The Nei’s Standard Genetic Distance in
Artificial Evolution
Yoshiaki Katada

Kazuhiro Ohkura

Kanji Ueda

Graduate School of
Science and Technology
Kobe University
Kobe 657-8501, JAPAN
Email: katada@rci.scitec.kobe-u.ac.jp

Faculty ofEngineering
Kobe University
Kobe 657-8501, JAPAN
Email: ohkura@rci.scitec.kobe-u.ac.jp

RACE (Research into Artifacts,
Center for Engineering)
The University of Tokyo
Meguro 153-8904, JAPAN
Email: ueda@race.u-tokyo.ac.jp

Abstract— In recent years, not only ruggedness but also
neutrality has been recognized as an important feature of afitness landscape for genetic search. Following that the concept ofneutrality in artificial evolution originates from Kimura’s neutral
theory in natural evolution, it is expected that the dynamics
of artificial evolution in the landscapes including neutrality
would be described by using techniques in population genetics.
Furthermore, new theoretical guidelines might be developedfor effective genetic search. In a recent paper [25], we havediscussed the use of the Nei’s standard genetic distance, which
originates from population genetics, for measuring neutralityof fitness landscapes. In our results, several consistencies withthe population genetics have been found by applying the Nei’sstandard genetic distance to a tunably neutral NK landscape. In
this paper, computer simulations are systematically conductedby using a standard genetic algorithm in order to clarify
the characteristics of the Nei’s standard genetic distance. The
terraced NK landscape is adopted as a test function.

I. I NTRODUCTION
Many EA researchers have been inspired by natural evolution and trying to model the process of natural evolution
in order to develop powerful optimization methods [1][2][3].
Therefore, the dynamics in artificial evolution would be explained by the theory of natural evolution. But until recently,
few theories which are applicable to artificial evolution havebeen found. Since the concept of neutrality was introduced
into the EA community, EA researchers have expected that the
dynamics of artificial evolution would be described by using
techniques in population genetics. This is because the conceptof neutrality in artificial evolution originates from Kimura’s
neutral theory in population genetics.
Therefore, neutrality has attracted much research interestin recent years [4][5]. This feature, due to highly redundant
mappings from genotype to phenotype or from phenotype tofitness, is also found in natural systems. From this point ofview, evolutionary theorists [6] and molecular biologists [7][8]also have investigated it.
Neutrality has been found in many real-world applicationsof artificial evolution, such as evolution of neural network
controllers in robotics [9][10][11][12], on-chip electronic circuit evolution [13][14][15]. Landscapes which include neutrality have been conceptualized as containing neutral networks[16][17][18]. Harvey first introduced the concept of neutral

