Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 409167 8 pageshttpdxdoiorg1011552013409167

Research ArticleA New Logistic Dynamic Particle Swarm OptimizationAlgorithm Based on Random Topology

Qingjian Ni12 and Jianming Deng1

1 School of Computer Science and Engineering Southeast University Nanjing 211189 China2 Provincial Key Laboratory for Computer Information Processing Technology Soochow University Suzhou 215006 China

Correspondence should be addressed to Qingjian Ni niqingjiangmailcom

Received 17 April 2013 Accepted 19 May 2013

Academic Editors P Agarwal S Balochian and V Bhatnagar

Copyright copy 2013 Q Ni and J Deng This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Population topology of particle swarm optimization (PSO) will directly affect the dissemination of optimal information during theevolutionary process and will have a significant impact on the performance of PSO Classic static population topologies are usuallyused in PSO such as fully connected topology ring topology star topology and square topology In this paper the performance ofPSO with the proposed random topologies is analyzed and the relationship between population topology and the performance ofPSO is also explored from the perspective of graph theory characteristics in population topologies Further in a relatively new PSOvariant which named logistic dynamic particle optimization an extensive simulation study is presented to discuss the effectivenessof the random topology and the design strategies of population topology Finally the experimental data are analyzed and discussedAnd about the design and use of population topology on PSO some useful conclusions are proposed which can provide a basis forfurther discussion and research

1 Introduction

Particle swarm optimization (briefed as PSO) is a kind ofbionic evolutionary algorithm which rooted in imitation ofbehavioral mechanisms in populations such as birds andfish stocks and has been widely used in engineering field asoptimization method [1ndash3]

In the PSO algorithms the particles evolve according totheir own experience and the experience of the neighborhoodparticles During the evolutionary process particles identifytheir own neighborhood according to the population topol-ogy and then learn from each other and update the positionsof the particles Therefore population topology determinesthe form of information sharing among particles and thushas a very important impact on the solving performance ofPSO algorithms Therefore it is important to explore thepopulation topologies of PSO algorithms This will producea deep understanding of the working mechanism of PSOalgorithms and thus improve the solving performance

In the PSOalgorithms themost commonused static pop-ulation topologies are the fully connected topology (Gbest

model) and the ring topology (lbest model) which are alsofirst proposed [4] Since then researchers have proposed dif-ferent population topologies in succession Kennedy carriedout a preliminary analysis of four static population topolo-gies [5] Suganthan adjusted the neighborhood structure ofparticles through calculating distances between particles inthe evolutionary process [6] Mendes et al detailly analyzedthe relationship between the population topology and a classof PSO variant [7ndash9] Clerc initially attempted to adopt therandom topology [10] However these studies concernedpopulation topologies paidmore attention to the static classicpopulation topology and research of random populationtopologies is relatively small As the population topologydirectly affect the exchange of information between theparticles it is necessary to design the suitable populationtopology according to the characteristics of different types ofapplications Therefore it is necessary to explore the popula-tion topologies in depth from a theoretical and experimentalpoint of view

In this paper in a relatively new PSO variant whichnamed logistic dynamic particle optimization we analyze the

2 The Scientific World Journal

linkages between population topologies and the performanceof PSO algorithms from graph theory and experimental pointof view The rest of the paper is organized as follows InSection 2 we describes the PSO variant which are used in thepaper Section 3 describes the classic population topologiesand introduces the proposed random population topologiesIn Section 4 we have presented the experimental analysisand comparative performance between the classic and theproposed randompopulation topologies Section 5 concludesthe paper

2 The PSO Variants

PSO is a population-based method which is similar to otherevolutionary computation methods The individual in thePSO population is called the particle and particles generallyhave the speed and position in most PSO variants

21 The Canonical PSO Based on the earlier version ofPSO Clerc developed the PSO with the compression factor[11] This PSO variants have been widely used in practicalapplications and the velocity and position update of particlesin this variant are as follows

V119894119889= 120594 lowast [V

119894119889+ 1198881lowast rand ( ) lowast (119901

119894119889minus 119909119894119889)

+ 1198882lowast Rand ( ) lowast (119901

119892119889minus 119909119894119889)]

(1)

120594 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic120593

2minus 4120593

1003816100381610038161003816100381610038161003816

(2)

119909119894119889= 119909119894119889+ V119894119889 (3)

In (1) V119894119889is the 119889th dimensional component of particle

119894s velocity attribute 1198881and 1198882are two positive acceleration

factors rand() and Rand() are random number generatingfunctions between 0 and 1 119909

119894119889is the 119889th dimensional

component of the particle 119894s position property 119901119894119889is the 119889th

dimensional component of the best position that particle 119894obtained and 119901

119892119889is the 119889th dimensional component of the

best position that the whole population obtained In actualuse usually 120594 in (2) is set to 0729 and 120593 is often set to 41

The right part of the equation1 can be understood asparticlersquos memory cognitive and social cognition Velocityof particles is precisely through this three-part interactioneffects thereby position of particles is updated

22 The Dynamic Probabilistic Particle Swarm OptimizationIn the previous PSO variants particles usually have bothvelocity attribute and position attribute Kennedy first pro-posed a new PSO variant without velocity attribute whichnamed Gaussian dynamic particle swarm optimization [12]Ni and Deng carried out a further study on the PSO variantswithout velocity [13] This type of algorithm variants canbe called dynamic probabilistic particle swarm optimization(briefed as DPPSO) In the DPPSO algorithms particles have

no velocity attributes and the update of particlesrsquo positions isreorganized as follows

119883119894 (119905 + 1) = 119883

119894 (119905) + 120572 lowast (119883

119894 (119905) minus 119883

119894 (119905 minus 1))

+ 120573 lowast CT119894 (119905) + 120574 lowast Gen ( ) lowastOT

119894 (119905)

(4)

CT119894119889 (

119905) =

119870

sum

119896=1

119875119896119889

119870 minus 119883119894119889 (

119905)

(5)

OT119894119889 (

119905) =

119870

sum

119896=1

1003816100381610038161003816

119875119894119889minus 119875119896119889

1003816100381610038161003816

119870

(6)

In (4) (5) and (6) 119905 represents the current evolutiongeneration of particle 119894 is the index number of particle119896 is the index number of particlersquos neighborhood 119870 rep-resents the number of particlersquos neighborhood particles119875119896is the optimum position of the particlersquos neighborhood

that numbered 119896 and 119889 is the number of the particles 119894sdimension CT

119894(119905) is an abbreviation of centralized tendency

which is a 119863-dimensional vector and is determined bythe particlersquos current location and neighborhood particlesrsquooptimal positions OT

119894(119905) is an abbreviation of outlier trend

which is also a 119863-dimensional vector and is determinedby the particlersquos current location and neighborhood par-ticlesrsquo optimal positions In (4) 120572 120573 and 120574 are gener-ally preferable to positive constants Gen() is a dynamicprobabilistic evolutionary operator which is the randomnumber generator function which satisfies a specific dis-tribution and this particular distribution may be providedby a Gaussian distribution or a logistic distribution and soforth

In (4) the calculation of particlersquos position in new gener-ation is decided jointly by the right four partsThe first part isthe memory of particles on the self-positionThe second partis the trend of the particles along the previous direction ofmovement of the ldquoflyingrdquo The third part means the influenceof neighborhood particlesrsquo experience to the new generationposition this part of the calculations needs neighborhoodparticlesrsquo experience and this part determines the degree ofinfluence of the neighborhood particlesrsquo experienceThe partIV reflects the impact of differences in best position betweenparticles on the next generation position

The performance of DPPSO variants is different whenusing different dynamic probabilistic evolutionary opera-tor Gen() [13] DPPSO-Gaussian (briefed as GDPS) hasfaster convergence speed in the early evolution DPPSO-Cauchy may get better solutions on certain issues but theperformance is unstable DPPSO-Logistic (logistic dynamicparticle swarm optimization briefed as LDPSO) still showsgood exploration ability in the later evolution For DPPSOvariants the calculation of CT

