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De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness Lands apes and Graphsin ombinatorial optimisationSeminar of ASAP group, university of NottinghamSébastien Verelsebastien.verel�inria.frDOLPHIN team - INRIA Lille-Nord Europe (Fran e)I3S laboratory - University of Ni e-Sophia Antipolis / CNRS (Fran e)http://www.i3s.uni e.fr/∼verelFebruary, 9 2010Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes : MotivationsWhy using �tness lands apes ?To analyse the stru ture of the sear h spa eTo study problem (sear h) di� ulty in ombinatorialoptimisation :information on runtime for a given problem and a lass of LSTo design e�e tive sear h algorithmsL. Barnett, U. Sussex, DPhil Diss. 2003"the more we know of the statisti al properties of a lass of �tnesslands apes, the better equipped we will be for the design ofe�e tive sear h algorithms for su h lands apes"Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes in biology
Biologi al s ien e :Wright 1930 [35℄Evolution :a metaphori al uphillstruggle a ross a "�tnesslands ape"mountain peaks representhigh "�tness", or ability tosurvive,valleys represent low �tness.evolution pro eeds :population of organismsperforms an "adaptive walk"Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes in biology In biology :Modelisation of spe iesevolutionUsed to model dynami alsystems :statisti al physi ,mole ular evolution,e ology, et Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes in biology2 sides for Fitness Lands apes :Powerful metaphor : most profound on ept in evolutionarydynami sgive pi tures of evolutionary pro essbe areful of misleading pi tures : "smooth lands ape withoutnoise"Quantitative on ept : predi t the evolutionary pathsQuasispe ies equation : mean �eld analysis with di�erentialequationsSto hasti pro ess : markov hainNetwork analysisSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksIn ombinatorial optimizationFitness lands ape (S,N , f ) :
S : set of admissible solutions,N : S → 2S : neighborhoodfun tion,f : S → IR : �tness fun tion.
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes for bla k-box optimisationFL : Tools for bla k-box optimisationBla kbox : We have only {(x , f (x)), ...}Sear h spa e analysis where "no" information is either not availableor needed on the de�nition of �tness fun tion.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksFitness lands apes in evolutionary omputation2 sides for Fitness Lands apes :Powerful metaphor : most profound on eptgive pi tures of the sear h dynami :"if the �tness lands apes have big valleys, I an use thisalgorithm"be areful of misleading pi tures : set of smooth mountainsQuantitative on ept : predi t the evolutionary dynami Quasispe ies equation : mean �eld analysis with di�erentialequationsSto hasti pro ess : markov hainNetwork analysisSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksWhat is a neighborhood ?N : S → 2S : neighborhood fun tion∀x ∈ S,N (x) = {y ∈ S | P(y = op(x)) > 0}or
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksWhat is a neighborhood ?N : S → 2S : neighborhood fun tion∀x ∈ S,N (x) = {y ∈ S | P(y = op(x)) > 0}orN (x) = {y ∈ S | d(y , x) ≤ 1}
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhoodSear h spa e : {0, 1}NAlgorithm : simple GA,hill- limbing, or simulatedannealing, et .x = 01101N (x) = {011010110001111010010010111101}
ImportantDe�nition of neighborhoood must bebased on the lo al sear h operatorused in the algorithmS = {0, 1}N
N (x) = {y ∈ S | dhamming(y , x) ≤1}Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhood
Traveling Salesman Problem :�nd the shortest tour whi h ross one time every townSear h spa e : set ofpermutationsAlgorithm : simple EA with2-opt mutation operator op2−opt
N (x) = {y ∈ S | P(y =op2−opt (x)) > 0}Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhood
Traveling Salesman Problem :�nd the shortest tour whi h ross one time every townSear h spa e : set ofpermutationsAlgorithm : simple EA with2-opt and 3-opt mutationoperators op2−opt and op3−opt