networks into the EA community [16]. He defined it asfollows: “A neutral network of a fitness landscape is defined
as a set of connected points of equivalent fitness, each representing a separate genotype: here connected means that there
exists a path of single (neutral) mutations which can traverse
the network between any two points on it without affecting
fitness.” For these years, several papers [19][20][17][18] havebeen published for investigating the evolutionary dynamics.
Population geneticists have been trying to explain evolutionary change quantitatively, that is, the change of genefrequency in the population. Statistical methods for estimating
the number of gene differences and the divergence time
between related species have been developed. These methods
use electrophoretic data for investigating protein variation.
The results are compared with the divergence time derived
from the fossil records. However, population geneticists cannot
get complete information about the genetic material throughelectrophoresis. In artificial evolution, EA researchers can get
all the genetic information of a population. Furthermore they
can define genetic operators. Therefore, the introduction ofsuch statistical methods for estimating the number of genesubstitutions would be helpful to understand the mechanism
of EAs for solving difficult optimization problems.
From a theoretical point of view, it would be beneficial toinvestigate whether the number of substitutions estimated in
EAs can be understood by the theory of natural evolution.
According to Kimura’s neutral theory [6] and Ohta’s nearly
neutral theory [21][22], the following assertions have been
made [23]:1) For each protein, the rate of evolution in terms of aminoacid substitutions is approximately constant per year persite for various lines, as long as the function and tertiary
structure of the molecule remain essentially unaltered.
2) Functionally less important molecules or parts of amolecule evolve (in terms of mutant substitutions) fasterthan more important ones.
3) Those mutant substitutions that disrupt less the existing
structure and function of a molecule (conservative substitutions) occur more frequently in evolution than moredisruptive ones.
4) Gene duplication must always precede the emergence ofa gene having a new function.
5) Selective elimination of definitely deleterious mutants
and random fixation of selectively neutral or veryslightly deleterious mutants occur far more frequently inevolution than positive Darwinian selection of definitely
advantageous mutants.
Recently, we have discussed the use of the Nei’s standard
genetic distance[24], which is one of such statistical methods
for estimating the number of substitutions, for measuring
neutrality of fitness landscapes 1 [25]. In our experiments,
several consistencies with population genetics have been foundby applying the Nei’s standard genetic distance to the resultsof evolution on tunably neutral NK landscapes. These can besummarized as follows:
Under small mutation rate per locus and fixed population size,• The number of gene substitutions increases with theincrease of neutrality.
• The number of gene substitutions decreases with theincrease of ruggedness where the landscape includes
neutrality.
• The number of gene substitutions is largest when random
sampling is applied with mutation.
To clarify whether these results hold for different population
sizes and to discuss the consistencies with population genetics,
a systematic investigation should be done.
This paper investigates the characteristics of the Nei’sstandard genetic distance in fitness landscapes including neutrality in various conditions. The next section describes the
Nei’s standard genetic distance. Section III applies the Nei’s
genetic distance to one of tunably neutral landscapes called
the terraced NK landscape and shows the results. Section IV
discusses the error threshold on the population size and themutation rate based on the obtained results. Conclusions are
given in the last section.

where, Ii = 1.0 if the two populations have the same allelesin identical frequencies, and Ii = 0.0 if they have no common
alleles. The normalized identity of genes between X and Y
with respect to the average in all loci is defined as

II. T HE N EI ’ S S TANDARD G ENETIC D ISTANCE

A terraced NK landscape was employed as the test functionin our computer simulations. This is the tunably neutral
NK landscape proposed by Newman and Engelhardt [26]. A
terraced NK landscape has three parameters: N , the lengthof the genotype; K, the number of epistatic linkages between
genes; and w, the contribution of a locus to the fitness of theentire genotype.
The fitness value is calculated as follows: The fitness contribution of the i-th locus, w i , is an integer generated randomly inthe range 0 ≤ wi < F, i = 1, · · · , N . To calculate the fitness,
W , of a genotype, the fitness contribution of each locus isaveraged, and then divided by F − 1, normalizing W to therange 0.0 to 1.0. More formally:

Genetic distance is a term of population genetics used for
estimating gene differences per locus between populations.
Although there are several definitions for this, the Nei’sstandard genetic distance [24] is adopted in this paper.
The Nei’s standard genetic distance is defined as follows.
Consider two populations, X and Y . Let x ik and yik be thefrequencies of the k-th alleles (i = 1, · · · , N , k ∈ {1, 2} in abinary coded GA) in X and Y , respectively. The probabilityof identity of two randomly chosen genes is j xi = x2i1 +x2i2 in22
the population X, while it is j yi = yi1+ yi2
in the population
Y . The probability of identity of a gene from X and a genefrom Y is jxyi = xi1 yi1 + xi2 yi2 . The normalized identity ofgenes between X and Y with respect to a locus is defined as
jxyi
Ii = √  ,jxi jyi(1)

1 Since the assertion 1), 2) and 3) can be interpreted as the number of genesubstitutions of each genotype increases with the increase of neutrality, thenumber of gene substitutions could be an index of neutrality.

