Deceptiveness and Neutrality - the ND family of fitness landscapes

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by  William Beaudoin

Deceptiveness and Neutrality
The ND Family of Fitness Landscapes
William Beaudoin, Se´bastien Verel, Philippe Collard, Cathy Escazut
UniversityofNice-SophiaAntipolis
I3SLaboratory
SophiaAntipolis,France
{beaudoin,verel,pc,escazut}@i3s.unice.fr
9 ABSTRACT dynamics of populations evolving on such neutral landscapes are
0
differentfromthoseonadaptivelandscapes:theyarecharacterized
0 Whenaconsiderable number of mutationshaveno effectson fit-
by long periods of fitness stasis (population stated on a ’neutral
2 nessvalues,thefitnesslandscapeissaidneutral. Inordertostudy
network’) punctuated by shorter periods of innovation withrapid
the interplaybetween neutrality, whichexistsin manyreal-world
n fitnessincreases[3].Inthefieldofevolutionarycomputation,neu-
applications,andperformancesofmetaheuristics,itisusefultode-
a trality plays an important role in real-world problems: in design
sign landscapes which make it possible to tune precisely neutral
J
of digital circuits [4] [5] [6], in evolutionary robotics [7] [8]. In
degree distribution. Eventhough manyneutral landscape models
3 thoseproblems,neutralityisimplicitlyembeddedinthegenotype
have already been designed, none of them are general enough to
2 tophenotypemapping.
create landscapes with specific neutral degree distributions. We

propose three steps to design such landscapes: first using an al-
] 1.1 Neutrality
I gorithm we construct a landscape whose distribution roughly fits
A
thetargetone,thenweuseasimulatedannealingheuristictobring Werecallafewfundamental concepts about fitnesslandscapes
. closer thetwodistributionsand finallyweaffect fitnessvalues to andneutrality(see[9]foramoredetailedtreatment). Alandscape
s
c eachneutralnetwork. Thenusingthisnewfamilyoffitnessland- isatriplet(S,V,f) where S isaset of potential solutions i.e. a
[ scapesweareabletohighlighttheinterplaybetweendeceptiveness searchspace,V : S → 2S,aneighbourhood structure,isafunc-
andneutrality. tion that assigns to every s ∈ S a set of neighbours V(s), and
1 f :S →IRisafitnessfunctionthatcanbepicturedasthe“height”
v
Categories andSubject Descriptors ofthecorrespondingpotentialsolutions.Theneighbourhoodisof-
9
tendefinedbyanoperator likebitflipmutation. Aneutral neigh-
6 I.2.8[ArtificialIntelligence]:ProblemSolving,ControlMethods, bour of sisaneighbour withthesamefitnessf(s). Theneutral
7 andSearch. degree of a solution is the number of its neutral neighbours. A
3
fitness landscape is neutral if there are many solutions with high
.
1 General Terms neutral degree. A neutral network, denoted NN, is a connected
0 graphwhereverticesaresolutionswiththesamefitnessvalueand
Algorithms,performance,design,experimentation.
9 twoverticesareconnectediftheyareneutralneighbours.
0
: Keywords 1.2 Fitness Landscapes withNeutrality
v
i Fitnesslandscapes,geneticalgorithms,search,benchmark. Inordertostudytherelationshipbetweenneutrality,dynamicsof
X
EvolutionaryAlgorithms(EA)andsearchdifficulty, somebench-
r 1. INTRODUCTION marksofneutrallandscapeshavebeenproposed. Moreoftenneu-
a
tralityiseitheranadd-onfeature,asinNK-landscapes,oraninci-
TheAdaptativeLandscapemetaphorintroducedbyS.Wright [1]
dentalproperty,asinRoyal-Roadfunctions. Inmostcasesthede-
hasdominatedtheviewofadaptiveevolution: anuphillwalkofa
signactsupontheamountofsolutionswiththesamefitness.Royal-
populationonamountainousfitnesslandscapeinwhichitcanget
Roadfunctions [10] aredefinedon binarystringsof lengthN =
stuckonsuboptimalpeaks. Resultsfrommolecularevolutionhas
n.k wherenisthenumberofblocksandkthesizeofoneblock.
