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Int. J. Appl. Math. Comput. Sci., 2015, Vol. 25, No. 3, 499–512
DOI: 10.1515/amcs-2015-0037
A STUDY ON MEME PROPAGATION IN MULTIMEMETIC ALGORITHMS
RAFAEL NOGUERAS a,CARLOS COTTA a, ∗
aDepartment of Computer Science and Programming Languages
Higher Technical School of Computer Engineering
University of Málaga, Campus de Teatinos, 29071 Málaga, Spain
e-mail: ccottap@lcc.uma.es
Multimemetic algorithms (MMAs) are a subclass of memetic algorithms in which memes are explicitly attached to geno-
types and evolve alongside them. We analyze the propagation of memes in MMAs with a spatial structure. For this purpose
we propose an idealized selecto-Lamarckian model that only features selection and local improvement, and study under
which conditions good, high-potential memes can proliferate. We compare population models with panmictic and toroidal
grid topologies. We show that the increased takeover time induced by the latter is essential for improving the chances
for good memes to express themselves in the population by improving their hosts, hence enhancing their survival rates.
Experiments realized with an actual MMA on three different complex pseudo-Boolean functions are consistent with these
findings, indicating that memes are more successful in a spatially structured MMA, rather than in a panmictic MMA, and
that the performance of the former is significantly better than that of its panmictic counterpart.
Keywords: memetic algorithms, spatial structure, meme propagation.
1. Introduction
Four decades ago, Richard Dawkins (1976) put forward
the definition of a meme in analogy to the biological
concept of a gene. Memes were broadly characterized as
units of imitation, that is, ideas or pieces of knowledge
that jump from brain to brain, thriving and proliferating in
some cases and becoming extinct in others. Even more
interestingly, memes are not static objects but dynamic
entities that mutate over their lifetime; these mutations
can make them more interesting/useful/stronger/etc., thus
boosting their propagation, or can have the opposite
effect, causing that particular mutation to fade away.
This plasticity explains their comparatively faster rate of
adaptation with respect to biological genes.
Inspired by this notion of a meme, Moscato
(1989) conceived a new optimization paradigm: memetic
algorithms (MAs). MAs are a family of population-based
optimization techniques that blend together ideas from
different metaheuristics, most notably the orchestrated
interplay between global (population-based) search and
local (individual-based) search. The most popular
incarnation of MAs features an evolutionarysearch engine
∗Corresponding author
endowed with local search add-ons. The notion of
memetic evolution is here captured by the Lamarckian
lifetime learning which solutions are subject to, via the
use of some local search operators. Incidentally, the
suggestion has been made (Norman and Moscato, 1989)
to use the term agent, rather than an individual or a
solution in this context, to emphasize the fact that they
are active entities that purposefully try to optimize the
problem under consideration. We refer to Hart et al.
(2005), Krasnogor and Smith (2005), Moscato and Cotta
(2010), Neri and Cotta (2012), or Neri et al. (2012) for a
broader discussion of the vibrant field of MAs and their
numerous applications.
While memes are typically fixed in classical MAs
(i.e., they are given by the particular choice of localsearch
operators), it was already envisioned in the 1990s that
MAs could eventually work in at least two levels and two
time scales (Moscato, 1999): during the short-time scale
a set of agents would be searching in the search space
associated to the problem, whereas during the long-time
scale they would adapt their search strategies. Indeed,
several models trying to make memes change during the
optimization process have been proposed, constituting
the central tenet in the notion of memetic computing
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R. Nogueras and C. Cotta
whichisdefinedas“...aparadigmthatuses thenotion of
meme(s) as units of information encodedin computational
representations for the purpose of problem solving” (see
Ong et al., 2010; Chen et al., 2011; Chen and Ong, 2012).
These models can be accomplished at a variety of levels
(Ong et al., 2006). A simple possibility is the so-called
‘meta-Lamarckian learning’ (Ong and Keane, 2004), in
which the MA has a collection of local search operators
(memes) available apriori, and a mechanism is used to
decide which of them is applied to which solution and
when (note the connection with hyperheuristics (Cowling
et al., 2008; Chakhlevitch and Cowling, 2008)). A more
complex approach features self-adaptation of the memes
themselves. An example of this kind of self-adaptation
is provided by multi-memetic algorithms (MMAs), in
which each solution carries “genes” that indicate which
local search has to be applied to it. These can range
from simple pointers to existing local search operators,
to the parametrization of a general local search template
(Krasnogor et al., 2002; Neri et al., 2007) or even to
the definition of a grammar to specify a new complex
local search operator (Krasnogor, 2004; Krasnogor and
Gustafson, 2004).