N (x) = {y ∈ S | P(y =op2−opt (x)) > 0 or P(y =op3−opt (x)) > 0}Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhoodAlgorithm : memeti algorithm, EA with hill- limbing mutationoperator opHCN (x) = {y ∈ S | y = opHC (x)}
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhoodAlgorithm : memeti algorithm, EA with hill- limbing mutationoperator opHCN (x) = {y ∈ S | y = opHC (x)}Algorithm : memeti algorithm, EA with hill- limbing mutationoperator opHC and bit-�ip mutation2 possibilities :Study 2 lands apes :one for HC operator, one for bit-�ip mutationStudy 1 lands ape :
N (x) = {y ∈ S | y = opHC (x) or P(y = opbit−�ip(x)) > 0}Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksExample of neighborhoodAlgorithm : memeti algorithm, EA with hill- limbing mutationoperator opHCN (x) = {y ∈ S | y = opHC (x)}Algorithm : memeti algorithm, EA with hill- limbing mutationoperator opHC and bit-�ip mutation2 possibilities :Study 2 lands apes :one for HC operator, one for bit-�ip mutationStudy 1 lands ape :
N (x) = {y ∈ S | y = opHC (x) or P(y = opbit−�ip(x)) > 0}It depends on what you want to knowSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes study"geometry" of �tness lands ape⇒ dynami of a lo al sear h algorithmgeometry is linked to the problem hardness :probability or time to have a �tness level for a given lo alsear h heuristi Study of the geometry of the lands ape allows to study the di� ulty
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To hoose the algorithm :analysis of global geometry of the lands apeWhi h algorithm an I use ?Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To hoose the algorithm :analysis of global geometry of the lands apeWhi h algorithm an I use ?3 To tune the parameters :o�-line analysis of stru ture of �tness lands apeWhi h is the mutation rate ? the size of the population ? et .Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To hoose the algorithm :analysis of global geometry of the lands apeWhi h algorithm an I use ?3 To tune the parameters :o�-line analysis of stru ture of �tness lands apeWhi h is the mutation rate ? the size of the population ? et .4 To ontrol the parameters during the run :on-line analysis of stru ture of �tness lands apeWhi h is the optimal mutation rate a ording to theestimation of stru ture ?Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima NetworksGoal of the �tness lands apes studyStudy of the geometry of the lands ape allows to study thedi� ulty, and design a good optimisation algorithmFitness lands ape is a graph (S,N , f ) where the nodes have avalue (�tness) : an be "pi tured" as a "real" lands apeTwo main geometries have been studied :multimodal and ruggednessneutral Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesLo al optima s∗ :no neighbor solution with higher �tness value∀s ∈ N (s∗), f (s) < f (s∗)
Search space
Fitness
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesAdaptive walk : (s0, s1, . . .) where si+1 ∈ N (si ) and f (si ) < f (si+1)Hill-Climbing (HC) algorithmChoose initial solution s ∈ Srepeat hoose s ′ ∈ N (s) su h that f (s ′) = maxx∈N (s) f (x)if f (s) < f (s ′) thens ← s ′end ifuntil s is a Lo al optimumBasin of attra tion of s∗ :{s ∈ S | HillClimbing(s) = s∗}.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesSearch space
Fitness
Optimisation di� ulty :number and size of attra tivebasins (Garnier et al [10℄)The idea :if the size of attra tive basinof global optima is relatively"small"the problem is di� ult tooptimizeSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesWalking on �tness lands apesSearch space
Fitness
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Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesRugged/smooth �tness lands apes 350
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E [(f (si )− f )(f (si+n)− f )]var(f (si ))auto orrelation length τ = 1ρ(1)small τ : rugged lands apelong τ : smooth lands apeSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesResults on rugged �tness lands apes (Stadler 96 [26℄)Problem parameter ρ(1)symmetri TSP n number of towns 1− 4nanti-symmetri TSP n number of towns 1− 4n−1Graph Coloring Problem n number of nodes 1− 2α