JXY
I=√ √ ,
JX JY

(2)

N
N
where, JX = i=1 jxi /N , JY = i=1 jyi /N and JXY =N
i=1 jxyi /N . The genetic distance between X and Y is
defined as
D = − loge I,(3)

under the assumption that the mutation rate per locus issufficiently small. However, the above definition cannot beapplied to the standard GA directly, because it is assumed thata new allele always appears on a locus when a mutation occurs,
while “back mutations [21]” frequently occur in the standardGA, due to the binary coding scheme. Therefore, the genetic
distance between the population at the initial generation andthe one at the last generation is calculated as:
Df inal =T−1Dt,t+1(4)

1where T is the number of the last generation and D t,t+1 isthe genetic distance between the population in the t-th and the(t + 1)-th generation. The rate of gene substitution is definedas the genetic distance per generation.
III. T HE N EI ’ S S TANDARD G ENETI C D ISTANCE IN A
T UNABLY N EUTRAL NK L ANDSC APE
A. A Terraced NK Landscape

N

W =


1
wi .
N (F − 1)

(5)

i=1

The neutrality of the landscape can be tuned by changing thevalue of F . The neutrality of the landscape is maximized whenF = 2, and is effectively non-existent as F → ∞.

1

0

2
Number of substitutions

2
Number of substitutions

Number of substitutions

2

1

0
0

500

1000

1500

2000

1

0
0

500

1000

1500

2000

0

500

1000

Generation

Generation

Generation(a) K = 0(b) K = 2(c) K = 6

1

0

1

0
0

500

1000
Generation

1500

2000

0

500

(d) K = 12
Fig. 1.

2000

2
Number of substitutions

Number of substitutions

2

1500

1000
Generation

1500

2000

(e) K = 19

Number of substitutions at each generation for the SGA with q = 0.008 and M = 50 for F = ∞ in 50 runs

B. Simulation Conditions

C. Existence And Non-existence of Neutrality
The first experiments were conducted to investigate theeffect of the existence of neutrality on the transition of thegenetic distance and the number of substitutions. Fig. 1 shows
the number of substitutions of the SGA for F = ∞, where
q was set at 0.008 based on the assumption of eq.(3). They
2 For F = ∞, the NK fitness landscape[27] was employed instead of theterraced NK landscape as [5], which results in practically non-existence ofneutrality.

Number of substitutions

We applied two genetic algorithms: the standard GA(SGA) and the (random-sampling, q)-algorithm. The (randomsampling, q)-algorithm employs standard bit mutation at therate of q as the genetic operation and random sampling as
a selection method where M offsprings are sampled fromM ancestors with replacements. This model was used toinvestigate the effect of random sampling with mutation on
the genetic distance. This is approximately equivalent with
Kimura’s stochastic genetic models to study random genetic
drift and the expected time of fixation of a mutant gene [6].
Computer simulations were conducted by varying the landscape parameters, the population size, M , and the mutation
rate, q. The SGA used standard bit mutation as the genetic
operation. Crossover was not employed. Tournament selection
was adopted for the SGA. The tournament size was set at 2
for the SGA. Each run lasted 2,000 generations. We conducted
50 independent runs for each problem under the landscape parameters, N = 20, K ∈ {0, 2, 6, 12, 19}, F ∈ {2, 3, 4, 6, ∞ 2}.
The results were averaged over 50 runs.

10
8
6
4
2
0
0

500

1000
Generation

1500

2000

Fig. 2. Number of substitutions at each generation for the (random-sampling,q)-algorithm with q = 0.008 and M = 50 in 50 runs

leveled off in the very early generations. This means that the
population converged to a certain point in the genotype spacethen the genetic distance between the generations (D t,t+1 ineq.(4)) became zero. In contrast, the number of substitutionsof the (random-sampling, q)-algorithm (Fig. 2) and the SGA
for F = ∞ (for instance, the results for F = 2 are shownin Fig. 3.) with q = 0.008 increased approximately linearlyover generations in all runs. This differentiates between theexistence and the non-existence of neutrality in the fitness
landscape. That is, the increase of the number of substitutions
over generations indicates the presence of neutrality in thefitness landscape.
In the remainder of this paper, the rates of substitution forthe (random-sampling, q)-algorithm and the SGA for F = ∞are shown by using the method of least squares on the resultsof all the runs because the rate of substitution is equivalent to
the gradient of the number of substitutions over generations.