An interesting issue that arises in the context of
MMAs is how memes propagate and spread over the
population. While population dynamics have been well
studied in the case of evolutionary algorithms (e.g., Alba
and Luque, 2004; Giacobini et al., 2005; Karcz-Dul˛eba,
2004; Rudolph and Sprave, 1995; Sarma and De Jong,
1997), the scenario is more complex in the case of
memes: unlike genotypes (which correspond to solutions
and thus can be evaluated according to the problem under
consideration), memes can be only indirectly assessed via
the effect they have on genotypes. Furthermore, memes
evolve in MMAs alongside solutions by being attached to
them. Since this attachment is part of the self-adaptive
process, it is up to the algorithm to discover good fits
between individual pairs of genotypes and memes, and
this is commonly done using only information about the
genetic quality of solutions (i.e., fitness information). The
work presented here studies how memes propagate in
such an environment driven by genetic selection and the
spatial structure. To this end, we consider and analyze an
idealized model of MMAs (Nogueras and Cotta, 2013).
This model is described and studied in Sections 2 to
4. Later on, we turn our attention to an actual MMA
in Section 5 and analyze its behavior on different test
problems in order to corroborate the theoretical findings.
2. Model description
Let us consider an abstract model of MMAs in which each
agent is characterized by a pair g, m∈D2,forsome
D⊂R. The first member of the pair, g, represents the
genotype, which we equate to fitness for simplicity. As
for the second member, m, it represents a meme. More
precisely, this value captures the improvement potential
of that meme, that is, a measure of how good solutions
can get by applying the meme. We assume there is a
function f:D2→Dmonotonically increasing in its first
parameter, encapsulating the application of a meme to a
genotype, i.e., the effect of a single epoch of Lamarckian
learning. Thus, an agent g, mbecomes f(g, m),m
after the application of the meme, where
lim
n→∞ fn(g, m)=mif g<m, (1)
f(g, m)=gif g≥m. (2)
Here fn(g, m)is the n-th composition
of the function on the first argument, i.e.,
f(f(···f(f(g, m),m),··· ,m),m). Intuitively, these
conditions indicate that the fitness of a solution can
be improved when a good meme is applied on it,
asymptotically approaching the potential of the latter.
Certainly, this is a very idealized characterization of the
potential of a meme since in general this potential is
not going to be absolute but may depend on a complex
match between the meme, the genotype and the problem
landscape, but it serves as an initial attempt to study
several issues related to meme propagation in the agent
pool.
The population Pof the MMA is thus a collection
of μsuch agents, P=[g1,m
1,...,gμ,m
μ],
endowed with a spatial structure that constrains agent
communication. Let this spatial structure be characterized
by a μ×μBoolean matrix S,whereSij is true if,
and only if, the agent placed in the i-th location can
communicate with the agent placed in the jlocation
(note that exactly one agent is placed in each location).
Since we are interested in observing the dynamics of
the propagation of memes, we consider an extension
of the selection-only model of evolutionary algorithms
(i.e., using only selection/replacement and no variation
operator) in which we add the local improvement stage
of memetic algorithms. A scheme of the model is shown
below in Algorithm 1.
Algorithm 1. Selecto-Lamarckian model.
1: for i∈{1,··· ,μ}do
2: INITIALIZEgi,m
i
3: end for
4: while ¬CONVERGED (P)do
5: i←URAND(1,μ){Pick random location}
6: g, m←SELECTION(P, S, i)
7: g←f(g, m){Local improvement}
8: P←REPLACE(P, S, i, g,m)
9: end while
After suitably initializing the contents of the
population, the algorithm engages in a cycle of selection
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plus improvement until the agents converge. Convergence
is here approached from a memetic perspective, that is, we
terminate the algorithm when the population comprises a
homogeneous collection of memes (regardless of whether
there is still diversity at the genetic level or not). As for
the inner working of the algorithm, it resembles the new
random sweep strategy of cellular automata (Schönfisch
and de Roos, 1999), in which the next location to
be activated is selected uniformly at random without
replacement.