(α−1)nα number of olorsNK lands apes N number of proteins 1− K+1NK number of epistasis linksRuggedness de reases with the size of thoses problems :small variation has less e�e t on the �tness valuesSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness Distan e orrelation (FDC) (Jones 95 [15℄)Correlation between distan e to global optimum and �tness 0
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De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Fitness Lands apesNeutral theory (Kimura ≈ 1960 [17℄)Theory of mutation and random driftA onsiderable number of mutations have no e�e ts on �tnessvaluesgenotypes space
Fitness plateausneutral degreeneutral networks[S huster 1994 [25℄,RNA folding℄Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Fitness Lands apesCombinatorial optimizationRedundant problem (symetries, ...) (Goldberg 87 [12℄)Problem �not well� de�ned or dynami environment (Torres 04[14℄)genotypes space
Fitness Appli ative problems :Robot ontrolerCir uit designgeneti programmingProtein Foldinglearning problemsSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyIn our knowledge, there is no de�nitive answerabout neutrality / problem hardnessCertainly, it is dependent on the nature of neutrality of the�tness lands ape
⇒ Sharp des ription of the geometry of neutral �tness lands apesis neededSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyNo information is better than Bad information :Hard trap fun tions are more di� ult thanneedle-in-a-haysta k fun tionsGood information is better than No information
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyNo information is better than Bad information :Hard trap fun tions are more di� ult thanneedle-in-a-haysta k fun tionsGood information is better than No informationWhen there is No information : you should have a goodmethod to �nd it !
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesIn the followingDes ription of neutral �tness lands apes :Neutral sets :set of solutions with the same �tnessNeutral networks :add neighborhood information
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Density Of StatesSearch space
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De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Density Of States 0
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FitnessDensity of state (D.O.S.)Informations given :Performan e of randomsear hthe tail of the distribution isan indi ator of di� ulty :the faster the de ay, theharder the problemBut do not are about theneighborhood relation
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Fitness Cloud 0
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(S,F ,P) : probability spa eop : S → S sto hasti operator of the lo al sear hX (s) = f (s)Y (s) = f (op(s))Fitness Cloud of op onditional probability densityfun tion of Y given XSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loud : Measure of evolvability
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EvolvabilityAbility to evolve : �tnessin the neighborhood ompared to the �tness ofthe solutionProbability of �ndingbetter solutionsAverage �tness ofbetter neighborsolutionsAverage and standarddeviation of �tnessesSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loudPredi tion of evolution (CEC 2003)
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Shape of the GHC Fitness CloudAverage trajectory of GHC
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral �tness lands apesNeutral setsset of solutions with same �tness→ no stru tureOne tool : Fitness loudneighborhood relation between neutral setsNeutral NetworksIntrodu tion of neighborhood stru ture into the neutral setsSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral networks (S huster 1994 [25℄)
genotypes space
Fitness
Fitness
Neutral Network
Doors
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesDe�nitionstest of neutrality isNeutral : S × S → {true, false}For example, isNeutral(s1, s2) is true if :f (s1) = f (s2).|f (s1)− f (s2)| ≤ 1/M with M is the sear h population size.|f (s1)− f (s2)| is under the evaluation error.neutral neighborhoodof s is the set of neighbors whi h have the same �tness f (s)