changed this picture: Kimura’s model [2] assumes that the over-
Thefitnessfunctioncorrespondstothenumberofblockswhichare
whelming majority of mutations are either effectively neutral or
setwithk bitsvalue1andneutralityincreaseswithk. Numerous
lethalandinthelattercasepurgedbynegativeselection. Thisas-
landscapesarevariantofNK-Landscapes[11].Thefitnessfunction
sumption iscalled theneutral hypothesis. Under thishypothesis,
of anNK-landscape isafunction f : {0,1}N → [0,1) defined
onbinarystringswithN bits. An’atom’withfixedepistasislevel
isrepresentedbyafitnesscomponentfi :{0,1}K+1 →[0,1)as-
sociatedtoeachbiti. Itdependsonthevalueatbitiandalsoon
Permission tomake digital orhardcopies ofall orpartofthis workfor the values at K other epistatic bits. The fitness f is the average
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare ofthevaluesoftheN fitnesscomponents fi. Severalvariantsof
notmadeordistributedforprofitorcommercialadvantageandthatcopies NK-landscapes try to reduce the number of fitness values in or-
breepaurbthliisshn,ototicpeosatnodnthseerfvuelrlscoitrattoiornedoinsttrhibeufitersttoplaisgtes.,rTeoquciorpeysporthioerrwspiesec,ifitoc dertoaddsomeneutrality. InNKp-landscapes[12],fi(x)hasa
permissionand/orafee. probabilityptobeequalto0; inNKq-landscapes[13],fi(x)is
Copyright200XACMX-XXXXX-XX-X/XX/XX...$5.00. uniformlydistributedintheinterval[0,q−1]∩IN;inTechnological
Landscapes [14], tunedbyanaturalnumber M, f(x)isrounded
sothatitcanonlytakeM differentvalues. Forallthoseproblems,
neutralityistunedbyoneparameteronly: neutralityincreasesac-
cording top and decreases with q or M (see for example Figure
2).
Dynamics of a population on a neutral network are complex,
evenonflatlandscapesasshownbyDerrida[15]. Theworks[16]
[17][18][19],attheinterplayofmolecularevolutionandoptimiza- 00000 00010 00100 10101 11001
tion,studytheconvergenceofapopulationonneutralnetworks.In
thecase ofinfinitepopulation under mutationand selection, they
showdistributiononaNN isonlydeterminedbythetopologyof 00001 10000 01000 00011 10010 00101 01001 01010 00110 10100 10111 11101 11011
thisnetwork. Thatistosay,thepopulationconvergestothesolu-
tionsintheNN withhighneutraldegree.Thus,theneutraldegree
distributionisanimportantfeatureofneutrallandscapes. 10001 11000 01100 00111 10011 11010 01101 01011 01110
Inordertostudymorepreciselyneutrality,forinstancelinkbe-
tweenneutralityandsearchdifficulty,weneedfor“neutrality-driven
design” whereneutralityreallyguidesthedesignprocess. Inthis 10110 11100 01111
paper we propose to generate a family of landscapes where it is
possibletotuneaccuratelytheneutraldegreeofsolutions.
11110 11111
2. ND-LANDSCAPES
In this section, we first present an algorithm to create a land- Figure1: ExampleofatinyND-Landscape. Eachnoderepre-
scapewithagivenneutraldegreedistribution.Thenwewillrefine sentsasolutionand twonodesareconnected iftheyhave the
themethodtoobtainmoreaccuratelandscapesandfinallywewill samefitnessvalueandareHammingneighbours.Inthisexam-
studytimeandspacecomplexityofthealgorithm. pletherearefiveneutralnetworks.