8
6
4
2
0

5
Number of substitutions

5
Number of substitutions

Number of substitutions

10

4
3
2
1
0

0

500

1000

1500

2000

500

(a) K = 0

1000

1500

1

2000

0

500

1000

1500

Generation(b) K = 2(c) K = 6

2000

5Number of substitutions

Number of substitutions

2

Generation

5
4
3
2
1

4
3
2
1

0

0
0

500

1000
Generation

1500

2000

0

(d) K = 12

H

3

0
0

Generation

Fig. 3.

4

500

1000
Generation

1500

2000

(e) K = 19

Number of substitutions at each generation for the SGA with q = 0.008 and M = 50 for F = 2 in 50 runs

TABLE IT HE RATE OF SUBSTITUTION FOR THE SGA WITH q = 0.008 AND

TABLE IIIT HE RATE OF SUBSTITUTION FOR THE SGA WITH q = 0.008 AND

M = 50M = 200

H K
HH

F

2
3
4
6

0

2

6

12

19

0.002042
0.0013070.001063
0.000738

0.000666
0.000421
0.000318
0.000235

0.0002200.000173
0.0001300.0001280.000131
0.0001020.000099
0.000090

0.0001060.0001040.000087
0.000087

TABLE II
T HE RATE OF SUBSTITUTION FOR THE SGA WITH q = 0.008 AND
M = 100

HH K
F HH
2
3
4
6

0

2

6

12

19

0.001370
0.000843
0.000607
0.000405

0.000401
0.000255
0.000164
0.0001280.000148
0.000096
0.000073
0.000056

0.000079
0.000051
0.000057
0.000048

0.000068
0.000050
0.000050
0.000044

D. Neutrality And Selective Constraint
With respect to the assertions 1), 2) and 3) in Section I,
Kimura has suggested that the rate of gene substitution islargest when the selective advantage of a new mutation over
the original allele is zero except that the new mutation isdeleterious in a small population [6]. Thus, it seems likelythat the number of substitutions increases with the increaseof neutrality and that for the (random-sampling, q)-algorithm
is largest, because random-sampling can be considered completely neutral for selection. In addition to this, according to

H

H K
HH

F

2
3
4
6

0

2

6

12

19

0.000781
0.000486
0.000359
0.000238

0.000263
0.000161
0.0001190.000067

0.000086
0.000051
0.000041
0.000031

0.000040
0.000039
0.000029
0.000025

0.000026
0.000028
0.000027
0.000026

TABLE IV
T HE RATE OF SUBSTITUTION FOR THE SGA WITH q = 0.008 AND
M = 400

HH K
F HH
2
3
4
6

0

2

6

12

19

0.000434
0.000261
0.000193
0.000127

0.000146
0.000087
0.000062
0.000050

0.000043
0.000032
0.000019
0.000017

0.000032
0.000025
0.000014
0.000017

0.000024
0.000013
0.000019
0.000015

Ohta’s nearly neutral theory [21][22], the stronger the selective
constraint on the molecule is, the lower its rate of evolution
becomes. That is, the number of substitutions is likely todecrease with the increase of selective constraint, K.
Table I and Fig. 4 show the rate of substitution for the
SGA with q = 0.008 and M = 50. Notice first that therate of substitution increased with the decrease of F for all
Ks. This means that the rate of substitution increases with
the increase of neutrality as predicted. Secondly, the rate of
substitution decreased with the increase of K for all F s.

0.0020

F=2
F=3
F=4
F=6

0.0015Rate of substitution

Rate of substitution

0.00200.00100.00050.00002

460.00100.00054

6

8 10 12 14 16 18
K

F=2
F=3
F=4
F=6

0.00150.00100.00058 10 12 14 16 18
K

0.0020

M=50
M=100
M=200
M=400

0.00150.00100.00050

2

4

6

8 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 and M = 400

Fig. 7.

0.0020Rate of substitution

2

Rate of substitution for the SGA with q = 0.008 and M = 200

Rate of substitution

6

0.000000.0000

M=50
M=100
M=200
M=400

0.00150.00100.00050.00002

4

6

8 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 for F = 2

0.0020

M=50
M=100
M=200
M=400

0.00150.00100.00050Fig. 9.