Nneut(s) = {s ′ ∈ N (s) | isNeutral(s, s ′)}neutral degreeof a solution is the number of its neutral neighborsnDeg(s) = ♯(Nneut (s)− {s}).Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesDe�nitionsneutral walkWneut = (s0, s1, . . . , sm)for all i ∈ [0,m − 1], si+1 ∈ N (si )for all (i , j) ∈ [0,m]2 , isNeutral(si , sj) is true.Neutral Networkgraph G = (N,E )N ⊂ S : for all s and s ′ from V , there is a neutral walk belongingto V from s to s ′ ,Two verti es are onne ted by an edge of E if they are neutralneighbors. A �tness lands ape is neutralif there are many solutions with high neutral degree.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Inside Metri sClassi al metri s of graph to des ribe NN :1 Size of NN : number of nodes of NN,2 Neutral degree distribution :measure of the quantity of "neutrality"3 Auto orrelation of neutral degree during neutral random walk(Bastolla 03 [3℄) : omparaison with random graph,measure of the orrelation stru ture of NNSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Outside Metri sNeutral Network
Fitness
Neutral random walk
S0S3S2
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1 rate of innovation(Huynen 96 [13℄) :The number of newa essible stru tures(�tness) per mutation2 Auto orrelation ofevolvability [32℄ :auto orrelation of thesequen e(evol(s0), evol(s1), . . .).Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Multimodal and rugged �tness lands apesNeutral �tness lands apesSummary of metri sNeutral degrees distribution :"How neutral is the �tness lands ape ?"Auto orrelation of neutral degrees : network �stru ture�High
0.20.0 0.35 0.6 1.0
Middle strongLowrate of innovation :low information for ombinatorial optimizationAuto orrelation of maximal evolvability :information on the links between NNSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesMotivation and general idea : Levels of des riptionFDC, autocorrelation, etc.High level
Medium level
Low level
Local Optima Network
Fitness landscape
One metric
local optima, basins of attraction
SolutionsFitness lands apes : based on an huge number of solutionsOne metri : based on one real number, or urve to at h allthe omplexityLo al optima Network : based on lo al optimaSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesOverview and MotivationBring the tools of omplex networks analysis to the study thestru ture of ombinatorial �tness lands apesGoals : Understand problem di� ulty, design e�e tive heuristi sear h algorithmsMethodology : Extra t a network that represents the lands ape(Inspiration from energy lands apes (Doye, 2002 )1)Verti es : lo al optimaEdges : a notion of adja en y between basinsCondu t a network analysisRelate (exploit ?) network features to sear h algorithm design1J. P. K. Doye, The network topology of a potential energy lands ape : astati s ale-free network., Phys. Rev. Lett., 88 :238701, 2002.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesSmall − world networks (Watts and Strogatz, 1998)Neither ordered nor ompletely randomNodes highly lustered yet path length is smallNetwork topologi al measures :C : lustering oe� ient, measure of lo al densityl : shortest path length global measure of separationS ale − free networks (Barabasi and Albert, 1999)The distribution of the number of neighbours (the degreedistribution) is right − skewed with a heavy tailMost of the nodes have less-than-average degree, whilst asmall fra tion of hubs have a large number of onne tionsDes ribed mathemati ally by a power-lawSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesEnergy surfa e and inherent networks (Doye, 2002)a Model of 2D energy surfa eb Contour plot, partition ofthe on�guration spa e intobasins of attra tionsurrounding minima lands ape as a networkInherent network :Nodes : energy minimaEdges : two nodes are onne ted if the energy barrierseparating them is su� iently low (transition state)Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Bit strings of length N = 626 = 64 solutionsone point = one solution