119894(119905) and OT

119894(119905) will use

the experience of each neighborhood particles which canalso be seen from (5) and (6) The optimal informationof neighborhood particles could be fully utilized so theresearch of population topology in DPPSO variants is moreimportant

The Scientific World Journal 3

(a) (b) (c)

Figure 1 (a) Fully connected topology (b) ring topology (c) star topology

3 Population Topologies

31 The Classic Population Topologies Figure 1 shows thefully connected topology ring topology and star topologyFully connected topology and ring topology are two com-monly used topologies which are also called Gbest modeland Lbest model In the fully connected topology particlersquosneighborhood contains all particles in the population Andin the evolutionary process only the particle that obtains theoptimal position is considered in the entire population PSOalgorithms with this topology converge very fast but easy tofall into local optimum

For a ring topology typically particlersquos neighborhoodincludes the particles on both sides of one or a few particlesIn this topology the exchange of information is relativelyslow within the population but once a particle searched foran optimal location the information eventually will slowlyspread to the entire population

For a star topology one particle is connected with allthe other particles and other particles only connect with theparticle In addition to a central particle other particles areindependent of each other the dissemination of informationmust be passed through the central particle

However the information dissemination mechanism ofthe social groups is not static throughout the whole evo-lutionary process and tends to have a certain degree ofrandomness and dynamic characteristics Mendes studiedrandom population topology and confirmed that populationtopology directly affects the execution performance of PSOalgorithms [9]

32 GraphTheory Characteristics of Population Topology Thepopulation topology of the PSO algorithm can be abstractedinto a connected undirected graph represented by the symbol119866(119881 119864) where119881 is the set of vertices119864 is the set of edges andthe number of vertices is denoted by 119899 For any two points 119906and V in 119866 119889(119906 V) denotes the distance from 119906 to V that isthe length of the shortest path between two points

Definition 1 (average degree) The degree of the vertex V is thenumber of its adjacent vertices denoted by 119896V Average degree119896 of undirected connected graph is calculated by

119870 = sum

Visin119881

119896V

119899

(7)

Definition 2 (average clustering coefficient) The local clus-tering coefficient 119888(V) of vertex V equals to the number ofedges which can be connected between vertices associatedwith vertex V divided by the maximum number of edgesbetween these vertices The average clustering coefficient 119888of a graph is the arithmetic mean of the local clusteringcoefficient of all vertices which can be calculated by

119888 = sum

Visin119881

119888 (V)

119899

(8)

The average degree of population topology means theaverage number of particlesrsquo neighborhood particles it rep-resents the degree of socialization of population A smallnumber of neighborhood particles means that the particleis both difficult to obtain information from the populationand difficult to influence other particles On the contrary aparticle which have large number of neighborhood particlescan get a lot of information available in the population andsuch a particle has a greater influence in the populationIn the common used population topologies fully connectedtopology has the largest average degree which is equal to thepopulation size minus 1The ring topology has the minimumaverage degree the average degree of a ring topology suchas in Figure 1 is equal to 2 The local clustering coefficientis the ratio of the number of connections between theactual existence and the possible existence and the averageclustering coefficient represents the degree of aggregationof the vertices in a graph which is the average of the localclustering coefficient of all vertices In this paper we willanalyze the role of the population topology based on theprevious graph theory characteristics

33 Random Population Topologies Clerc initially attemptedto proposed a method of random population topology [10]The basic idea is to generate a random topology by selectingthe neighborhood particles randomly for each particle Theconcrete steps could be described as Algorithm 1

In the resulting matrix 119871 of Algorithm 1 119871(119906 V) = 1

means that the particles 119906 and V are connected By this

4 The Scientific World Journal

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

119898 from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 1(3) For the matrix 119871 If 119871(119894 119898) = 1 then let 119871 (119898 119894) = 1

Algorithm 1 Clercrsquos generating method of random population topology

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

m (119898 = 119894) from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 119871 (119898 119894) = 1(3) Use the Dijkstra algorithm to calculate the distance between the particles and stored in the 119878 lowast 119878matrix119863(4)While The graph corresponding to the generating random topology is not connected do(5) Scan the matrix119863(6) IF two particles 119906 and V are not connected(7) Let 119871 (119906 V) = 1

Algorithm 2 The proposed generating method of random population topology

method a random topology could be generated with anaverage degree slightly larger than119870 In the random topologyby this method the distribution of degree is the sum of 119878 minus 1

independent Bernoulli random variables which is describedin [10]

prob (119884 = 119899) = 119862

119899minus1

119878minus1(

119870

119878

)

119899minus1

(1 minus

119870

119878

)

119904minus119899

(9)

In this paper in order to ensure the connectivity ofundirected graph which is corresponding to the generatingrandom topology we proposed a new generating method ofrandom topology based on Clercrsquos method The basic idea isin selecting the neighborhood particles for each particle if theselection is itself reselect in order to reduce the probability ofparticles isolated after the random population is generateduse the Dijkstra algorithm to compute the distance betweenthe particles if there exist the unconnected particles inthe generated topology then add an edge between theseunconnected particles and retest The improved randompopulation topology generating method is as Algorithm 2

4 Experiment and Analysis

41 Experiment Setting Two sets of experiments were con-ducted which used the canonical PSO (briefed as CPSO)and the DPPSO-Logistic respectively that are described inSection 2 For CPSO set 119888

1= 1198882= 205 120594 = 07298 For

DPPSO-Logistic set 120572 = 0729 120573 = 2187 and 120574 = 05The algorithms were used to solve five benchmark func-

tions which is defined in Table 1 These functions consist ofAckley Schwefel Schafferrsquos F6 Rastrigin and Sphere Table 2shows the settings of these functions

In the experiment the population size is set to 20 inaddition to the the Schafferrsquos F6 function of the dimension2 the remaining functions are carried out in the case of 30-dimensional test and the frequency of repeated experimentsis 50

The performance of the algorithms will be evaluated bythe following aspects

(i) in the case of a certain number of iterations comparethe accuracy (briefed as Perform) of the optimalfitness value in each case These values reflect thequality of the optimal solution obtained in the last

(ii) in the case of a certain number of iterations comparethe success rates (briefed as Prop) which means thatthe algorithms achieve the accuracy (accepted error)that is defined in Table 2 These data reflect thestability of the algorithms

(iii) in the case of a certain number of iterations comparethe evolutionary trends of various algorithms thesefigures reflect the evolution of the optimal solution inthe evolutionary process

42 Comparison between Random Topologies and DifferentAverage Degrees For the random topology the first set ofexperiments used the canonical PSO algorithm to solve thefive benchmark functions By changing the 119870 value wegenerated random topologies with different average degreesfor comparison and evaluation Experimental results areshown in Table 3

As can be seen fromTable 3 with the increase in the valueof 119870 the indicators of Perform and Prop have improvedFor multimodal functions such as Schwefel Schafferrsquos F6Rastrigin and Ackley function with the increase in the valueof119870 when the119870 value is from 3 to 4 (ie the average degreeof the random topology is between 5 and 7 substantially inabout 6) the PSO algorithm could get better performanceAnd when the 119870 value is larger the performance is usuallypoor For multimodal functions when a particle has found alocal optimal solution if the interaction between particles ismore the dissemination of information will be very quicklyand the entire population is susceptible to rapid convergenceto the local optimal solution Therefore too large average

The Scientific World Journal 5

Table 1 Definition of benchmark functions

Benchmark function Formula

Sphere 119865() =

119899

sum

119894=1

119909

2

119894

Schafferrsquos F6 119865() = 05 +

(sin2radic1199092 + 119910

2) minus 05

(10 + 0001 (119909

2+ 119910

2))

2

Rastrigin 119865() = 10 lowast 119899 +

119899

sum

119894=1

(119909

2

119894minus 10 cos(2120587119909

119894))

Schwefel 119865() = 4189829 lowast 119899 +

119899

sum

119894=1

119909119894sin(radic|119909

119894|)