2.1 An algorithmto design
smallND-Landscape
Wenow introduce asimplemodel ofneutral landscapes called
ND-LandscapeswhereNreferstothenumberofbitsofasolution
andDtotheneutraldegreedistribution.Inthisfirststepouraimis
toprovideanexhaustivedefinitionofthelandscapeassigningone
fitnessvaluetoeachsolution.WefixNto16bitsandsothesizeof
searchspaceis216. BuildingaND-Landscapeisdonebysplitting
thesearchspaceintoneutralnetworks. Howeverthefitnessvalue
ofeachneutralnetworkhasnoinfluenceontheneutrality. Thisis
whythesefitnessvaluesarerandomlychosen.
LetDbeanarrayofsizeN+1representinganeutraldegreedis-
tribution. NandDaregivenasinputsandthealgorithm(seealgo- Algorithm1GenerationofND-Landscapes
rithm1) returnsafitnessfunction f from{0,1}N toIRsuch that
∀s∈S, f[s]←unaffected
theneutraldegreedistributionofthefitnesslandscapeissimilarto
randomlychooseonesolutions .
0
D.For moresimplicity, wechosetogiveadifferentfitnessvalue
CandidatesList←Ssortedbydistancefroms .
0
toeachneutralnetwork.WedefineRouletteWheel(D)asarandom
whilenotempty(CandidatesList) do
variablewhosedensityisgivenbydistributionD.Itisdirectlyin-
s←head(CandidatesList)
spiredfromthegeneticalgorithmselectionoperator. Forexample:
ford=0toNdo
let∆bethefollowingdistribution:
ifscan’thavedneutralneighbours
∆[0]=0 ∆[1]=0.25 ∆[2]=0.5 ∆[3]=0.25.RouletteWheel(∆)
thenD’[d]←0
willreturnvalue1in25%ofthetime,2in50%ofthetimeand
elseD’[d]←D[d]
3in25%ofthetime. Figure1showstheneutralnetworksofan
endfor
idealND-Landscape(size=25)forthedistribution∆.
n←RouletteWheel(D’[d])
2.2 AmetaheuristictoimprovetheNDdesign Giveavaluetosomeunaffectedneighbourssothatshasex-
actlynneutral neighbours and sothat theneutral degreesof
Usingalgorithm1,exhaustivefitnessallocationdoesnotcreate
alreadychosensolutions(6∈CandidatesList)areunchanged.
alandscapewithaneutraldegreedistributioncloseenoughtothe
D[n]←D[n]- 1
inputdistribution. Thereasonisthefitnessfunctioniscompletely 2N
CandidatesList←next(CandidatesList)
defined beforetheneutral degree ofevery solutionhasbeen con-
endwhile
sidered. Hence,weuseasimulatedannealingmetaheuristictoim-
prove the landscape created by algorithm 1. Here, simulated an-
nealingisnotusedtofindagoodsolutionofaND-Landscapebut
toadjustthelandscapebymodifyingthefitnessofsomesolutions
suchasneutraldistributionofaND-Landscapebeclosertothein-
put distribution. The local operator is the changement of fitness
valueofonesolutionofthelandscape,whichcanalteratmostN+1
0.35 0.25
0.3
0.2
0.25
Frequency 0 0.1.25 Frequency 0 0.1.15
0.1
0.05
0.05
0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12
Neutral Degree Neutral Degree
NKpwithN =16,K =5,p=0.8 NKqwithN =16,K =4,q=2
average:5.16stddev:2.37 average:3.94stddev:1.74
0.25 0.35
0.3
0.2
0.25
Frequency 0 0.1.15 Frequency 0 0.1.25
0.1
0.05
0.05
0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16
Neutral Degree Neutral Degree
TechnologicallandscapewithN =16,K =4,M =20 RoyalRoadwithN =16,n=4,k=4
average:5.49stddev:1.99 average:14.0stddev:2.00
Figure2:Neutraldegreedistributionforsomeneutrallandscapes
neutraldegrees.Theacceptanceofatransitionisdeterminedbythe ConsequentlywecanonlyconstructND-Landscapeswithasmall
differencebetweenthedistancetotheinputdistributionbeforeand N(≤ 16)butwewillseeinsection4howtocreateAdditiveEx-
afterthistransition. Thedistanceweusetocomparetwodistribu- tendedND-Landscapeswithfargreatersearchspaces.