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Bit strings of length N = 6Neighborhood size = 6Line between points =solutions are neighborsHamming distan es betweensolutions are saved (ex eptfor at the border of the ube)Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Color represent �tness valuered = the �tness is highblue = the �tness is low
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Color represent �tness valueRed arrow = toward thesolution with the highest�tness in the neighborhood(if better)remember that the size ofthe neighborhood is 6Why not make a Hill-Climbingwalk on it ?Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Ea h olor orrespond to onebasin of attra tionBasins of attra tion are veryimbri ateBasins have no "interior"
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2
Basin of attra tion are very imbri ateA lot of neighor's solution outside the basinSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesLo al optima network0.76
0.185
0.65
0.29
0.270.4
0.55 0.05
0.33
Nodes : lo al optimaWeighted edges : probabilityto pass from randomsolution of one basin toanotherSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesBasin of attra tionHill-Climbing (HC) algorithmChoose initial solution s ∈ Srepeat hoose s ′ ∈ N (s) su h that f (s ′) = maxx∈N (s) f (x)if f (s) < f (s ′) thens ← s ′end ifuntil s is a Lo al optimumBasin of attra tion of s∗ :{s ∈ S | HillClimbing(s) = s∗}.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapeslo al optima networkLo al optima networkNodes : set of lo al optima S∗Edges : notion of onne tivity between basins of attra tioneij between i and j if there is at least a pair of neighbours siand sj ∈ N (si ) su h that si ∈ bi and sj ∈ bj (GECCO 2008[21℄)weights wij is atta hed to the edges, a ount for transitionprobabilities between basins (ALIFE 2008 [33℄, Phys. Rev. E2008 [30℄)Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesWeights of edgesFrom ea h s and s ′ , p(s → s ′) = P(s ′ = op(s)) theprobability to pass from s to s ′For example, S = {0, 1}N and bit-�ip operatorif s ′
∈ N (s) , p(s → s ′
) = 1Nif s ′
6∈ N (s) , p(s → s ′
) = 0
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesWeights of edgesFrom ea h s and s ′ , p(s → s ′) = P(s ′ = op(s)) theprobability to pass from s to s ′For example, S = {0, 1}N and bit-�ip operatorif s ′
∈ N (s) , p(s → s ′
) = 1Nif s ′
6∈ N (s) , p(s → s ′
) = 0Probability that a on�guration s ∈ S has a neighbor in abasin bj p(s → bj) =∑s′∈bj p(s → s ′)
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesWeights of edgesFrom ea h s and s ′ , p(s → s ′) = P(s ′ = op(s)) theprobability to pass from s to s ′For example, S = {0, 1}N and bit-�ip operatorif s ′
∈ N (s) , p(s → s ′
) = 1Nif s ′
6∈ N (s) , p(s → s ′
) = 0Probability that a on�guration s ∈ S has a neighbor in abasin bj p(s → bj) =∑s′∈bj p(s → s ′)wij : Total probability of going from basin bi to basin bj is theaverage over all s ∈ bi of the transition prob. to s ′ ∈ bj :p(bi → bj ) =1
♯bi ∑s∈bi p(s → bj)⇒ lo al optima network : weighted oriented graphSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesNK �tness lands apes : ruggedness and epistasisK from 0 to N − 1, NK lands apes an be tuned from smoothto rugged (easy to di� ult respe tively)K = 0 no orrelations, f is an additive fun tion, and there is asingle maximumK = N − 1 lands ape ompletely random, the expe tednumber of lo al optima is 2NN+1Intermediate values of K interpolate between these twoextreme ases and have a variable degree of epistasis (i.e. geneintera tion) Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesMethodsExtra ted and analysed networks for N = 14, 16 and 18,K = 2, 4, . . . ,N − 2,N − 1 (30 random instan es for ea h ase)Measures :Statisti s on basins sizes and �tness of optimaNetwork features : lustering oe� ient, shortest path to theglobal optimum, weight distribution, disparity, boundary ofbasins
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesGlobal optimum basin size versus K 1e-05
0.0001
0.001
0.01
0.1
1
2 4 6 8 10 12 14 16 18
Nor
mal
ized
siz
e of
the
glob
al o
ptim
a’s
basi
n
K
N=16N=18
Size of the basin orresponding tothe global maximum for ea h KTrend : the basin shrinksvery qui kly with in reasingK.for higher K, more di� ultfor a sear h algorithm tolo ate the basin of attra tionof the global optimum