Ackley 119865() = minus20 sdot exp(minus02radic 1

119899

sdot

119899

sum

119894=1

119909

2

119894) minus exp(1

119899

sdot

119899

sum

119894=1

cos (2120587119909119894)) + 20 + exp(1)

Table 2 Settings of benchmark functions

Benchmark function Dimension Optimal value Optimal solution Range Accepted errorSphere 30 0 (0 0 0 0) (minus100 100) 001Schafferrsquos F6 2 0 (0 0) (minus100 100) 000001Rastrigin 30 0 (0 0 0 0) (minus512 512) 100Schwefel 30 0 (0 0 0 0) (minus500 500) 6000Ackley 30 0 (0 0 0 0) (minus30 30) 5

degree of population topology is not conducive to find theglobal optimal solution for a multimodal function

Taking these factors together when the value of 119870 isat about 4 the PSO algorithm has a more satisfactoryperformance for most benchmark functions Accordinglythe following experiments will generate random topologieswhich the 119870 value is 4 and compare these topologies withother classic static population topologies

43 Comparison between Random Topologies and ClassicTopologies The second set of experiments used the DPPSO-Logistic algorithm to solve the five benchmark functions onthe fully connected topology ring topology star topologyand random topology wherein 119870 value is set to 4 togenerate random topologies Comparison and evaluation areconducted by the evolutionary trends of algorithms withvarious population topologies In the figures of evolution-ary trends different line types expressed different DPPSO-Logistic algorithms with various population topologies

For Sphere function (Figure 2) the performance of ringtopology and star topology is poor and the fully connectedtopology and random topology show better search abilityFor Schafferrsquos F6 function (Figure 2) random topologyis significantly better than the three classic neighborhoodtopologies

For Rastrigin function (Figure 3) random topology issuperior to the three classic population topologies in theearly stages of evolution the convergence speed of therandom topology is almost the same as the fully connectedtopology however in the later stage of evolution randomtopology shows a larger advantage and the end result is betterthan the fully connected topology

For Schwefel function (Figure 3) the performance ofthe ring topology is poor the fully connected topologyconvergence fast in the early stage of evolution but the endresult is poor the star topology has achieved good results theconvergence speed of random topology is second only to thefully connected topology in the early evolution stage and thefinal result of random topology is better than the other classictopologies

For Ackley function (Figure 4) the fully connectedtopology and the star topology show poor performance thering topology performs better the random topology showsobvious advantage in both the convergence speed and thefinal result

Overall according to the convergence speed randomtopology is relatively stable in the early stages of evolutionfaster in the midstages of evolution and shows a distinctadvantage in the late stages of evolution From the viewof final result the PSO algorithms using random topologydemonstrate remarkable performance

For unimodal function (such as Sphere function)because there is no problem of falling into a local optimumthe close ties between particles can make faster convergenceand achieve better results Therefore the performance ofthe fully connected topology and random topology with arelatively high average degree is ideal In the case of randomtopology for unimodal function the convergence speed andthe solution will be better with the greater average degree ofpopulation topology

For multimodal function it can be seen that the con-vergence speed will be faster when the average degree ofpopulation topology is increasing If the average degree ofpopulation topology is too high it is easy to fall into localoptimum On the average degree after 6 the optimal solution

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

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2 The Scientific World Journal

linkages between population topologies and the performanceof PSO algorithms from graph theory and experimental pointof view The rest of the paper is organized as follows InSection 2 we describes the PSO variant which are used in thepaper Section 3 describes the classic population topologiesand introduces the proposed random population topologiesIn Section 4 we have presented the experimental analysisand comparative performance between the classic and theproposed randompopulation topologies Section 5 concludesthe paper

2 The PSO Variants

PSO is a population-based method which is similar to otherevolutionary computation methods The individual in thePSO population is called the particle and particles generallyhave the speed and position in most PSO variants

21 The Canonical PSO Based on the earlier version ofPSO Clerc developed the PSO with the compression factor[11] This PSO variants have been widely used in practicalapplications and the velocity and position update of particlesin this variant are as follows

V119894119889= 120594 lowast [V

119894119889+ 1198881lowast rand ( ) lowast (119901

119894119889minus 119909119894119889)

+ 1198882lowast Rand ( ) lowast (119901

119892119889minus 119909119894119889)]

(1)

120594 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic120593

2minus 4120593

1003816100381610038161003816100381610038161003816

(2)

119909119894119889= 119909119894119889+ V119894119889 (3)

In (1) V119894119889is the 119889th dimensional component of particle

119894s velocity attribute 1198881and 1198882are two positive acceleration

factors rand() and Rand() are random number generatingfunctions between 0 and 1 119909

119894119889is the 119889th dimensional

component of the particle 119894s position property 119901119894119889is the 119889th

dimensional component of the best position that particle 119894obtained and 119901

119892119889is the 119889th dimensional component of the

best position that the whole population obtained In actualuse usually 120594 in (2) is set to 0729 and 120593 is often set to 41

The right part of the equation1 can be understood asparticlersquos memory cognitive and social cognition Velocityof particles is precisely through this three-part interactioneffects thereby position of particles is updated

22 The Dynamic Probabilistic Particle Swarm OptimizationIn the previous PSO variants particles usually have bothvelocity attribute and position attribute Kennedy first pro-posed a new PSO variant without velocity attribute whichnamed Gaussian dynamic particle swarm optimization [12]Ni and Deng carried out a further study on the PSO variantswithout velocity [13] This type of algorithm variants canbe called dynamic probabilistic particle swarm optimization(briefed as DPPSO) In the DPPSO algorithms particles have

no velocity attributes and the update of particlesrsquo positions isreorganized as follows

119883119894 (119905 + 1) = 119883

119894 (119905) + 120572 lowast (119883

119894 (119905) minus 119883

119894 (119905 minus 1))

+ 120573 lowast CT119894 (119905) + 120574 lowast Gen ( ) lowastOT

119894 (119905)

(4)

CT119894119889 (

119905) =

119870

sum

119896=1

119875119896119889

119870 minus 119883119894119889 (

119905)

(5)

OT119894119889 (

119905) =

119870

sum

119896=1

1003816100381610038161003816

119875119894119889minus 119875119896119889

1003816100381610038161003816

119870

(6)

In (4) (5) and (6) 119905 represents the current evolutiongeneration of particle 119894 is the index number of particle119896 is the index number of particlersquos neighborhood 119870 rep-resents the number of particlersquos neighborhood particles119875119896is the optimum position of the particlersquos neighborhood

that numbered 119896 and 119889 is the number of the particles 119894sdimension CT

119894(119905) is an abbreviation of centralized tendency

which is a 119863-dimensional vector and is determined bythe particlersquos current location and neighborhood particlesrsquooptimal positions OT

119894(119905) is an abbreviation of outlier trend

which is also a 119863-dimensional vector and is determinedby the particlersquos current location and neighborhood par-ticlesrsquo optimal positions In (4) 120572 120573 and 120574 are gener-ally preferable to positive constants Gen() is a dynamicprobabilistic evolutionary operator which is the randomnumber generator function which satisfies a specific dis-tribution and this particular distribution may be providedby a Gaussian distribution or a logistic distribution and soforth

In (4) the calculation of particlersquos position in new gener-ation is decided jointly by the right four partsThe first part isthe memory of particles on the self-positionThe second partis the trend of the particles along the previous direction ofmovement of the ldquoflyingrdquo The third part means the influenceof neighborhood particlesrsquo experience to the new generationposition this part of the calculations needs neighborhoodparticlesrsquo experience and this part determines the degree ofinfluence of the neighborhood particlesrsquo experienceThe partIV reflects the impact of differences in best position betweenparticles on the next generation position

The performance of DPPSO variants is different whenusing different dynamic probabilistic evolutionary opera-tor Gen() [13] DPPSO-Gaussian (briefed as GDPS) hasfaster convergence speed in the early evolution DPPSO-Cauchy may get better solutions on certain issues but theperformance is unstable DPPSO-Logistic (logistic dynamicparticle swarm optimization briefed as LDPSO) still showsgood exploration ability in the later evolution For DPPSOvariants the calculation of CT

