University in

Zlín, where he studied Computer and

Communication Systems and obtained his

MSc degree in 2015. He is studying his

Ph.D. at the same university and the field

of his studies are: Artificial intelligence, data mining

and evolutionary algorithms. His email address is:

aviktorin@fai.utb.cz

TOMAS KADAVY was born in the Czech Republic,

and went to the Faculty of Applied

Informatics at Tomas Bata University in

Zlín, where he studied Information

Technologies and obtained his MSc degree

in 2016. He is studying his Ph.D. at the

same university and the fields of his

studies are: Artificial intelligence and evolutionary

algorithms. His email address is: kadavy@fai.utb.cz