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : basin size 0.1
1
10
100
1000
0 2000 4000 6000 8000 10000 12000
cum
ulat
ive
dist
ribut
ion
size of basin
exp.regr. line
Cumulative distribution of basinssizes for N = 18 and K = 4Trend : small number oflarge basin, large number ofsmall basinlog-normal umulativedistributionslope of orrelation in reaseswith Kwhen K large : basin sizesare nearly equalsSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : basin size 1e-20
1e-15
1e-10
1e-05
1
100000
0 100 200 300 400 500 600 700 800 900 1000
cum
ulat
ive
dist
ribut
ion
size of basin
K=2K=4K=6K=8
K=10K=12K=14K=16K=17
Trend : small number oflarge basin, large number ofsmall basinlog-normal umulativedistributionslope of orrelation in reaseswith Kwhen K large : basin sizesare nearly equalsSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : �tness vs. basin size 1
10
100
1000
10000
0.5 0.55 0.6 0.65 0.7 0.75 0.8
basi
n of
attr
actio
n si
ze
fitness of local optima
exp.regr. line
Correlation �tness of lo al optimavs. their orresponding basinssizesTrend : lear positive orrelation between the�tness values of maxima andtheir basins' sizesOn average, the globaloptimum easier to �nd thanone other lo al optimumBut more di� ult to �nd, asthe number of lo al optimain reases exponentially within reasing KSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesGeneral network statisti sWeighted lustering oe� ientlo al density of the network w (i) =1si(ki − 1) ∑j ,h wij + wih2 aijajhahiwhere si =
∑j 6=i wij , anm = 1 if wnm > 0, anm = 0 if wnm = 0 andki =∑j 6=i aij .Disparitydishomogeneity of nodes with a given degreeY2(i) =
∑j 6=i (wijsi )2Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesGeneral network statisti s N = 16K ♯ nodes ♯ edges Cw Y d2 3315 516358 0.960.0245 0.3260.0579 56144 17833 91292930 0.920.0171 0.1370.0111 12686 46029 417914690 0.790.0154 0.0840.0028 17038 89033 933844394 0.650.0102 0.0620.0011 194210 1, 47034 1621394592 0.530.0070 0.0500.0006 206112 2, 25432 2279122670 0.440.0031 0.0430.0003 207114 3, 26429 2907322056 0.380.0022 0.0400.0003 203115 3, 86833 3212032061 0.350.0022 0.0390.0004 2001Clustering Coe� ient : For high K, transition between a givenpair of neighboring basins is less likely to o urDisparity : For high K the transitions to other basins tend tobe ome equally likely, an indi ation of the randomness of thelands ape Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesWeights distribution 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0001 0.001 0.01 0.1 1
P(w
ij=W
)
W
K=2K=4
K=10K=14K=17
distribution of the networkweights wij for outgoing edgeswith j 6= i in log-x s ale, N = 18Weights (transition prob.between neighbouringbasins) are smallFor high K the de ay isfasterLow K has longer tailsOn average, the transitionprobabilities are higher forlow KSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesWeight distribution remain in the same basin 0
5e-05
0.0001
0.00015
0.0002
0.00025
0.01 0.1 1
P(w
ii =
W)
Wii
K=2K=6
K=14
Average weight wii a ording tothe parameter N and KWeights to remains in thesame are large ompare towij with i 6= jwii are higher for low Keasier to leave the basin forhigh K : high explorationBut : number of lo aloptima in reases fast with K
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesInterior and border size 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
2 4 6 8 10 12 14 16 18
aver
age
of th
e m
ean
size
K
N=14N=16N=18
Average of the mean size ofbasins interiorsDo basins look like a "montain"with interior and border ?solution is in the interior ifall neighbors are in the samebasinInterior is very smallNearly all solution are in theborder
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesShortest path length between lo al optima 0
50
100
150
200
250
300
2 4 6 8 10 12 14 16 18
aver
age
path
leng
th
K
N=14N=16N=18
Average distan e (shortest path)between nodesIn rease with N (♯ of nodesin reases exponentially)For a given N, in rease withK up to K = 10, thenstagnatesSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesShortest path length to global optima 0