119894(119905) and OT

119894(119905) will use

the experience of each neighborhood particles which canalso be seen from (5) and (6) The optimal informationof neighborhood particles could be fully utilized so theresearch of population topology in DPPSO variants is moreimportant

The Scientific World Journal 3

(a) (b) (c)

Figure 1 (a) Fully connected topology (b) ring topology (c) star topology

3 Population Topologies

31 The Classic Population Topologies Figure 1 shows thefully connected topology ring topology and star topologyFully connected topology and ring topology are two com-monly used topologies which are also called Gbest modeland Lbest model In the fully connected topology particlersquosneighborhood contains all particles in the population Andin the evolutionary process only the particle that obtains theoptimal position is considered in the entire population PSOalgorithms with this topology converge very fast but easy tofall into local optimum

For a ring topology typically particlersquos neighborhoodincludes the particles on both sides of one or a few particlesIn this topology the exchange of information is relativelyslow within the population but once a particle searched foran optimal location the information eventually will slowlyspread to the entire population

For a star topology one particle is connected with allthe other particles and other particles only connect with theparticle In addition to a central particle other particles areindependent of each other the dissemination of informationmust be passed through the central particle

However the information dissemination mechanism ofthe social groups is not static throughout the whole evo-lutionary process and tends to have a certain degree ofrandomness and dynamic characteristics Mendes studiedrandom population topology and confirmed that populationtopology directly affects the execution performance of PSOalgorithms [9]

32 GraphTheory Characteristics of Population Topology Thepopulation topology of the PSO algorithm can be abstractedinto a connected undirected graph represented by the symbol119866(119881 119864) where119881 is the set of vertices119864 is the set of edges andthe number of vertices is denoted by 119899 For any two points 119906and V in 119866 119889(119906 V) denotes the distance from 119906 to V that isthe length of the shortest path between two points

Definition 1 (average degree) The degree of the vertex V is thenumber of its adjacent vertices denoted by 119896V Average degree119896 of undirected connected graph is calculated by

119870 = sum

Visin119881

119896V

119899

(7)

Definition 2 (average clustering coefficient) The local clus-tering coefficient 119888(V) of vertex V equals to the number ofedges which can be connected between vertices associatedwith vertex V divided by the maximum number of edgesbetween these vertices The average clustering coefficient 119888of a graph is the arithmetic mean of the local clusteringcoefficient of all vertices which can be calculated by

119888 = sum

Visin119881

119888 (V)

119899

(8)

The average degree of population topology means theaverage number of particlesrsquo neighborhood particles it rep-resents the degree of socialization of population A smallnumber of neighborhood particles means that the particleis both difficult to obtain information from the populationand difficult to influence other particles On the contrary aparticle which have large number of neighborhood particlescan get a lot of information available in the population andsuch a particle has a greater influence in the populationIn the common used population topologies fully connectedtopology has the largest average degree which is equal to thepopulation size minus 1The ring topology has the minimumaverage degree the average degree of a ring topology suchas in Figure 1 is equal to 2 The local clustering coefficientis the ratio of the number of connections between theactual existence and the possible existence and the averageclustering coefficient represents the degree of aggregationof the vertices in a graph which is the average of the localclustering coefficient of all vertices In this paper we willanalyze the role of the population topology based on theprevious graph theory characteristics

33 Random Population Topologies Clerc initially attemptedto proposed a method of random population topology [10]The basic idea is to generate a random topology by selectingthe neighborhood particles randomly for each particle Theconcrete steps could be described as Algorithm 1

In the resulting matrix 119871 of Algorithm 1 119871(119906 V) = 1

means that the particles 119906 and V are connected By this

4 The Scientific World Journal

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

119898 from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 1(3) For the matrix 119871 If 119871(119894 119898) = 1 then let 119871 (119898 119894) = 1

Algorithm 1 Clercrsquos generating method of random population topology

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

m (119898 = 119894) from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 119871 (119898 119894) = 1(3) Use the Dijkstra algorithm to calculate the distance between the particles and stored in the 119878 lowast 119878matrix119863(4)While The graph corresponding to the generating random topology is not connected do(5) Scan the matrix119863(6) IF two particles 119906 and V are not connected(7) Let 119871 (119906 V) = 1

Algorithm 2 The proposed generating method of random population topology

method a random topology could be generated with anaverage degree slightly larger than119870 In the random topologyby this method the distribution of degree is the sum of 119878 minus 1

independent Bernoulli random variables which is describedin [10]

prob (119884 = 119899) = 119862

119899minus1

119878minus1(

119870

119878

)

119899minus1

(1 minus

119870

119878

)

119904minus119899

(9)

In this paper in order to ensure the connectivity ofundirected graph which is corresponding to the generatingrandom topology we proposed a new generating method ofrandom topology based on Clercrsquos method The basic idea isin selecting the neighborhood particles for each particle if theselection is itself reselect in order to reduce the probability ofparticles isolated after the random population is generateduse the Dijkstra algorithm to compute the distance betweenthe particles if there exist the unconnected particles inthe generated topology then add an edge between theseunconnected particles and retest The improved randompopulation topology generating method is as Algorithm 2

4 Experiment and Analysis

41 Experiment Setting Two sets of experiments were con-ducted which used the canonical PSO (briefed as CPSO)and the DPPSO-Logistic respectively that are described inSection 2 For CPSO set 119888

1= 1198882= 205 120594 = 07298 For

DPPSO-Logistic set 120572 = 0729 120573 = 2187 and 120574 = 05The algorithms were used to solve five benchmark func-

tions which is defined in Table 1 These functions consist ofAckley Schwefel Schafferrsquos F6 Rastrigin and Sphere Table 2shows the settings of these functions

In the experiment the population size is set to 20 inaddition to the the Schafferrsquos F6 function of the dimension2 the remaining functions are carried out in the case of 30-dimensional test and the frequency of repeated experimentsis 50

The performance of the algorithms will be evaluated bythe following aspects

(i) in the case of a certain number of iterations comparethe accuracy (briefed as Perform) of the optimalfitness value in each case These values reflect thequality of the optimal solution obtained in the last

(ii) in the case of a certain number of iterations comparethe success rates (briefed as Prop) which means thatthe algorithms achieve the accuracy (accepted error)that is defined in Table 2 These data reflect thestability of the algorithms

(iii) in the case of a certain number of iterations comparethe evolutionary trends of various algorithms thesefigures reflect the evolution of the optimal solution inthe evolutionary process

42 Comparison between Random Topologies and DifferentAverage Degrees For the random topology the first set ofexperiments used the canonical PSO algorithm to solve thefive benchmark functions By changing the 119870 value wegenerated random topologies with different average degreesfor comparison and evaluation Experimental results areshown in Table 3

As can be seen fromTable 3 with the increase in the valueof 119870 the indicators of Perform and Prop have improvedFor multimodal functions such as Schwefel Schafferrsquos F6Rastrigin and Ackley function with the increase in the valueof119870 when the119870 value is from 3 to 4 (ie the average degreeof the random topology is between 5 and 7 substantially inabout 6) the PSO algorithm could get better performanceAnd when the 119870 value is larger the performance is usuallypoor For multimodal functions when a particle has found alocal optimal solution if the interaction between particles ismore the dissemination of information will be very quicklyand the entire population is susceptible to rapid convergenceto the local optimal solution Therefore too large average

The Scientific World Journal 5

Table 1 Definition of benchmark functions

Benchmark function Formula

Sphere 119865() =

119899

sum

119894=1

119909

2

119894

Schafferrsquos F6 119865() = 05 +

(sin2radic1199092 + 119910

2) minus 05

(10 + 0001 (119909

2+ 119910

2))

2

Rastrigin 119865() = 10 lowast 119899 +

119899

sum

119894=1

(119909

2

119894minus 10 cos(2120587119909

119894))

Schwefel 119865() = 4189829 lowast 119899 +

119899

sum

119894=1

119909119894sin(radic|119909

119894|)