tionsistherootmeansquare:
2.4 Sizes ofthe generated Neutral Networks
N Figure4shows the diversityof sizesof neutral networks for 4
dist(D,D )= (D[i]−D [i])2 distributions. For every distribution we created 50 different ND-
0 v 0
uuXi=0 Landscapes. Graphicsontheleftshowtheinputandthemeanre-
Simulatedannealingappeartedtobeafastandefficientmethod sultingdistribution.Graphicsontherightshowallofthenetworks
of these landscapes sorted by decreasing size with a logarithmic
forthisparticulartask.Improvementsmadebysimulatedannealing
scale. Weclearlyseethattheneutraldegreedistributionisareally
areshowninfigure3.Easiestneutraldistributionstoobtainseemed
determiningparameterforthestructureofthegeneratedlandscape.
tobesmoothoneslikegaussiandistributions(whicharethemost
encounteredwhendealingwithrealorartificialneutralproblems).
Ontheotherhand,sharpdistributions(likethemiddle-rightoneof 3. TUNING DECEPTIVENESS
figure 3) are really hard to approach. In addition, independently
OF ND-LANDSCAPES
of theshape, distributionswithhigher average neutral degree are
hardertoapproximate. Oncewehavegeneratedalandscapewithaspecificneutralde-
greedistribution,wecanchangethefitnessvalueofallneutralnet-
2.3 Space and Timecomplexity
workswithoutchangingtheneutraldegreedistribution(aslongas
Tocreatealandscapewithasearchspaceofsize2N,weusean wedonotgivethesamefitnesstotwoadjacentnetworks). Hence,
array of size 2N containing fitness values and a list of forbidden foragivenneutraldistribution,wecantunethedifficultyofaND-
values for each solution. Thus we need a memory space of size Landscape. ForinstanceifeachNN havearandomfitnessvalue
O(2N ×N).Consequentlythespacecomplexityis:O(2N ×N) from[0,1]thenthelandscape isveryhardtooptimize. Here, we
Inordertoknowwhatarethepossibleneutraldegreesofanun- willusethewellknownTrapFunctions[20]toaffectfitnessesto
affectedsolutions,wemustconsidereveryinterestingvaluefors NN in order to obtain a ND-Landscape with tunable deceptive-
(thefitnessesofallneighbour solutionsandarandomvalue), and ness.
foreachofthesevalues, wemustfindoutallpossibleneutralde- Thetrapfunctionsaredefinedfromthedistancetooneparticular
grees.ThiscanbedoneinatimeO(N2).Weevaluatethepossible solution. They admit two optima, a global one and a local one.
neutraldegreesonceforeachsolution.Timeallowedforsimulated Theyareparametrizedbytwovaluesbandr.Thefirstone,ballows
annealing isproportional tothetimeelapsed duringconstruction. tosetthewidthoftheattractivebasinforeachoptima,andr sets
Thus,thetimecomplexityofthealgorithmisO(2NN2). theirsrelativeimportance. ThefunctionfT : {0,1}N → IRisso
0.25 D1 : construction D1 : construction
D2 : adjustment 0.3 D2 : adjustment
D0 : target distribution D0 : target distribution
0.2
0.25
0.15 0.2
0.15
0.1
0.1
0.05
0.05
0 0
0 2 4 6 8 10 0 2 4 6 8 10
dist(D ,D )=0.110 dist(D ,D )=0.119
1 0 1 0
dist(D ,D )=0.00338 dist(D ,D )=0.0175
2 0 2 0
0.25 D1 : construction 0.25 D1 : construction
D2 : adjustment D2 : adjustment
D0 : target distribution D0 : target distribution
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10
dist(D ,D )=0.0937 dist(D ,D )=0.151
1 0 1 0
dist(D ,D )=0.00246 dist(D ,D )=0.0693
2 0 2 0
0.25 D1 : construction D1 : construction
D2 : adjustment 0.2 D2 : adjustment
D0 : target distribution D0 : target distribution
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0 0
0 2 4 6 8 10 0 2 4 6 8 10
dist(D ,D )=0.0750 dist(D ,D )=0.0917
1 0 1 0
dist(D ,D )=0.0103 dist(D ,D )=0.00403
2 0 2 0
Figure3:NeutralDegreeDistributionsobtainedbyalgorithm1(D )andthenadjustedbysimulatedannealing(D ).Neutraldegrees
1 2
areonabcissa.Impulsesrepresentthetargetdistributions(D ).
