2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

Particle Swarm Optimization with Novel Concept of Complex Network
Haruna Matsushita† , Yoshifumi Nishio‡ and Toshimichi Saito††Dept. of Electrical and Electronics Engineering, Hosei University
3–7–2 Kajino-cho, Koganei, Tokyo 184–8584, Japan‡Dept. of Electrical and Electronics Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima, 774–8506 Japan
Email: †{haruna.matsushita.pd, tsaito}@k.hosei.ac.jp, ‡nishio@ee.tokushima-u.ac.jp
Abstract—The paper proposes a novel concept of acomplex network for the Particle Swarm Optimization(PSO); Independent-minded PSO (IPSO). Particles of thestandard PSO always ﬂy toward its own past best position(pbest) and the best position among the swarm (gbest). On
the other hand, in the proposed IPSO, whether each particle
and the swarm aﬀected each other is stochastically decided
according to a ﬁxed parameter called “Cooperativeness”.
We conﬁrm behavior of IPSO and its eﬀectiveness by applying it to various benchmarks.

We apply the proposed IPSO to four benchmark optimization functions containing unimodal and multimodal
functions, and investigate the behavior of IPSO by carryingout the simulation using diﬀerent the probability of independence. IPSO is compared with various PSO, and fromresults, we conﬁrm that IPSO can eﬀectively enhance thesearching eﬃciency for the multimodal functions. The results show that it is better that the particles value own information and are sometimes aﬀected by the swarm, instead ofbeing completely dependent on others.

1. Introduction

2. Independent-minded Particle Swarm Optimizer(IPSO)

Particle Swarm Optimization (PSO) [1] is an algorithm
to simulate the movement of ﬂocks of birds. Due to thesimple concept, easy implementation and quick convergence, PSO has attracted much attention and is used towide applications in diﬀerent ﬁelds in recent years. Eachparticle position is updated according to its personal best
position pbest and the best particle position among thewhole swarm gbest. In other words, all the particles are
fully-connected and always inﬂuence each other.
Meanwhile, in the real world, we are spending our life
inﬂuencing each other in the human community, and it isimportant not to just depend on other people, but also tohave own sense of independence.
On the other hand, various topological neighborhoods
for PSO have been considered by researches [2]–[7]. In
these search, each particle shares its best position amongneighboring particles on the network. In other words,
it is an application of the network topology to the particle swarm. Thus, investigations of the suitable network for PSO, especially using complex networks such as
small-world network [8], have attracted attention in theseyears [9], [10].
In this study, we propose a novel application of thecomplex network to PSO; an Independent-minded Particle
Swarm Optimization (IPSO). The most important featureof IPSO is that it is decided stochastically that each particle
depends on gbest or becomes independent from the swarm
and moves depending only on pbest. In other words, theparticles are not always connected each other, and they act
with self-reliance.

In the algorithm of the PSO, multiple potential solutions
called “particles” coexist. At each time step, each particle
ﬂies toward its own past best position (pbest) and the best
position among all particles (gbest). In other words, they
always inﬂuence each other. In this study, we propose thenovel concept of the complex network; the Independentminded PSO (IPSO). The particles of IPSO have independence, thus, it is decided stochastically whether they areconnected to others at every step. In other words, they are
not always aﬀected by gbest and their pbest does not always aﬀect the swarm.
Each particle has two informations; position and velocity. The position vector of each particle i and its velocity vector are represented by Xi = (xi1 , · · · , xid , · · · , xiD )and V i = (vi1 , · · · , vid , · · · , viD ), respectively, where (d =1, 2, · · · , D), (i = 1, 2, · · · , M).
(Step1) (Initialization) Let a generation step t = 0. Randomly initialize the particle position Xi (xid ∈ [xmin , xmax ]),initialize its velocity V i to zero, and initialize Pi =(pi1 , pi2 , · · · , piD ) with a copy of Xi . Evaluate the objective function f (Xi ) for each particle i and ﬁnd Pg with thebest function value among all the particles.
(Step2) Decide whether each particle i is connected tothe others according to randi which is a random number(∈ (0, 1)) for the particle i. If randi ≤ C, the particle i isconnected to other particles. If not, the particle i is isolated from the swarm, then, it and others does not aﬀect
each other. C is a constant cooperativeness coeﬃcientwhich is the independence probability of the particles.