Ackley 119865() = minus20 sdot exp(minus02radic 1

119899

sdot

119899

sum

119894=1

119909

2

119894) minus exp(1

119899

sdot

119899

sum

119894=1

cos (2120587119909119894)) + 20 + exp(1)

Table 2 Settings of benchmark functions

Benchmark function Dimension Optimal value Optimal solution Range Accepted errorSphere 30 0 (0 0 0 0) (minus100 100) 001Schafferrsquos F6 2 0 (0 0) (minus100 100) 000001Rastrigin 30 0 (0 0 0 0) (minus512 512) 100Schwefel 30 0 (0 0 0 0) (minus500 500) 6000Ackley 30 0 (0 0 0 0) (minus30 30) 5

degree of population topology is not conducive to find theglobal optimal solution for a multimodal function

Taking these factors together when the value of 119870 isat about 4 the PSO algorithm has a more satisfactoryperformance for most benchmark functions Accordinglythe following experiments will generate random topologieswhich the 119870 value is 4 and compare these topologies withother classic static population topologies

43 Comparison between Random Topologies and ClassicTopologies The second set of experiments used the DPPSO-Logistic algorithm to solve the five benchmark functions onthe fully connected topology ring topology star topologyand random topology wherein 119870 value is set to 4 togenerate random topologies Comparison and evaluation areconducted by the evolutionary trends of algorithms withvarious population topologies In the figures of evolution-ary trends different line types expressed different DPPSO-Logistic algorithms with various population topologies

For Sphere function (Figure 2) the performance of ringtopology and star topology is poor and the fully connectedtopology and random topology show better search abilityFor Schafferrsquos F6 function (Figure 2) random topologyis significantly better than the three classic neighborhoodtopologies

For Rastrigin function (Figure 3) random topology issuperior to the three classic population topologies in theearly stages of evolution the convergence speed of therandom topology is almost the same as the fully connectedtopology however in the later stage of evolution randomtopology shows a larger advantage and the end result is betterthan the fully connected topology

For Schwefel function (Figure 3) the performance ofthe ring topology is poor the fully connected topologyconvergence fast in the early stage of evolution but the endresult is poor the star topology has achieved good results theconvergence speed of random topology is second only to thefully connected topology in the early evolution stage and thefinal result of random topology is better than the other classictopologies

For Ackley function (Figure 4) the fully connectedtopology and the star topology show poor performance thering topology performs better the random topology showsobvious advantage in both the convergence speed and thefinal result

Overall according to the convergence speed randomtopology is relatively stable in the early stages of evolutionfaster in the midstages of evolution and shows a distinctadvantage in the late stages of evolution From the viewof final result the PSO algorithms using random topologydemonstrate remarkable performance

For unimodal function (such as Sphere function)because there is no problem of falling into a local optimumthe close ties between particles can make faster convergenceand achieve better results Therefore the performance ofthe fully connected topology and random topology with arelatively high average degree is ideal In the case of randomtopology for unimodal function the convergence speed andthe solution will be better with the greater average degree ofpopulation topology

For multimodal function it can be seen that the con-vergence speed will be faster when the average degree ofpopulation topology is increasing If the average degree ofpopulation topology is too high it is easy to fall into localoptimum On the average degree after 6 the optimal solution

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

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Advances in

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The Scientific World Journal 3

(a) (b) (c)

Figure 1 (a) Fully connected topology (b) ring topology (c) star topology

3 Population Topologies

31 The Classic Population Topologies Figure 1 shows thefully connected topology ring topology and star topologyFully connected topology and ring topology are two com-monly used topologies which are also called Gbest modeland Lbest model In the fully connected topology particlersquosneighborhood contains all particles in the population Andin the evolutionary process only the particle that obtains theoptimal position is considered in the entire population PSOalgorithms with this topology converge very fast but easy tofall into local optimum

For a ring topology typically particlersquos neighborhoodincludes the particles on both sides of one or a few particlesIn this topology the exchange of information is relativelyslow within the population but once a particle searched foran optimal location the information eventually will slowlyspread to the entire population

For a star topology one particle is connected with allthe other particles and other particles only connect with theparticle In addition to a central particle other particles areindependent of each other the dissemination of informationmust be passed through the central particle

However the information dissemination mechanism ofthe social groups is not static throughout the whole evo-lutionary process and tends to have a certain degree ofrandomness and dynamic characteristics Mendes studiedrandom population topology and confirmed that populationtopology directly affects the execution performance of PSOalgorithms [9]

32 GraphTheory Characteristics of Population Topology Thepopulation topology of the PSO algorithm can be abstractedinto a connected undirected graph represented by the symbol119866(119881 119864) where119881 is the set of vertices119864 is the set of edges andthe number of vertices is denoted by 119899 For any two points 119906and V in 119866 119889(119906 V) denotes the distance from 119906 to V that isthe length of the shortest path between two points

Definition 1 (average degree) The degree of the vertex V is thenumber of its adjacent vertices denoted by 119896V Average degree119896 of undirected connected graph is calculated by

119870 = sum

Visin119881

119896V

119899

(7)

Definition 2 (average clustering coefficient) The local clus-tering coefficient 119888(V) of vertex V equals to the number ofedges which can be connected between vertices associatedwith vertex V divided by the maximum number of edgesbetween these vertices The average clustering coefficient 119888of a graph is the arithmetic mean of the local clusteringcoefficient of all vertices which can be calculated by

119888 = sum

Visin119881

119888 (V)

119899

(8)

The average degree of population topology means theaverage number of particlesrsquo neighborhood particles it rep-resents the degree of socialization of population A smallnumber of neighborhood particles means that the particleis both difficult to obtain information from the populationand difficult to influence other particles On the contrary aparticle which have large number of neighborhood particlescan get a lot of information available in the population andsuch a particle has a greater influence in the populationIn the common used population topologies fully connectedtopology has the largest average degree which is equal to thepopulation size minus 1The ring topology has the minimumaverage degree the average degree of a ring topology suchas in Figure 1 is equal to 2 The local clustering coefficientis the ratio of the number of connections between theactual existence and the possible existence and the averageclustering coefficient represents the degree of aggregationof the vertices in a graph which is the average of the localclustering coefficient of all vertices In this paper we willanalyze the role of the population topology based on theprevious graph theory characteristics

33 Random Population Topologies Clerc initially attemptedto proposed a method of random population topology [10]The basic idea is to generate a random topology by selectingthe neighborhood particles randomly for each particle Theconcrete steps could be described as Algorithm 1

In the resulting matrix 119871 of Algorithm 1 119871(119906 V) = 1

means that the particles 119906 and V are connected By this

4 The Scientific World Journal

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

119898 from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 1(3) For the matrix 119871 If 119871(119894 119898) = 1 then let 119871 (119898 119894) = 1

Algorithm 1 Clercrsquos generating method of random population topology

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

m (119898 = 119894) from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 119871 (119898 119894) = 1(3) Use the Dijkstra algorithm to calculate the distance between the particles and stored in the 119878 lowast 119878matrix119863(4)While The graph corresponding to the generating random topology is not connected do(5) Scan the matrix119863(6) IF two particles 119906 and V are not connected(7) Let 119871 (119906 V) = 1

Algorithm 2 The proposed generating method of random population topology

method a random topology could be generated with anaverage degree slightly larger than119870 In the random topologyby this method the distribution of degree is the sum of 119878 minus 1

independent Bernoulli random variables which is describedin [10]

prob (119884 = 119899) = 119862

119899minus1

119878minus1(

119870

119878

)

119899minus1

(1 minus

119870

119878

)

119904minus119899

(9)

In this paper in order to ensure the connectivity ofundirected graph which is corresponding to the generatingrandom topology we proposed a new generating method ofrandom topology based on Clercrsquos method The basic idea isin selecting the neighborhood particles for each particle if theselection is itself reselect in order to reduce the probability ofparticles isolated after the random population is generateduse the Dijkstra algorithm to compute the distance betweenthe particles if there exist the unconnected particles inthe generated topology then add an edge between theseunconnected particles and retest The improved randompopulation topology generating method is as Algorithm 2