0
definedby: nessvalueoftheNN issetaccordingtoatrapfunction2 andthe
distanced. Inordertoensurethatalladjacentnetworkshavedif-
1.0− d(x) ifd(x)≤b, ferentfitnessvalues,itispossibletoaddawhitenoisetothefitness
fT(x)=(r(d1.(0x−)−bbb) elsewhere values of each NN. In the following experiments, the length of
bitstringisN = 16. ND-landscapesareconstructedwithuniform
where d(x) is the Hamming distance to the global optimum, di- neutraldegreedistributions.Weusethedistributionsdefinedby
videdbyN,betweenxandoneparticularsolution.Theproblemis
mostdeceptiveasrislowandbishigh.Inourexperiment,wewill
1/w ifi∈{p,p+w−1},
usetwokindsofTrapfunctionswithr = 0.9,onewithb = 0.25 Dp,w[i]=
andanotheronewithb=0.75(seefigure5(a)and(b)). (0 elsewhere
Toaffectafitnessvaluetoeachneutralnetwork,wefirstchoose
where p ∈ {0,7}, w ∈ {3,4}, and the two Trap functions de-
the optimum neutral network, denoted NNopt, (for example the finedinfigure5. ForeachdistributionandeachTrapfunction,30
onecontainingthesolution0N)andsetitsfitnesstothemaximal
landscapesweregenerated.
value1.0.Then,foreachneutralnetwork,wecomputethedistance
dbetweenitscentroid1 andthecentroidofNNopt ;finallythefit-
ofbitvalue1ateachposition.
1ThecentroidofaNN isthestringofthefrequencyofappearance 2Thistrapfunctionisdefinedforallrealnumbersbetween0andN
0.3
DD02 104
0.25
0.2 103
proportion 0.15 size (log) 102
0.1
10
0.05
0 1
0 2 4 6 8 10 12 14 16 1 10 102 103
neutral degree rank (log)
meanneutraldegree5.15
0.18
0.16 DD02 104
0.14
0.12 103
proportion 0 0.0.18 size (log) 102
0.06
0.04 10
0.02
0 1
0 2 4 6 8 10 12 14 16 1 10 102 103
neutral degree rank (log)
meanneutraldegree5.00
0.09
0.08 DD02 104
0.07
0.06 103
proportion 00..0045 size (log) 102
0.03
0.02 10
0.01
0 1
0 2 4 6 8 10 12 14 16 1 10 102 103
neutral degree rank (log)
meanneutraldegree5.50
0.3
DD02 104
0.25
0.2 103
proportion 0.15 size (log) 102
0.1
10
0.05
0 1
0 2 4 6 8 10 12 14 16 1 10 102 103
neutral degree rank (log)
meanneutraldegree4.00
Figure4:NeutralnetworkssizesforvariousND-landscapes
3.1 Fitness DistanceCorrelation where:
ofND-Landscapes
m
1
Toestimatethedifficultytosearchintheselandscapeswewill CFD = (fi−f)(di−d)
m
useameasureintroducedbyJones[21]calledfitnessdistancecor- i=1
X
relation(FDC).GivenasetF = {f1,f2,...,fm}ofmindividual isthecovarianceofF andDandσF,σD,f anddarethestandard
fitnessvaluesandacorrespondingsetD={d1,d2,...,dm}ofthe deviations and averages of F and D. Thus, by definition, FDC
mdistancestotheglobaloptimum,FDCisdefinedas:
stands in the range [−1,1]. As we hope that fitness increases as
distancetoglobaloptimumdecreases,weexpectthat,withanideal
fitnessfunction, FDCwillassumethevalueof−1. Accordingto
Jones[21],problemscanbeclassifiedinthreeclasses,depending
FDC = CFD onthevalueoftheFDCcoefficient:
σFσD
1 1
deceptive trap (a)
0.9 easy trap (b)
0.8
0.8
0.6
0.6
fitness 0.4 0.4
0.2
0.2 FDC 0
0 -0.2
0 2 4 6 8 10 12 14 16
distance -0.4
(a)
-0.6
1
0.9 -0.8
0.8 -1
0 1 2 3 4 5 6 7 8 9
average neutral degree
0.6
fitness 0.4 Figure6:AverageandstandarddeviationofFDCasafunction
ofaverageneutraldegreeforND-landscapescreatedfromeasy
0.2 anddeceptiveTrapfunctions.