50
100
150
200
250
2 4 6 8 10 12 14 16 18
aver
age
path
leng
th to
the
optim
um
K
N=14N=16N=18
Average path length to the globaloptimum from all the other basinsMore relevant foroptimisationIn rease steadily within reasing KSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesSummary on lo al optima networkMedium level of des ription : proposed hara terization of ombinatorial lands apes as networksNew �ndings about basin's stru ture : sizes, �tness vs. size,et .Related some network features to sear h di� ulty
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesFuture on lo al optima networkDe�ne basin for neutral �tness lands apes :NK with neutrality (under submission)Test on more realisti ombinatorial �tness lands apes :Quadrati Assignment Problem (under submission)Generalised the on ept of basin :the �rst improvment heuristi s (under writing)Compare the properties of Lo . Opt. Network and the optimaltradeo� between exploration and exploitationStudy the LON like a �tness lands apeDedu e some approximation of the runtime from theproperties of LONSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesSummary on �tness lands apesFitness lands ape is a representation ofnotion of neighborhood�tness of solutions
Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesSummary on �tness lands apesFitness lands ape is a representation ofnotion of neighborhood�tness of solutionsGoal :lo al des ription : �tness between neighbor solutionsRuggedness, lo al optima, �tness loud, neutral networks, lo aloptima networks...and to dedu e global results :Di� ulty !to de ide a good hoi e of the representation, operator and�tness fun tion Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesOpen questionsHow to ontrol the parameters and/or operators of thealgorithm with the lo al des ription of �tness lands ape ?Can �tness lands ape des ribe the dynami s of a population ofsolutions ?Links between neutrality and �tness di� ulty ?Whi h intermediate des ription shows relevant properties ofthe optimization problem a ording to the lo al sear hheuristi ?What is the �tness lands apes for a multiobje tive problem ?Integration of the FL tools into the open framework paradisEOhttp://paradiseo.gforge.inria.frSébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesL. Barnett.Ruggedness and neutrality - the NKp family of �tnesslands apes.In C. Adami, R. K. Belew, H. Kitano, and C. Taylor, editors,ALIFE VI, Pro eedings of the Sixth International Conferen eon Arti� ial Life, pages 18�27. ALIFE, The MIT Press, 1998.Lionel Barnett.Net rawling - optimal evolutionary sear h with neutralnetworks.In Pro eedings of the 2001 Congress on EvolutionaryComputation CEC2001, pages 30�37, COEX, World TradeCenter, 159 Samseong-dong, Gangnam-gu, Seoul, Korea, 27-302001. IEEE Press.U. Bastolla, M. Porto, H. E. Roman, and M. Vendrus olo.Statis al properties of neutral evolution.Journal Mole ular Evolution, 57(S) :103�119, August 2003.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesMeriema Belaidouni and Jin-Kao Hao.An analysis of the on�guration spa e of the maximal onstraint satisfa tion problem.In PPSN VI : Pro eedings of the 6th International Conferen eon Parallel Problem Solving from Nature, pages 49�58,London, UK, 2000. Springer-Verlag.P. Collard, M. Clergue, and M. Defoin Platel.Syntheti neutrality for arti� ial evolution.In Arti� ial Evolution : Fourth European Conferen e AE'99,pages 254�265. Springer-Verlag, 2000.Sele ted papers in Le ture Notes in Computer S ien es 1829.J. C. Culberson.Mutation- rossover isomorphisms and the onstru tion ofdis rimination fun tion.Evolutionary Computation, 2 :279�311, 1994.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesJ. P. K. Doye.The network topology of a potential energy lands ape : a stati s ale-free network.Phys. Rev. Lett., 88 :238701, 2002.J. P. K. Doye and C. P. Massen.Chara terizing the network topology of the energy lands apesof atomi lusters.J. Chem. Phys., 122 :084105, 2005.Ri ardo Gar ia-Pelayo and Peter F. Stadler.Correlation length, isotropy, and meta-stable states.Physi a D, 107 :240�254, 1997.Santa Fe Institute Preprint 96-05-034.Josselin Garnier and Leila Kallel.E� ien y of lo al sear h with multiple lo al optima.SIAM Journal on Dis rete Mathemati s, 15(1) :122�141, 2002.