An experimental study was performed to explore whether it
is neutrality itself or simply the larger neighborhoods associ-
ated with neutral representations that influence the results
achieved by evolutionary algorithms on NK fitness landscape
problems. Markov chains were used to model the behaviour
of a stochastic hill-climber on NK fitness landscapes, using
two different types of representation: a neutral network rep-
resentation which exhibits neutrality and a redundant rep-
resentation without neutrality which implements the same
neighborhood induced by the corresponding neutral repre-
sentation

Correia, Marisol B.

Fonseca, C. M.

Universidade do Algarve

On the Roles of Redundancy and Neutrality in Evolutionary
Optimization: An Experimental Study
Marisol B. Correia
CSI – Centro de Sistemas Inteligentes & ESGHT
Universidade do Algarve
8005-117 Faro - Portugal
mcorreia@ualg.pt
Carlos M. Fonseca
CSI – Centro de Sistemas Inteligentes
Universidade do Algarve
8005-139 Faro - Portugal
cmfonsec@ualg.pt
ABSTRACT
An experimental study was performed to explore whether it
is neutrality itself or simply the larger neighborhoods associ-
ated with neutral representations that influence the results
achieved by evolutionary algorithms on NK fitness landscape
problems. Markov chains were used to model the behaviour
of a stochastic hill-climber on NK fitness landscapes, using
two different types of representation: a neutral network rep-
resentation which exhibits neutrality and a redundant rep-
resentation without neutrality which implements the same
neighborhood induced by the corresponding neutral repre-
sentation.
Categories and Subject Descriptors
I.2.8 [Artificial Intelligence]: Problem Solving, Control
Methods, and Search; I.6.5 [Simulation and Modeling]:
Model Development
General Terms
Design, experimentation
Keywords
Redundancy, neutral network, NK fitness landscape, hill-
climber, Markov chain
1. NEUTRAL VERSUS NON-NEUTRAL REP-
RESENTATIONS
The family of binary representations inspired by error-
control codes proposed in [1] exhibits various levels of neu-
trality, connectivity, synonymity and locality. In the same
work, another family of redundant, but non-neutral, binary
representations was proposed, which can be defined in terms
of a linear transformation between the genotypic and the
phenotypic spaces:
u = G · v (1)
where the columns of the binary matrix G consist of the
phenotypes which are reachable from the all-zero phenotype
through single gene mutations. Therefore, the connectivity
of the representation can be explicitly designed to match
the connectivity of given neutral representations. In this
Copyright is held by the author/owner(s).
GECCO’07, July 7–11, 2007, London, England, United Kingdom.
ACM 978-1-59593-697-4/07/0007.
0 100 200 300 400 500 600 700
0
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1
Number of iterations
Pr
ob
ab
ilit
y
Neutral
Non−neutral
Non−redundant
0 100 200 300 400 500 600 700
0
0.1
0.2
0.3
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Non−neutral
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Figure 1: Results on two different NK-landscape in-
stances
way, pairs of analogous representations differing mainly with
respect to neutrality may be obtained.
2. EXPERIMENTAL RESULTS
As the NK fitness landscapes can be gradually tuned from
smooth to rugged, it is a good fitness model to study differ-
ent types of neutral networks. The behaviour of a stochastic
hill-climber on an NK fitness landscape was modeled as a
Markov chain, and the long-term probability of reaching the
global optimum was computed for neutral network represen-
tations, their non-neutral analogues, and the original, non-
redundant binary representation, on various NK-landscape
instances. Figure 1 presents the results obtained using one
particular neutral network with three redundant bits on two
different NK(11, 1) landscape instances.
3. CONCLUSIONS
The results obtained suggest that the influence of the
neutral representations considered on the behaviour of a
stochastic hill-climber on NK-landscapes may not only be
due to the larger neighbourhoods induced by neutral net-
works, and that neutrality itself may improve the probabil-
ity of finding the global optimum. However, the choice of a
suitable neutral representation instance for a given problem
remains an open question.
Acknowledgments
The authors would like to thank Prof. Fernando Lobo for
his useful suggestions.
4. REFERENCES
[1] C. M. Fonseca and M. B. Correia. Developing redundant binary
representations for genetic search. Proceedings of the 2005
IEEE Congress on Evolutionary Computation (CEC05),
2:1675–1682, 2005.