3. Meme propagation
Having defined the general model in the previous section,
let us consider some qualitative features of meme
propagation that can be extracted from it. Let us assume
that selection is done by binary tournament, i.e., once a
location iselected, a neighboring location jis selected
from N(i)={j|Sij }, and the agent with the best fitness
is retained. Regarding replacement, let us assume that the
improved agent replaces the agent that lost the previous
tournament.
We are interested in analyzing the number of copies
of each meme in the meme pool, so let us denote by
N(m, g, t)the number of instances of meme mattached
to genotype gat time t, assuming for simplicity that D
is some discrete domain. If we divide this quantity by
the pool size μ, we obtain p(m, g, t), the fraction of the
population comprising meme mattached to genotype gat
time t. In each passing iteration of the system the number
of copies can be estimated as
N(m, g, t +1)
=N(m, g, t)+C(m, g, t)−D(m, g, t),(3)
where C(m, g, t)and D(m, g, t)represent the expected
number of copies of meme mattached to genotype g
that are created or destroyed at time t. The creation of
a new copy can be accomplished by the combined effect
of selection of a suitable agent with meme mand the
application of the meme to the corresponding genotype.
Let us express this as
C(m, g, t)=
g
σ(m, g,t)p(gm
−→ g),(4)
where σ(m, g,t)is the probability of selecting an agent
carrying meme mand genotype gat time tand p(gm
−→
g)is the probability that the application of meme mon
genotype gresults in genotype g. The first quantity can
be computed as the probability of the binary tournament
picking two agents with meme mand genotype gor only
one agent with this structure but with a better fitness than
its competitor:
σ(m, g, t)
=p(m, g, t)2+2p(m, g, t)[1−p(m, g , t)]
×mg<g p(m,g
,t)
1−p(m, g, t),(5)
where the last factor is the probability that the fitness of
the competitor is worse than gprovided it is not an g, m
agent. This expression assumes that the global distribution
of memes/genotypes across the whole population is the
same as for local neighborhoods. Obviously, this holds for
the panmictic case in which any two agents are neighbors,
so we can assume this case initially and consider it a first
approximation towards more general situations.
Concerning the destruction of a copy of a particular
pair meme/genotype, it can arise via the selection of such a
pair and the subsequent application of local improvement
(which will alter the genotype) or via replacement by an
agent of higher fitness. The first case also requires that
the other agent chosen in the tournament be a copy of the
same pair, so that it is later substituted by the improved
agent. Thus,
D(m, g, t)=
g=g
p(m, g, t)2p(gm
−→ g)+˜σ(m, g, t).
(6)
The replacement probability ˜σ(m, g, t)can be expressed
as
˜σ(m, g, t)=2p(m, g , t)[1−p(m, g, t)]
×mg>g p(m,g
,t)
1−p(m, g, t).(7)
Let us now consider the evolution of the system in
the early-term and mid-term, before a particular meme
starts to saturate the population. In this situation memes
are widely spread across genotypes, so p(m, g, t)1
and hence we can neglect quadratic terms p(m, g , t)2.We
thus have
σ(m, g, t)=2p(m, g, t)
m
g<g
p(m,g
,t),(8)
˜σ(m, g, t)=2p(m, g, t)
m
g>g
p(m,g
,t).(9)
Substituting back into Eqns. (4) and (6), we get
C(m, g, t)=2
g,m
p(m, g,t)
×
g<g
p(m,g
,t)p(gm
−→ g),(10)
D(m, g, t)=2p(m, g, t)
m
g>g
p(m,g
,t).(11)
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Since p(gm
−→ g)=0for g<g
or m<g, Eqn. (10)
reduces to
C(m, g, t)
=2
g<gg
m
p(m, g,t)p(m,g
,t)p(gm
−→ g).
(12)
If m≤g,thenp(gm
−→ g)is 1 if g=gand 0 otherwise.