2460.00008 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 for F = 3

0.0020Rate of substitution

0

Rate of substitution

4

0.0020

F=2
F=3
F=4
F=6

0.00152

Rate of substitution for the SGA with q = 0.008 and M = 100

Fig. 5.

0.0000

M=50
M=100
M=200
M=400

0.00150.00100.00050.00000

Fig. 10.

0.00050Rate of substitution

Rate of substitution

0.0020Fig. 8.

0.00108 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 and M = 50

Fig. 6.

0.00150.00000

Fig. 4.

F=2
F=3
F=4
F=6

2

4

6

8 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 for F = 40Fig. 11.

2

4

6

8 10 12 14 16 18
K

Rate of substitution for the SGA with q = 0.008 for F = 6

TABLE V
T HE RATE OF SUBSTITUTION FOR THE (random-sampling, q)- ALGORITHM

50
0.004576

100
0.003194200
0.001971

400
0.001120

This means that not only neutrality but also ruggedness has
an influence on the rate of substitution. This tendency is
consistent with Ohta’s results for NK landscapes with weak
selection based on the nearly neutral theory, where the numberof substitutions decreases with the increase of K [22][28].
Similar behavior to M = 50 is shown for each population
size M = {100, 200, 400} (Table II, III and IV, and Fig. 5, 6
and 7).
The rate of substitution for the (random-sampling, q)algorithm with each M is shown in Table V. It is confirmedthat for each M , the rate of substitution for the (randomsampling, q)-algorithm was always larger than any others for
the SGA with K and F (from Table I to IV ). This agrees with
our expectation.
E. Varying The Population Size
In the next experiments, the analysis was extended byvarying the population size. According to Ohta’s nearly neutral theory [21][22], population movement depends on the
population size. That is, mutant dynamics becomes slower
by increasing the population size. This is demonstrated from
Fig. 8 to 11 and Table V. With the increase of the population
size, the rate of substitution decreased for each K and F .
Therefore, the larger the population size becomes, the slower
the population moves. This tendency is also consistent with
Ohta’s results for NK landscapes with weak selection basedon the nearly neutral theory, where the number of substitutions
decreases with the increase of the population size [22][28].
Table V shows the rate of substitution for the (randomsampling, q)-algorithm with each M . The rate of substitution
also decreased with the increase of the population size.
F. Varying The Mutation Rate
In population genetics, it is assumed that the mutation rate
per locus is sufficiently small as mentioned in Section II. In thelast series of experiments, the transition of the Nei’s genetic
distance were observed by varying the mutation rate from q =0.005 to 0.010 and 0.1 for the SGA with M = 50.
Fig.
12
shows
the
resultswithq={0.005, 0.006, 0.007, 0.008, 0.009, 0.010}. In this range,
the rate of substitution increased with the increase of themutation rate for each K and F . For each q, similar behaviors
were observed to the results with q = 0.008 in the previous
subsections. In contrast, the results with q = 0.1 show
the different behaviors (Fig. 13). Surprisingly, the rate of
substitution increased with the increase of K for all F s.
In addition to this, no significant differences were foundbetween the graphs of different F s. The rate of substitution
for the (random-sampling, q)-algorithm with q = 0.1 wasRate of substitutionM
rate

0.0020

F=2
F=3
F=4
F=6

0.00150.00100.00050.00000

8 10 12 14 16 18
K
Fig. 12. Rate of substitution for the SGA with M = 50: The solid lines,
from left to right, correspond to the rate of substitution for F = 2 withq = {0.005, 0.006, 0.007, 0.008, 0.009, 0.010}. Similaly, the dashed lines
correspond to the rate of substitution for F ∈ {3, 4, 6}.