On the roles of redundancy and neutrality in evolutionary optimization

Sapientia

On the Roles of Redundancy and Neutrality in EvolutionaryOptimization: An Experimental StudyMarisol B. CorreiaCSI – Centro de Sistemas Inteligentes & ESGHTUniversidade do Algarve8005-117 Faro - Portugalmcorreia@ualg.ptCarlos M. FonsecaCSI – Centro de Sistemas InteligentesUniversidade do Algarve8005-139 Faro - Portugalcmfonsec@ualg.ptABSTRACTAn experimental study was performed to explore whether itis neutrality itself or simply the larger neighborhoods associ-ated with neutral representations that influence the resultsachieved by evolutionary algorithms on NK fitness landscapeproblems. Markov chains were used to model the behaviourof a stochastic hill-climber on NK fitness landscapes, usingtwo different types of representation: a neutral network rep-resentation which exhibits neutrality and a redundant rep-resentation without neutrality which implements the sameneighborhood induced by the corresponding neutral repre-sentation.Categories and Subject DescriptorsI.2.8 [Artificial Intelligence]: Problem Solving, ControlMethods, and Search; I.6.5 [Simulation and Modeling]:Model DevelopmentGeneral TermsDesign, experimentationKeywordsRedundancy, neutral network, NK fitness landscape, hill-climber, Markov chain1. NEUTRAL VERSUS NON-NEUTRAL REP-RESENTATIONSThe family of binary representations inspired by error-control codes proposed in [1] exhibits various levels of neu-trality, connectivity, synonymity and locality. In the samework, another family of redundant, but non-neutral, binaryrepresentations was proposed, which can be defined in termsof a linear transformation between the genotypic and thephenotypic spaces:u = G · v (1)where the columns of the binary matrix G consist of thephenotypes which are reachable from the all-zero phenotypethrough single gene mutations. Therefore, the connectivityof the representation can be explicitly designed to matchthe connectivity of given neutral representations. In thisCopyright is held by the author/owner(s).GECCO’07, July 7–11, 2007, London, England, United Kingdom.ACM 978-1-59593-697-4/07/0007.0 100 200 300 400 500 600 70000.10.20.30.40.50.60.70.80.91Number of iterationsProbabilityNeutralNon−neutralNon−redundant0 100 200 300 400 500 600 70000.10.20.30.40.50.60.70.80.91Number of iterationsProbabilityNeutralNon−neutralNon−redundantFigure 1: Results on two different NK-landscape in-stancesway, pairs of analogous representations differing mainly withrespect to neutrality may be obtained.2. EXPERIMENTAL RESULTSAs the NK fitness landscapes can be gradually tuned fromsmooth to rugged, it is a good fitness model to study differ-ent types of neutral networks. The behaviour of a stochastichill-climber on an NK fitness landscape was modeled as aMarkov chain, and the long-term probability of reaching theglobal optimum was computed for neutral network represen-tations, their non-neutral analogues, and the original, non-redundant binary representation, on various NK-landscapeinstances. Figure 1 presents the results obtained using oneparticular neutral network with three redundant bits on twodifferent NK(11, 1) landscape instances.3. CONCLUSIONSThe results obtained suggest that the influence of theneutral representations considered on the behaviour of astochastic hill-climber on NK-landscapes may not only bedue to the larger neighbourhoods induced by neutral net-works, and that neutrality itself may improve the probabil-ity of finding the global optimum. However, the choice of asuitable neutral representation instance for a given problemremains an open question.AcknowledgmentsThe authors would like to thank Prof. Fernando Lobo forhis useful suggestions.4. REFERENCES[1] C. M. Fonseca and M. B. Correia. Developing redundant binaryrepresentations for genetic search. Proceedings of the 2005IEEE Congress on Evolutionary Computation (CEC05),2:1675–1682, 2005.

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On the roles of redundancy and neutrality in evolutionary optimization

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