4 Experiment and Analysis

41 Experiment Setting Two sets of experiments were con-ducted which used the canonical PSO (briefed as CPSO)and the DPPSO-Logistic respectively that are described inSection 2 For CPSO set 119888

1= 1198882= 205 120594 = 07298 For

DPPSO-Logistic set 120572 = 0729 120573 = 2187 and 120574 = 05The algorithms were used to solve five benchmark func-

tions which is defined in Table 1 These functions consist ofAckley Schwefel Schafferrsquos F6 Rastrigin and Sphere Table 2shows the settings of these functions

In the experiment the population size is set to 20 inaddition to the the Schafferrsquos F6 function of the dimension2 the remaining functions are carried out in the case of 30-dimensional test and the frequency of repeated experimentsis 50

The performance of the algorithms will be evaluated bythe following aspects

(i) in the case of a certain number of iterations comparethe accuracy (briefed as Perform) of the optimalfitness value in each case These values reflect thequality of the optimal solution obtained in the last

(ii) in the case of a certain number of iterations comparethe success rates (briefed as Prop) which means thatthe algorithms achieve the accuracy (accepted error)that is defined in Table 2 These data reflect thestability of the algorithms

(iii) in the case of a certain number of iterations comparethe evolutionary trends of various algorithms thesefigures reflect the evolution of the optimal solution inthe evolutionary process

42 Comparison between Random Topologies and DifferentAverage Degrees For the random topology the first set ofexperiments used the canonical PSO algorithm to solve thefive benchmark functions By changing the 119870 value wegenerated random topologies with different average degreesfor comparison and evaluation Experimental results areshown in Table 3

As can be seen fromTable 3 with the increase in the valueof 119870 the indicators of Perform and Prop have improvedFor multimodal functions such as Schwefel Schafferrsquos F6Rastrigin and Ackley function with the increase in the valueof119870 when the119870 value is from 3 to 4 (ie the average degreeof the random topology is between 5 and 7 substantially inabout 6) the PSO algorithm could get better performanceAnd when the 119870 value is larger the performance is usuallypoor For multimodal functions when a particle has found alocal optimal solution if the interaction between particles ismore the dissemination of information will be very quicklyand the entire population is susceptible to rapid convergenceto the local optimal solution Therefore too large average

The Scientific World Journal 5

Table 1 Definition of benchmark functions

Benchmark function Formula

Sphere 119865() =

119899

sum

119894=1

119909

2

119894

Schafferrsquos F6 119865() = 05 +

(sin2radic1199092 + 119910

2) minus 05

(10 + 0001 (119909

2+ 119910

2))

2

Rastrigin 119865() = 10 lowast 119899 +

119899

sum

119894=1

(119909

2

119894minus 10 cos(2120587119909

119894))

Schwefel 119865() = 4189829 lowast 119899 +

119899

sum

119894=1

119909119894sin(radic|119909

119894|)

Ackley 119865() = minus20 sdot exp(minus02radic 1

119899

sdot

119899

sum

119894=1

119909

2

119894) minus exp(1

119899

sdot

119899

sum

119894=1

cos (2120587119909119894)) + 20 + exp(1)

Table 2 Settings of benchmark functions

Benchmark function Dimension Optimal value Optimal solution Range Accepted errorSphere 30 0 (0 0 0 0) (minus100 100) 001Schafferrsquos F6 2 0 (0 0) (minus100 100) 000001Rastrigin 30 0 (0 0 0 0) (minus512 512) 100Schwefel 30 0 (0 0 0 0) (minus500 500) 6000Ackley 30 0 (0 0 0 0) (minus30 30) 5

degree of population topology is not conducive to find theglobal optimal solution for a multimodal function

Taking these factors together when the value of 119870 isat about 4 the PSO algorithm has a more satisfactoryperformance for most benchmark functions Accordinglythe following experiments will generate random topologieswhich the 119870 value is 4 and compare these topologies withother classic static population topologies

43 Comparison between Random Topologies and ClassicTopologies The second set of experiments used the DPPSO-Logistic algorithm to solve the five benchmark functions onthe fully connected topology ring topology star topologyand random topology wherein 119870 value is set to 4 togenerate random topologies Comparison and evaluation areconducted by the evolutionary trends of algorithms withvarious population topologies In the figures of evolution-ary trends different line types expressed different DPPSO-Logistic algorithms with various population topologies

For Sphere function (Figure 2) the performance of ringtopology and star topology is poor and the fully connectedtopology and random topology show better search abilityFor Schafferrsquos F6 function (Figure 2) random topologyis significantly better than the three classic neighborhoodtopologies

For Rastrigin function (Figure 3) random topology issuperior to the three classic population topologies in theearly stages of evolution the convergence speed of therandom topology is almost the same as the fully connectedtopology however in the later stage of evolution randomtopology shows a larger advantage and the end result is betterthan the fully connected topology

For Schwefel function (Figure 3) the performance ofthe ring topology is poor the fully connected topologyconvergence fast in the early stage of evolution but the endresult is poor the star topology has achieved good results theconvergence speed of random topology is second only to thefully connected topology in the early evolution stage and thefinal result of random topology is better than the other classictopologies

For Ackley function (Figure 4) the fully connectedtopology and the star topology show poor performance thering topology performs better the random topology showsobvious advantage in both the convergence speed and thefinal result

Overall according to the convergence speed randomtopology is relatively stable in the early stages of evolutionfaster in the midstages of evolution and shows a distinctadvantage in the late stages of evolution From the viewof final result the PSO algorithms using random topologydemonstrate remarkable performance

For unimodal function (such as Sphere function)because there is no problem of falling into a local optimumthe close ties between particles can make faster convergenceand achieve better results Therefore the performance ofthe fully connected topology and random topology with arelatively high average degree is ideal In the case of randomtopology for unimodal function the convergence speed andthe solution will be better with the greater average degree ofpopulation topology

For multimodal function it can be seen that the con-vergence speed will be faster when the average degree ofpopulation topology is increasing If the average degree ofpopulation topology is too high it is easy to fall into localoptimum On the average degree after 6 the optimal solution

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 The Scientific World Journal

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

119898 from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 1(3) For the matrix 119871 If 119871(119894 119898) = 1 then let 119871 (119898 119894) = 1

Algorithm 1 Clercrsquos generating method of random population topology

(1) For the population that the size is 119878 build a 119878 lowast 119878matrix and let 119871 (119894 119894) = 1(2) Select a119870 value for each row 119894 of the matrix 119871 generate a uniformly distributed random number

m (119898 = 119894) from 1 119898 (119898may be selected repeatedly) let 119871 (119894 119898) = 119871 (119898 119894) = 1(3) Use the Dijkstra algorithm to calculate the distance between the particles and stored in the 119878 lowast 119878matrix119863(4)While The graph corresponding to the generating random topology is not connected do(5) Scan the matrix119863(6) IF two particles 119906 and V are not connected(7) Let 119871 (119906 V) = 1

Algorithm 2 The proposed generating method of random population topology

method a random topology could be generated with anaverage degree slightly larger than119870 In the random topologyby this method the distribution of degree is the sum of 119878 minus 1

independent Bernoulli random variables which is describedin [10]

prob (119884 = 119899) = 119862

119899minus1

119878minus1(

119870

119878

)

119899minus1

(1 minus

119870

119878

)

119904minus119899

(9)

In this paper in order to ensure the connectivity ofundirected graph which is corresponding to the generatingrandom topology we proposed a new generating method ofrandom topology based on Clercrsquos method The basic idea isin selecting the neighborhood particles for each particle if theselection is itself reselect in order to reduce the probability ofparticles isolated after the random population is generateduse the Dijkstra algorithm to compute the distance betweenthe particles if there exist the unconnected particles inthe generated topology then add an edge between theseunconnected particles and retest The improved randompopulation topology generating method is as Algorithm 2