2

4

6

0.017
Rate of substitution

WITH q = 0.008 FOR EACH POPULATION SIZE

0.016
0.015
0.014
0.013
F=2
F=3
F=4
F=6

0.012
0.011
0.010
0

2

4

6

8 10 12 14 16 18
K
Fig. 13. Rate of substitution for the SGA with q = 0.1 and M = 50

0.0135733. Thus, the rate of substitution for the SGA washigher than that for the (random-sampling, q)-algorithm for
K > 2 and all F s. This implies that artificial evolution haschanged into random search, caused by the mutation rate
which is larger than the error threshold[17].
From the above, we confirmed that the Nei’s genetic distance depends on the mutation rate, and can be used as longas the mutation rate is sufficiently small compared with theerror threshold.
IV. D ISCUSS ION
In the previous section, it has been shown that population
movement depends on the population size. If the population
size is small, the population for the SGA moves quickly. This
would have the advantage of flexibility. As pointed out in [22],evolution would become more flexible for a small population
size than for a large population size, particularly when theenvironment is not static, that is, the fitness landscape changes
occasionally. On the other hand, it has been reported that as the
population size becomes too small, it becomes easier for the
population to lose the current best individuals through random
sampling or mutation and fall to lower neutral networks [20].
This phenomenon is due to the influence of the error thresholdon the population size 3 . This would be more understandable
3 It has been known that there are two kinds of error threshold: on themutation rate and on the population size [20][17].

by considering the population movement. Due to the small
population size, the population on the neutral networks moves
too quickly to keep the current neutral network. This implies
that there exists the optimal population size that keeps thefastest speed as well as avoids the influence of the error
threshold.
The same discussion can be applied to the error thresholdon the mutation rate mentioned in Section III-F [17][18].
V. C ONCLUSIONS
We have investigated the characteristics of the Nei’s standard genetic distance by applying it to the Terraced NK
landscapes, and shown the consistencies of the results with
the neutral theory and the nearly neutral theory in population
genetics. Based on the presented results, we discussed theinfluence of the error threshold on the population size andthe mutation rate.
The characteristics of the number of substitutions estimatedby the Nei’s genetic distance can be summarized as follows:
When the mutation rate per locus is small,• Random sampling with mutation results in the largest
number of substitutions.
• The number of substitutions increases with the increaseof neutrality.
• The number of substitutions decreases with the increaseof ruggedness where the landscape includes neutrality.
• The number of substitutions decreases with the increaseof the population size.
These results can be predicted mainly by the assertion of theneutral theory and the nearly neutral theory, “functionally lessimportant molecules or parts of a molecule evolve faster thanmore important ones”. Consequently, these allow us to understand the evolutionary dynamics of GAs from the viewpoint of
population genetics using the Nei’s standard genetic distance.
Therefore, this method will play a significantly important rolethat connects artificial evolution and natural evolution.
R EFERENCE S[1] J. H. Holland, Adaptation in Natural and Artificial Systems, Universityof Michigan Press, 1975.
[2] T. Bäck and H. -P. Schwefel, ”An Overview of Evolutionary Algorithms for Parameter Optimization,” Evolutionary Computation, 1(1):123, 1993.
[3] H. Mühlenbein, ”Predictive Models for the Breeder Genetic Algorithm,”
Evolutionary Computation, 1(1):25–49, 1993.
[4] M. Ebner, P. Langguth, J. Albert, M. Shackleton and R. Shipman, ”On
Neutral Networks and Evolvability,” In Proceedings of the 2001 IEEE
Congress on Evolutionary Computation: CEC2001, IEEE Press pp. 1–8,
2001.
[5] T. Smith, P. Husbands, P. Layzell and M. O’Shea, ”Fitness Landscapesand Evolvability,” Evolutionary Computation, 10(1):1-34, 2002.
[6] M. Kimura, The Neutral Theory of Molecular Evolution, Cambridge
University, Press, New York, 1983.
[7] C. V. Forst, C. Reidys and J. Weber, ”Evolutionary Dynamics and Optimization: Neutral Networks as Model-Landscapes for RNA SecondaryStructure Folding-Landscapes,” In Proceedings of the Third European
Conference on Artificial Life ECAL95, pp. 128–147, 1995.
[8] M. Huynen, P. Stadler and W. Fontana, ”Smoothness within ruggedness:
The role of neutrality in adaptation,” In Proceedings of the National
Academy of Science USA, 93, pp. 397–401, 1996.