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Table 1: Four Test Functions.
Function name

Test FunctionD∑
f1 (x) =xd2 ,Sphere func.;

f2 (x) =4th De Jong’s func.;

d=1
D
∑

dxd4 ,

d=1
D−1 (
∑

20 + e − 20e−0.2

Initialization Space

Criterionx ∈ [−5.12, 5.12]D ,0.01

x ∈ [−1.28, 1.28]D ,0.01

x ∈ [−30, 30]D ,1.0

x ∈ [−10, 10]D

10

√

2 )
0.5(xd2 +xd+1

Ackley’s func.;

f3 (x) =Stretched V sine wave func.;)−e0.5(cos(2πxd )+cos(2πxd+1 )) ,
D−1
∑
(
)
2
2
f4 (x) =(xd2 + xd+1)0.25 1 + sin2 (50(xd2 + xd+1
)0.1 ) ,
d=1

d=1

(a)

(b)

(c)

(d)

Figure 1: Test functions with two variables. First and second variables are on the x- and y-axis, respectively, and z-axis
shows its function value. (a) Sphere function. (b) 4th De Jong’s function. (c) Ackley’s function. (d) Stretched V sine wave
function.

(Step3) Evaluate the ﬁtness f (Xi ) for each particle i. Update the personal best position (pbest) as Pi = Xi if f (Xi ) <f (Pi ).
(Step4) Let Pg represents the best position with the best
pbest among particles being connected to others (gbest).
Update gbest Pg = (pg1 , pg2 , . . . , pgD ) according to
g = arg min f (Pi ), randi ≤ C.
i

(1)

In other words, even if the f (Pi ) is the minimum pbest
among all the particles, gbest is not updated if i is not connected to others.
(Step5) Update V i and Xi of each particle i according toV i (t + 1) =

wV i (t) + c1 r1 (Pi − Xi (t)) + c2 r2 (Pg









wV i (t) + c1 r1 (Pi − Xi (t)),
Xi (t + 1) = Xi (t) + V i (t + 1),−Xi (t)),randi ≤ C
randi > C(2)
(3)

where w is the inertia weight determining how much of theprevious velocity of the particle is preserved. c1 and c2 are
two positive acceleration coeﬃcients, generally c1 = c2 .
r1 and r2 are d-dimensional uniform random number vectors from U(0, 1). These equations mean that whether eachparticle is aﬀected by gbest is decided at random with thecooperativeness C. When C = 0, all the particles move depending only on own pbest, and when C = 1, the algorithm
is completely the same as the standard PSO.
(Step6) Let t = t + 1 and go back to (Step2).

3. Simulation
In order to evaluate the performance of IPSO, we use 4
benchmark optimization problems summarized in Table 1.
f1 and f2 are unimodal functions, and f3 and f4 are multimodal functions with numerous local minima. The optimum solution X∗ of all the functions are [0, 0, . . . , 0], andits optimum value f (X∗ ) is 0. All the functions have D variables, in this study, D = 30. The landscape maps of benchmark functions with two variables are shown in Fig. 1.
We compare IPSO with three PSOs; the standard PSO(PSO), PSO whose two acceleration coeﬃcients are different (PSO2) and IPSO which use either gbest or pbest(IPSO2). Features of each algorithm are follows:
PSO: This is the standard PSO and is completely same as
IPSO with C = 1. The velocities of all the particles are updated depending on its pbest and gbest at every generation.
PSO2: This is the standard PSO with diﬀerent accelerationcoeﬃcients. All the particles are updated depending on its
pbest and gbest at every generation, however the accelera-- 198 -

0

10

0

10
f (Pg)f (Pg)

PSO
PSO2
IPSO
IPSO2

PSO
PSO2
IPSO
IPSO2

−50

10

−50

10

−100

−2

Achievement rate [%]

−1

10

10

10

10

0

100
PSO
PSO2
IPSO
IPSO2

50
0 −3
10

−2

−1

10
10
Cooperativeness C

10

−3

−2

10
Achievement rate [%]

−3

10

0

−1

10

10

10

0

100
PSO
PSO2
IPSO
IPSO2

50
0 −3
10

−2

−1

10
10
Cooperativeness C

(a)

10

0

(b)
80
PSO
PSO2
IPSO
IPSO2

60

2

f (Pg)f (Pg)

10

PSO
PSO2
IPSO
IPSO2

10

−3

−2

10
Achievement rate [%]

20

−1

10

10

10

0 −3
10

0

Achievement rate [%]

0

100
50
0 −3
10

40

PSO
PSO2
IPSO
IPSO2
−2

−1

10
10
Cooperativeness C

10

0

(c)

−2

−1

10

10

10

0

100
50
0 −3
10

PSO
PSO2
IPSO
IPSO2

−2

−1

10
10
Cooperativeness C

10

0

(d)

Figure 2: Simulation results of three PSOs using diﬀerent C to four benchmarks. (a) Sphere function. (b) 4th De Jong’s
function. (c) Ackley’s function. (d) Stretched V sine wave function.