0
0 2 4 6 8 10 12 14 16
distance
(b) degree 5, the landscapes are fully deceptive (a) or fullyeasy (b).
Betweenneutraldegree5and7,thedeceptivenessandtheeasiness
Figure5:Twotrapfunctions:Deceptiveonewithb=0.25and decrease.Afterneutraldegree7,thetwotrapfunctionshavenearly
r=0.9(a),Easyonewithb=0.75andr=0.9(b) asamegoodsuccessrate(0.7). WhereastheFDCcoefficientgave
noinformationaboutperformances,examiningthescatterplotfit-
ness/distance allowed to predict these good performances (figure
7(c)). Whenneutralityincreasesondeceptivetrap,performances
• Deceptive problems (FDC ≥ 0.15), in which fitness in-
increasewhereasoneasytraptheydecrease.Theseresultsconfirm
creaseswithdistancetooptimum.
conclusionsfoundinsection3.1fromFDCmeasures.
• Hardproblems(−0.15 < FDC < 0.15)inwhichthereis
nocorrelationbetweenfitnessanddistance. 4. ADDITIVEEXTENDED
ND-LANDSCAPES
• Easyproblems(FDC ≤ −0.15)inwhichfitnessincreases
astheglobaloptimumapproaches. Exhaustivefitnessallocationallowsonlytogeneratelandscapes
withsmallsearchspace(inourexperiments216solutions). Hence
Hardproblemsareinfacthardtopredict,sinceinthiscase,the
wemustfindawaytoconstructsimilarproblemsonalargerscale.
FDCbringslittleinformation.Thethresholdinterval[−0.15,0.15]
We propose here to concatenate several small ND-Landscapes to
has been empirically determined by Jones. When FDC does not
create an additive ND-Landscape. Even though we are not able
give a clear indication i.e., inthe interval [−0.15,0.15], examin-
tocreateanadditiveND-Landscapedirectlyfromaneutraldegree
ing the scatterplot of fitness versus distance can be useful. The
distributionwecanchosethemean,thestandarddeviationandless
FDChasbeencriticizedonthegroundsthatcounterexamplescan
precisely the shape of his neutral degree distribution. Moreover,
beconstructedforwhichthemeasuregiveswrongresults[22,23,
thismethodallowsustoknowexactlytheneutraldegreedistribu-
24].