[9] I. Harvey, ”Artificial Evolution for Real Problems,” In Evolutionary
Robotics: From Intelligent Robots to Artificial Life (ER’97), T. Gomi,
Ed. AAI Books, 1997.
[10] T. Smith, P. Husbands and M. O’Shea, ”Neutral Networks and Evolvability with Complex Genotype-Phenotype Mapping,” In Proceedingsof the European Conference on Artificial Life: ECAL2001, pp. 23–36,
2001.
[11] T. Smith, P. Husbands and M. O’Shea, ”Neutral Networks in an
Evolutionary Robotics Search Space,” In Proceedings of the 2001 IEEE
Congress on Evolutionary Computation, pp. 136–145, 2001.
[12] T. Smith, A. Philippides, P. Husbands and M. O’Shea, ”Neutrality andRuggedness in Robot Landscapes,” In Proceedings of the 2002 IEEE
Congress on Evolutionary Computation, pp. 1348–1353, 2002.
[13] A. Thompson, ”An Evolved Circuit, Intrinsic in Silicon, Entwinedwith Physics,” In Proceedings of the first International Conference on
Evolvable Systems: From Biology to Hardware, pp. 390–405, 1996.
[14] V. K. Vassilev, T. C. Fogarty and J. F. Miller, ”Information Characteristics and the Structure of Landscapes,” Evolutionary Computation,
8(1):31–60, 2000.
[15] V. K. Vassilev and J. F. Miller, ”The Advantages of Landscape Neutrality
in Digital Circuit Evolution,” In Proceedings of the Third International
Conference on Evolvable Systems: From Biology to Hardware, pp. 252263, 2000.
[16] I. Harvey and A. Thompson, ”Through the Labyrinth Evolution Finds
a Way: A Silicon Ridge,” In Proceedings of the First International
Conference on Evolvable Systems: From Biology to Hardware, pp. 406–
422, 1996.
[17] E. Nimwegen, J. Crutchfield and M. Mitchell, ”Statistical dynamics ofthe royal road genetic algorithm,” In Theoretical Computer Science,Vol. 229, No. 1, pp. 41-102, 1999.
[18] L. Barnett, ”Netcrawling - Optimal Evolutionary Search with Neutral
Networks,” In Proceedings of the 2001 IEEE Congress on Evolutionary
Computation, pp. 30–37, 2001.
[19] L. Barnett, ”Tangled Webs: Evolutionary Dynamics on Fitness Landscapes with Neutrality,” In MSc. dissertation, School of Cognitive andComputing Sciences, Sussex University, UK, 1997.
[20] E. Nimwegen and J. Ctrutchfield, ”Optimizing epochal evolutionary
search: Population-size dependent theory,” In SFI Working Paper 9810090, Santa Fe Institute, 1998.
[21] T. Ohta, ”The nearly neutral theory of molecular evolution,” Annu. Rev.
Ecol. Syst., 23:263-286, 1992.
[22] T. Ohta, ”Evolution by nearly-neutral mutations,” In Genetica, 102/103,
pp. 83-90, 1998.
[23] M. Kimura and T. Ohta, ”On Some Principles Governing Molecular
Evolution,” In Proc. Nat. Acad. Sci., Vol. 71, No. 7, pp.2848–2852,
1974.
[24] M. Nei, ”Genetic Distance between Populations,” In The American
Naturalist, Vol. 106, pp. 283-292, 1972.
[25] Y. Katada, K. Ohkura and K. Ueda, ”Measuring Neutrality of Fitness
Landscapes Based on the Nei’s Standard Genetic Distance,” In Proceedings of 2003 Asia Pacific Symposium on Intelligent and Evolutionary
Systems: Technology and Applications, pp. 107-114, 2003.
[26] M. Newman and R. Engelhardt, ”Effect of neutral selection on theevolution of molecular species,” In Proceedings of the Royal Societyof London B, Morgan Kaufmann, 256, pp. 1333-1338, 1998.
[27] S. Kauffman, The origins of order, Oxford University Press, 1993.
[28] T. Ohta, ”Role of random genetic drift in the evolution of interactive
systems,” In Journal of Molecular Evolution, Vol. 44, pp. S9-S14, 1997.