4 Experiment and Analysis

41 Experiment Setting Two sets of experiments were con-ducted which used the canonical PSO (briefed as CPSO)and the DPPSO-Logistic respectively that are described inSection 2 For CPSO set 119888

1= 1198882= 205 120594 = 07298 For

DPPSO-Logistic set 120572 = 0729 120573 = 2187 and 120574 = 05The algorithms were used to solve five benchmark func-

tions which is defined in Table 1 These functions consist ofAckley Schwefel Schafferrsquos F6 Rastrigin and Sphere Table 2shows the settings of these functions

In the experiment the population size is set to 20 inaddition to the the Schafferrsquos F6 function of the dimension2 the remaining functions are carried out in the case of 30-dimensional test and the frequency of repeated experimentsis 50

The performance of the algorithms will be evaluated bythe following aspects

(i) in the case of a certain number of iterations comparethe accuracy (briefed as Perform) of the optimalfitness value in each case These values reflect thequality of the optimal solution obtained in the last

(ii) in the case of a certain number of iterations comparethe success rates (briefed as Prop) which means thatthe algorithms achieve the accuracy (accepted error)that is defined in Table 2 These data reflect thestability of the algorithms

(iii) in the case of a certain number of iterations comparethe evolutionary trends of various algorithms thesefigures reflect the evolution of the optimal solution inthe evolutionary process

42 Comparison between Random Topologies and DifferentAverage Degrees For the random topology the first set ofexperiments used the canonical PSO algorithm to solve thefive benchmark functions By changing the 119870 value wegenerated random topologies with different average degreesfor comparison and evaluation Experimental results areshown in Table 3

As can be seen fromTable 3 with the increase in the valueof 119870 the indicators of Perform and Prop have improvedFor multimodal functions such as Schwefel Schafferrsquos F6Rastrigin and Ackley function with the increase in the valueof119870 when the119870 value is from 3 to 4 (ie the average degreeof the random topology is between 5 and 7 substantially inabout 6) the PSO algorithm could get better performanceAnd when the 119870 value is larger the performance is usuallypoor For multimodal functions when a particle has found alocal optimal solution if the interaction between particles ismore the dissemination of information will be very quicklyand the entire population is susceptible to rapid convergenceto the local optimal solution Therefore too large average

The Scientific World Journal 5

Table 1 Definition of benchmark functions

Benchmark function Formula

Sphere 119865() =

119899

sum

119894=1

119909

2

119894

Schafferrsquos F6 119865() = 05 +

(sin2radic1199092 + 119910

2) minus 05

(10 + 0001 (119909

2+ 119910

2))

2

Rastrigin 119865() = 10 lowast 119899 +

119899

sum

119894=1

(119909

2

119894minus 10 cos(2120587119909

119894))

Schwefel 119865() = 4189829 lowast 119899 +

119899

sum

119894=1

119909119894sin(radic|119909

119894|)

Ackley 119865() = minus20 sdot exp(minus02radic 1

119899

sdot

119899

sum

119894=1

119909

2

119894) minus exp(1

119899

sdot

119899

sum

119894=1

cos (2120587119909119894)) + 20 + exp(1)

Table 2 Settings of benchmark functions

Benchmark function Dimension Optimal value Optimal solution Range Accepted errorSphere 30 0 (0 0 0 0) (minus100 100) 001Schafferrsquos F6 2 0 (0 0) (minus100 100) 000001Rastrigin 30 0 (0 0 0 0) (minus512 512) 100Schwefel 30 0 (0 0 0 0) (minus500 500) 6000Ackley 30 0 (0 0 0 0) (minus30 30) 5

degree of population topology is not conducive to find theglobal optimal solution for a multimodal function

Taking these factors together when the value of 119870 isat about 4 the PSO algorithm has a more satisfactoryperformance for most benchmark functions Accordinglythe following experiments will generate random topologieswhich the 119870 value is 4 and compare these topologies withother classic static population topologies

43 Comparison between Random Topologies and ClassicTopologies The second set of experiments used the DPPSO-Logistic algorithm to solve the five benchmark functions onthe fully connected topology ring topology star topologyand random topology wherein 119870 value is set to 4 togenerate random topologies Comparison and evaluation areconducted by the evolutionary trends of algorithms withvarious population topologies In the figures of evolution-ary trends different line types expressed different DPPSO-Logistic algorithms with various population topologies

For Sphere function (Figure 2) the performance of ringtopology and star topology is poor and the fully connectedtopology and random topology show better search abilityFor Schafferrsquos F6 function (Figure 2) random topologyis significantly better than the three classic neighborhoodtopologies

For Rastrigin function (Figure 3) random topology issuperior to the three classic population topologies in theearly stages of evolution the convergence speed of therandom topology is almost the same as the fully connectedtopology however in the later stage of evolution randomtopology shows a larger advantage and the end result is betterthan the fully connected topology

For Schwefel function (Figure 3) the performance ofthe ring topology is poor the fully connected topologyconvergence fast in the early stage of evolution but the endresult is poor the star topology has achieved good results theconvergence speed of random topology is second only to thefully connected topology in the early evolution stage and thefinal result of random topology is better than the other classictopologies

For Ackley function (Figure 4) the fully connectedtopology and the star topology show poor performance thering topology performs better the random topology showsobvious advantage in both the convergence speed and thefinal result

Overall according to the convergence speed randomtopology is relatively stable in the early stages of evolutionfaster in the midstages of evolution and shows a distinctadvantage in the late stages of evolution From the viewof final result the PSO algorithms using random topologydemonstrate remarkable performance

For unimodal function (such as Sphere function)because there is no problem of falling into a local optimumthe close ties between particles can make faster convergenceand achieve better results Therefore the performance ofthe fully connected topology and random topology with arelatively high average degree is ideal In the case of randomtopology for unimodal function the convergence speed andthe solution will be better with the greater average degree ofpopulation topology

For multimodal function it can be seen that the con-vergence speed will be faster when the average degree ofpopulation topology is increasing If the average degree ofpopulation topology is too high it is easy to fall into localoptimum On the average degree after 6 the optimal solution

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 5

Table 1 Definition of benchmark functions

Benchmark function Formula

Sphere 119865() =

119899

sum

119894=1

119909

2

119894

Schafferrsquos F6 119865() = 05 +

(sin2radic1199092 + 119910

2) minus 05

(10 + 0001 (119909

2+ 119910

2))

2

Rastrigin 119865() = 10 lowast 119899 +

119899

sum

119894=1

(119909

2

119894minus 10 cos(2120587119909

119894))

Schwefel 119865() = 4189829 lowast 119899 +

119899

sum

119894=1

119909119894sin(radic|119909

119894|)

Ackley 119865() = minus20 sdot exp(minus02radic 1

119899

sdot

119899

sum

119894=1

119909

2

119894) minus exp(1

119899

sdot

119899

sum

119894=1

cos (2120587119909119894)) + 20 + exp(1)

Table 2 Settings of benchmark functions

Benchmark function Dimension Optimal value Optimal solution Range Accepted errorSphere 30 0 (0 0 0 0) (minus100 100) 001Schafferrsquos F6 2 0 (0 0) (minus100 100) 000001Rastrigin 30 0 (0 0 0 0) (minus512 512) 100Schwefel 30 0 (0 0 0 0) (minus500 500) 6000Ackley 30 0 (0 0 0 0) (minus30 30) 5

degree of population topology is not conducive to find theglobal optimal solution for a multimodal function

Taking these factors together when the value of 119870 isat about 4 the PSO algorithm has a more satisfactoryperformance for most benchmark functions Accordinglythe following experiments will generate random topologieswhich the 119870 value is 4 and compare these topologies withother classic static population topologies

43 Comparison between Random Topologies and ClassicTopologies The second set of experiments used the DPPSO-Logistic algorithm to solve the five benchmark functions onthe fully connected topology ring topology star topologyand random topology wherein 119870 value is set to 4 togenerate random topologies Comparison and evaluation areconducted by the evolutionary trends of algorithms withvarious population topologies In the figures of evolution-ary trends different line types expressed different DPPSO-Logistic algorithms with various population topologies