tion of the resulting landscape. Let be P = (E ,V ,f ) and
Figure6showstheaverageandstandarddeviationofFDCover 1 1 1 1
P = (E ,V ,f )twofitnesslandscapes.Wedefinetheextended
ND-Landscapesforeachsetofparameters,neutraldistributionand 2 2 2 2
landscapeP =P ⊕P = (E,V,f)suchas:
deceptiveness. TheabsolutevalueofFDCdecreasesaswegener- 1 2
ateND-Landscapeswithmoreandmoreneutrality. Whenadding
• E =E ×E
1 2
neutrality,thelandscapesareincreasinglyflatterandthuslesseasy
ordeceptive.Soneutralitysmoothescorrelation.Addingneutrality • ∀(x ,y )∈E2,∀(x ,y )∈E2,
1 1 1 2 2 2
toadeceptivelandscapemakesiteasierandaddingneutralitytoa (x ,x ) ∈ V(y ,y ) ⇐⇒ (x ∈ V (y ) and x =
1 2 1 2 1 1 1 2
easylandscapemakesitharder. y )or(x =y andx ∈V (y ))
2 1 1 2 2 2
3.2 Genetic Algorithm Performances on ND- • f(x ,x )=f (x )+f (x )
1 2 1 1 2 2
Landscapes
ThesizeofthelargerND-landscapewillbetheproductofthesizes
Inthissection, difficultyismeasuredbygeneticalgorithmper-
ofthesmallones. Theneutraldegreedistributionoftheresulting
formanceswhichisthesuccessrateover10independent runs. In
landscapeistheconvolutionproductofthetwocomponentsdistri-
order to minimize the influence of the random creation of ND-
butions. TheconvolutionproductoftwodistributionsD andD
1 2
Landscapes, we consider 30 different landscapes for each distri-
isthedistributionD(seefigure9)suchas
bution D and each trap function. For the GA, one-bit mutation
andone-pointcrossoverareusedwithratesofrespectively0.8and n
∀n∈IN,D(n)= (D (i)×D (n−i))
0.2. Theevolution, without elitismandwith3-tournament selec- 1 2
tionofapopulationof50individualstookplaceduring50gener- Xi=0
ations. Figure8showsaverageandstandarddeviationofGAper- Wehavethefollowingproperties:
formances over the 30 ND-landscapes. Until the average neutral
1
deceptive trap (a)
1 easy trap (b)
0.9 0.8
0.8
0.6
s 00..67 Success rate 0.4
s
e 0.5
n
fit 0.4 0.2
0.3
0
0.2 1 2 3 4 5 6 7 8 9
average neutral degree
0.1
Figure 8: Average with standard deviation of success rate of
0
aGAonND-landscapescreatedfromeasyanddeceptivetrap
0 2 4 6 8 10 12 14 16
functions
distance
(a)
0.3 0.3
1
0.25 0.25
0.9
0.2 0.2
0.8 0.15 0.15
0.7 0.1 0.1
0.6 0.05 0.05
s
nes 0.5 00 2 4 6 8 10 12 14 00 2 4 6 8 10 12 14
fit 0.4 (D3,4) (D9,4)
0.3
0.3
0.25
0.2
0.2
0.1
0.15
0
0 2 4 6 8 10 12 14 16 0.1
distance 0.05
(b) 0
0 5 10 15 20 25 30
1 convolutionproduct(D ⊕D )
3 9
0.9
0.8 Figure9:Exampleofconvolutionproduct
0.7
0.6 average(D)=average(D1)+average(D2)
ess 0.5 andσ(D) = σ2(D1)+σ2(D2)whereσisthestandardde-
n viation.
fit 0.4 p
Theconvolutionproductoftwonormal(resp. χ2,Poisson)dis-
0.3
tributionsisanormal(resp.χ2,Poisson)distribution.
0.2 Concatenation iscommutative, soit issimple toconcatenate a
number of small landscapes. Hence, we can tack together many
0.1
small ND-Landscapes generated exhaustively to obtain a bigger
0 onewithknownneutraldegreedistribution. Moreover,ithasbeen
0 2 4 6 8 10 12 14 16 provenbyJones[21]thattheFDCcoefficientisunchangedwhen
distance multiple copies of a problem are concatenated to form a larger
(c) problem.
Figure7:ND-Landscapes:FDCscatterplotfromeasytrap(cf Conclusion
Figure5a)withrespectiveaverageneutraldegree(a)1.5(b)6
(c)8.5 This paper presents the family of ND-Landscapes as a model of
neutralfitnesslandscapes.Mostoftheacademicfitnesslandscapes
foundintheliteraturedealswithneutrality,eitherasanadd-onfea- [9] P.F.Stadler.Fitnesslandscapes.InM.La¨ssigandValleriani,
tureorasanincidentalproperty. Theoreticalstudiesinmolecular editors,BiologicalEvolutionandStatisticalPhysics,volume
evolution,aswellasinoptimization,showthatthedistributionof 585ofLectureNotesPhysics,pages187–207,Heidelberg,
neutral degree isa key factor inthe evolution of population on a 2002.Springer-Verlag.
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