tion coeﬃcient of gbest depends on the value of C, as

3.1. Experimental Results

V i (t + 1) = wV i (t) + c1 r1 (Pi − Xi (t)) + c2 r2C(Pg − Xi (t)).
IPSO: This is the proposed PSO. It is decided according to
the cooperativeness C whether each particle is connected
to others. If the particle is connected to others, it is updated depending both on pbest and gbest, but otherwise, it
depends only on own pbest, according as Eq. (2).
IPSO2: In IPSO2, it is decided according to the cooperativeness C whether each particle is connected to others, as
the original IPSO. However, each particle is updated depending either on pbest or gbest aswV i (t) + c2 r2 (Pg − Xi (t)), randi ≤ C
V i (t + 1) = wV i (t) + c1 r1 (Pi − Xi (t)), randi > C.
For all PSOs, the population size M is set to 36, and theparameters are set as w = 0.7 and c1 = c2 = 1.6. In order to investigate the behavior of PSO2, IPSO and IPSO2,
we carry out simulations using diﬀerent cooperativeness Cfrom 0 to 1.0. The maximum generation are set to 3000
for all the benchmarks, and the results are evaluated in an
achievement rate of the criterion attainment over 100 trials.

Figure 2 shows respective mean results and theirachievement rate over 100 runs in diﬀerent cooperativenessC. Note that the standard PSO used C = 1.0 for all the simulations. The best mean result among diﬀerent C, and its
value of C and achievement rate [%] are listed in Table 2.
As shown in Figs. 2(a) and (b), in IPSO for the unimodal
functions, the case that all the particles are connected eachother at every generations was the most eﬀective, namely
C = 1.0. In other words, the standard PSO was the mostsuitable to unimodal functions. Then, let us consider theimportances of gbest and pbest for the unimodal functions
from the results of PSO2 and IPSO2. In the results of
PSO2, the larger acceleration rate of gbest was able to obtain better result, especially, it obtained eﬀective results
when the acceleration rate of gbest was about from 90 to
100 percent of that of pbest (C = 0.9–1.0). Moreover, for
IPSO2, the performance was eﬀective when the particles
moved depending on gbest in almost generations and on
pbest in about every 10 generations. From these results, we
can say that quick communication to the swarm is more im-- 199 -

Table 2: Comparison results of four PSOs on 4 test functions.
D

f
f1

f2
30
f3

f4

Best avg.
C
Achievement
Best avg.
C
Achievement
Best avg.
C
Achievement
Best avg.
C
Achievement

PSO

PSO2

IPSO

IPSO2

4.73e-51
100%
9.08e-76
100%
16.87
2%
18.6
7%

4.73e-51
1.0
100%
5.07e-99
0.9
100%
16.87
1.0
2%
18.68
1.0
7%

4.73e-51
1.0
100%
9.08e-76
1.0
100%
0.98
0.5
85%
6.92
0.06
86%

1.12e-220.9
100%3.02e-39
0.9
100%
8.01
0.5
25%
7.69
0.04
78%

portant for the unimodal functions than a particle diversity.
Meanwhile, for f1 and f2 , there were insigniﬁcant diﬀerences between the best mean results of four PSOs, as shownin Table 2. All the PSOs obtained 100% achievement rate.
In contrast, the changes in results depending on C for themultimodal functions were distinctly diﬀerent from that for
the unimodal functions. When the particles were little affected by gbest, the proposed IPSO obtained more eﬀective
results than when it was fully-connected (C = 1.0). As theresults of IPSO2, IPSO with about C = 0.5 was eﬀective
for Ackley function f3 , and C = 0.02–0.8 was suitable forStretched V sine wave function f4 . These results mean thatthe particle diversity is more important for the multimodal
functions than the quick communication. At the same time,
the proposed IPSO obtained high achievement rate as 85
and 86% for f3 and f4 , respectively. It is the signiﬁcant improved performances from the standard PSO, and the standard PSO has never obtained the same results although its
parameters were changed as PSO2.
From these results, we can say that instead of connecting to all the particles, we can obtain better performancethat the particles value own information and are sometimes
aﬀected by others. It is the interesting result, and the sameis equally true of the human relationship in the real world.

the multimodal functions, IPSO, whose particles were little aﬀected by the swarm, signiﬁcantly improved the fullyconnected PSO. This is because PSO with C < 1.0 brings
diversity to the swarm and the particles can easily get out
from the local optima. From these results, we can say thatinstead of connecting to all the particles, it is better that theparticles value own information and are sometimes aﬀectedby others. In addition, we have conﬁrmed a possibility ofan appropriate value of C depending on the kinds of optimization problems. Our future problem is to investigate
the association between the performance of IPSO and its
parameters such as the cooperativeness C and the population size M.
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4. Conclusions
In this study, we have proposed the Independent-minded
Particle Swarm Optimization (IPSO). The particles of
IPSO act with its sense of independence. Whether eachparticle is aﬀected by the swarm and itself also aﬀects theswarm is decided stochastically according to the ﬁxed parameter “Cooperativeness”.
We have applied IPSO to various benchmark functions
containing unimodal and multimodal functions. From results, we have conﬁrmed that the fully-connected IPSO,
namely the standard PSO, was suitable for the unimodal
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