For Sphere function (Figure 2) the performance of ringtopology and star topology is poor and the fully connectedtopology and random topology show better search abilityFor Schafferrsquos F6 function (Figure 2) random topologyis significantly better than the three classic neighborhoodtopologies

For Rastrigin function (Figure 3) random topology issuperior to the three classic population topologies in theearly stages of evolution the convergence speed of therandom topology is almost the same as the fully connectedtopology however in the later stage of evolution randomtopology shows a larger advantage and the end result is betterthan the fully connected topology

For Schwefel function (Figure 3) the performance ofthe ring topology is poor the fully connected topologyconvergence fast in the early stage of evolution but the endresult is poor the star topology has achieved good results theconvergence speed of random topology is second only to thefully connected topology in the early evolution stage and thefinal result of random topology is better than the other classictopologies

For Ackley function (Figure 4) the fully connectedtopology and the star topology show poor performance thering topology performs better the random topology showsobvious advantage in both the convergence speed and thefinal result

Overall according to the convergence speed randomtopology is relatively stable in the early stages of evolutionfaster in the midstages of evolution and shows a distinctadvantage in the late stages of evolution From the viewof final result the PSO algorithms using random topologydemonstrate remarkable performance

For unimodal function (such as Sphere function)because there is no problem of falling into a local optimumthe close ties between particles can make faster convergenceand achieve better results Therefore the performance ofthe fully connected topology and random topology with arelatively high average degree is ideal In the case of randomtopology for unimodal function the convergence speed andthe solution will be better with the greater average degree ofpopulation topology

For multimodal function it can be seen that the con-vergence speed will be faster when the average degree ofpopulation topology is increasing If the average degree ofpopulation topology is too high it is easy to fall into localoptimum On the average degree after 6 the optimal solution

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

Table 3 Comparison between random topologies with different average degrees

Benchmarkfunction 119870

Averagedegree

Clusteringcoefficient Perform Prop

Sphere

1 1958 0753 138119864 minus 15 07685072 3700 0541 153119864 minus 30 08176933 5244 0498 370119864 minus 32 0826244 6640 0510 592119864 minus 33 08339935 7928 0534 415119864 minus 33 08369736 9040 0568 809119864 minus 34 0841787 10116 0606 100119864 minus 33 0847028 11092 0641 168119864 minus 34 0842789 11814 0671 918119864 minus 34 084446

Schafferrsquos F6

1 1942 0753 0002915 04973932 3700 0545 0001757 0639163 5228 0497 0001749 06818474 6626 0508 000136 07160535 7926 0530 0001943 063036 8998 0563 0003303 05586137 10070 0602 000272 0595388 11078 0643 0003313 05427079 11854 0673 0002915 0558667

Rastrigin

1 1956 0751 6365224 08685872 3688 0546 5810571 09239333 5198 0500 5864275 09159934 6694 0507 6268228 09342075 7940 0532 5880197 0941446 9018 0565 6254299 0937827 10118 0608 5910044 09446738 10982 0638 6190622 09086879 11916 0674 6351805 0925493

Schwefel

1 1956 0752 4773649 0913882 3694 0548 4366848 09424273 5234 0500 4388532 094364 6616 0502 4399952 09435335 7922 0532 4286961 09506336 9046 0571 4397724 09509877 10074 0602 4412291 09323278 11016 0641 442019 09510079 11734 0667 4397566 0933313

Ackley

1 1948 0751 1224299 09607532 3690 0549 1030908 09697333 5282 0493 0985613 09723334 6672 0507 1256866 0973045 7872 0539 1238158 09739476 9028 0568 1593962 09740877 10134 0611 1755415 0974328 10922 0640 1919407 09748479 11876 0671 1962931 097564

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

Iteration

Log1

0(fit

ness

)

Sphere

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

Iteration

Log1

0(fit

ness

)

Schafferrsquos F6

Fully connectedRing

StarRandom

(b)

Figure 2 Comparison of evolutionary trend between four topologies (Sphere and Schafferrsquos F6)

0 500 1000 1500 2000 2500 300045

5

55

6

65

7

Iteration

Log1

0(fit

ness

)

Rastrigin

Fully connectedRing

StarRandom

(a)

0 500 1000 1500 2000 2500 300084

86

88

9

92

94

96

98

Iteration

Log1

0(fit

ness

)

Schwefel

Fully connectedRing

StarRandom

(b)

Figure 3 Comparison of evolutionary trend between four topologies (Rastrigin and Schwefel)

quality of most algorithms begins to decrease When theaverage degree is between 5 and 7 (119870 is set to 4) theperformance of algorithms is usually ideal

5 Conclusion

In this paper we propose an improved method of generatingrandom topology based on previous research And we carryout in-depth research of the performance of the algorithmsusing random topology based on the canonical PSO andDPPSO-Logistic respectively Combined with experimental

results we conduct the analysis and interpretation of theperformance of the algorithms from the perspective of graphtheory And empirical laws in generating random topologiesare given according to our experimental results and theoreti-cal analysis

On the whole relative to the three classic populationtopologies (fully connected topology ring topology andstar topology) the algorithm has obvious advantages whichis using the proposed random topology Further workwill include the theoretical analysis of different populationtopologies as well as dynamic population topology strategy

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

0 500 1000 1500 2000 2500 3000minus30

minus25

minus20

minus15

minus10

minus5

0

5

Iteration

Log1

0(fit

ness

)

Ackley

Fully connectedRing

StarRandom

Figure 4 Comparison of evolutionary trend between four topolo-gies (Ackley)

which is designed in accordance with the conclusions of thispaper

Acknowledgments

This paper is supported by Provincial Key Laboratory forComputer Information Processing Technology SoochowUniversity Suzhou China and NSFC (Grant no 61170164)

References

[1] M R AlRashidi and M E El-Hawary ldquoA survey of particleswarm optimization applications in electric power systemsrdquoIEEE Transactions on Evolutionary Computation vol 13 no 4pp 913ndash918 2009

[2] A A Esmin R A Coelho and S Matwin ldquoA review on particleswarm optimization algorithm and its variants to clusteringhighdimensional datardquo Artificial Intelligence Review 2013

[3] R V Kulkarni and G K Venayagamoorthy ldquoParticle swarmoptimization in wireless-sensor networks a brief surveyrdquo IEEETransactions on Systems Man and Cybernetics C vol 41 no 2pp 262ndash267 2011

[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[5] J Kennedy ldquoSmall worlds andmega-minds effects of neighbor-hood topology on particle swarm performancerdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo99) vol 3IEEE 1999

[6] P N Suganthan ldquoParticle swarm optimiser with neighbour-hood operatorrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 3 IEEE 1999

[7] R Mendes J Kennedy and J Neves ldquoThe fully informedparticle swarm simpler maybe betterrdquo IEEE Transactions onEvolutionary Computation vol 8 no 3 pp 204ndash210 2004

[8] M A M de Oca and T Stutzle ldquoConvergence behavior ofthe fully informed particle swarm optimization algorithmrdquoin Proceedings of the 10th Annual Genetic and EvolutionaryComputation Conference (GECCO rsquo08) pp 71ndash78 ACM July2008

[9] R Mendes Population topologies and their influence in parti-cle swarm performance [PhD dissertation] Universidade doMinho 2004

[10] M Clerc ldquoBack to random topologyrdquo Tech Rep 2007 2007[11] M Clerc and J Kennedy ldquoThe particle swarm-explosion sta-

bility and convergence in a multidimensional complex spacerdquoIEEE Transactions on Evolutionary Computation vol 6 no 1pp 58ndash73 2002

[12] J Kennedy ldquoDynamic-probabilistic particle swarmsrdquo in Pro-ceedings of the Conference on Genetic and Evolutionary Compu-tation Conference pp 201ndash207 ACM June 2005

[13] Q Ni and J Deng ldquoTwo improvement strategies for logisticdynamic particle swarm optimizationrdquo inAdaptive and NaturalComputing Algorithms pp 320ndash329 Springer 2011

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Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014