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesP. Git ho� and G. Wagner.Re ombination indu ed hypergraphs : A new approa h tomutation-re ombination isomorphism, 1996.David E. Goldberg and Philip Segrest.Finite markov hain analysis of geneti algorithms.In ICGA, pages 1�8, 1987.M. Huynen.Exploring phenotype spa e through neutral evolution.Journal Mole ular Evolution, 43 :165�169, 1996.E. Izquierdo-Torres.The role of nearly neutral mutations in the evolution ofdynami al neural networks.In J. Polla k and al, editors, Ninth International Conferen e ofthe Simulation and Synthesis of Living Systems (Alife 9), pages322�327. MIT Press, 2004.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesT. Jones.Evolutionary Algorithms, Fitness Lands apes and Sear h.PhD thesis, University of New Mexi o, Albuquerque, 1995.S. A. Kau�man.The Origins of Order.Oxford University Press, New York, 1993.M. Kimura.The Neutral Theory of Mole ular Evolution.Cambridge University Press, Cambridge, UK, 1983.J. Lobo, J. H. Miller, and W. Fontana.Neutrality in te hnology lands ape, 2004.M. Newman and R. Engelhardt.E�e t of neutral sele tion on the evolution of mole ularspe ies. Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesIn Pro . R. So . London B., volume 256, pages 1333�1338,1998.Erik Van Nimwegen, James P. Crut h�eld, and MartijnHuynen.Metastable evolutionary dynami s : Crossing �tness barriers ores aping via neutral paths ?Te hni al Report 99-07-041, SanteFe institute, 1999.Gabriela O hoa, Mar o Tomassini, Sébastien Verel, andChristian Darabos.A Study of NK Lands apes' Basins and Lo al OptimaNetworks.In Pro eedings of the 10th annual onferen e on Geneti andevolutionary omputation Geneti And EvolutionaryComputation Conferen e, pages 555�562, Atlanta États-Unisd'Amérique, 07 2008. ACM New York, NY, USA.best paper nomination.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesM. Defoin Platel.Homologie en Programmation Génétique - Appli ation à larésolution d'un problème inverse.PhD thesis, Université de Ni e Sophia Antipolis, Fran e, 2004.Eduardo Rodriguez-Tello, Jin-Kao Hao, and JoseTorres-Jimenez.A new evaluation fun tion for the minla problem.In Pro eedings of the MIC 2005, pages 796�801, ViennaAustria, 2005.Helge Rosé, Werner Ebeling, and Torsten Asselmeyer.The density of states - a measure of the di� ulty ofoptimisation problems.In Parallel Problem Solving from Nature, pages 208�217, 1996.P. S huster, W. Fontana, P. F. Stadler, and I. L. Hofa ker.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesFrom sequen es to shapes and ba k : a ase study in RNAse ondary stru tures.In Pro . R. So . London B., volume 255, pages 279�284, 1994.Peter F. Stadler.Lands apes and their orrelation fun tions.J. Math. Chem., 20 :1�45, 1996.Peter F. Stadler and W. S hnabl.The lands ape of the traveling salesmen problem.Phys. Letters, A(161) :337�344, 1992.Peter F. Stadler and Gunter P. Wagner.Algebrai theory of re ombination spa es.Evolutionary Computation, 5(3) :241�275, 1997.Terry Stewart.Extrema sele tion : A elerated evolution on neutral networks.Sébastien Verel Fitness lands apes
De�nition of �tness lands apeMultimodal, rugged and neutral �tness lands apesLo al Optima Networks Complex networksDe�nitionsLON of NK landsapesIn Pro eedings of the 2001 Congress on EvolutionaryComputation CEC2001, pages 25�29, COEX, World TradeCenter, 159 Samseong-dong, Gangnam-gu, Seoul, Korea, 27-30May 2001. IEEE Press.Mar o Tomassini, Sébastien Verel, and Gabriela O hoa.Complex-network analysis of ombinatorial spa es : The NKlands ape ase.Physi al Review E : Statisti al, Nonlinear, and Soft MatterPhysi s, 78(6) :066114, 12 2008.89.75.H ; 89.75.Fb ; 75.10.Nr.Vesselin K. Vassilev and Julian F. Miller.The advantages of lands ape neutrality in digital ir uitevolution.In ICES, pages 252�263, 2000.Sebastien Verel, Philippe Collard, and Manuel Clergue.Measuring the evolvability lands ape to study neutrality.Sébastien Verel Fitness lands apes
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