Subsequently, the difference Δm
g(t)=C(m, g, t)−
D(m, g, t)is in this case
Δm
g(t)=2p(m, g, t)
m
g<g
p(m,g
,t)
−2p(m, g, t)
m
g>g
p(m,g
,t)
=2p(m, g, t)
m
g<g
p(m,g
,t)
−
g>g
p(m,g
,t).
(13)
Focusing on the sign of the difference in the expression
above, we essentially obtain that inert memes (i.e., memes
that can no longer improve their hosts) can thrive by
hitchhiking, that is, if they attach themselves to agents
above the median of the population.
Let us, on the other hand, consider the case m>g.
In this situation, p(gm
−→ g)is 1 if g=f−1(g, m)and
0 otherwise, where we denote by f−1(g, m)the genotype
value such that f(f−1(g, m),m)=g.Usingg−mas a
shorthand for f−1(g, m),
Δm
g(t)=2p(m, g−m,t)
m
g<g−m
p(m,g
,t)
−2p(m, g, t)
m
g>g
p(m,g
,t)
=
m⎡
⎣2p(m, g−m,t)
g<g−m
p(m,g
,t)
−2p(m, g, t)
g>g
p(m,g
,t)⎤
⎦.
(14)
The sign of this expression depends on the balance
between the goodness of genotypes in the basin of
attraction of gand the badness of gitself (in both
cases, goodness/badness relative to the rest of the
population). Note that in general there is a reinforcement
between these quantities in the sense that, the better a
genotype in the basin of attraction of g, the better we can
expect gto be. This does not just mean that active memes
proliferate more and more when they attach themselves to
good solutions as one would expect, but also that memes
with high potential can find their way to the final stages of
the evolution provided they have enough time to improve
their hosts (recall that the goodness of solutions evolves
with time as an effect of the application of the meme).
This suggests that models with slower genetic
convergence can have a beneficial effect on the
propagation of good memes, allowing the latter enough
time to express themselves in the population and
overcome the hitchiking effect of bad memes. The next
section provides a more quantitative analysis of this effect
via numerical simulations.
4. Numerical simulations
The numerical experimentation aims to empirically
explore the dynamics of meme propagation and how
they are affected by factors such as the population size,
the relative improvement potential of memes and the
underlying spatial structure of the population. Regarding
population sizes, we have considered values μ∈
{100,256,400,625}. These values cover a broad range
of population sizes and are also perfect squares, which
is important in connection with one of the spatial
structures considered, namely, a square toroidal grid
with a von Neumann neighborhood: two locations (i, j)
and (i,j)are connected if their Manhattan distance
|i−i|+|j−j|r,whereris the neighborhood
radius. We have considered r=1, which leads to
the traditional North-South-East-West (plus the current
location) neighborhood. The other spatial structure
considered is the panmictic model in which all locations
are connected. In either case, we have considered the
function
f(g, m)=gif g≥m,
(g+m)/2if g<m (15)
to represent the action of memes. Intuitively, this
function provides smaller improvements for increasingly
good genotypes, much like it often happens in practice.
All experiments are averaged over 100 runs in order to
obtain representative results. Each run is terminated upon
convergence of memes, which for simplicity is determined
when all memes are equal to 2 decimal positions.
Let us firstly analyze meme propagationas a function
of the relative improvement potential of memes at the
beginning of the run. For this purpose, we take D=
[0,1] and consider a scenario in which genotypes and
memes are randomly initialized in this range, and another
scenario in which genotypes take initial values in [0,0.5]
whereas memes are randomly sampled from [0,1].
Figure 1 shows the distribution of memes at each
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Fig. 1. Meme maps for simulations with μ= 625. The upper row corresponds to panmictic connectivity and the lower row to a
von Neumann neighborhood. Similarly, the left column corresponds to genotypes initialized in [0,1] and the right one to
initialization in [0,0.5] (memes are initialized in [0,1] in both cases). Lighter shades of gray indicate higher meme values. The
evolution of the algorithm is depicted in each subfigure from left to right, each vertical slice representing the distribution of
memes at a certain time t. Note the different scale on the x-axis.
time step (the lighter the shade of gray1, the higher
the meme value). Focusing firstly on the upper row
(panmictic topology), note the clearly different behavior
depending on genotype initialization. When genes and
memes are both initialized in [0,1], the algorithm seldom
converges on a high-potential meme. Actually, such
memes temporarily proliferate in the initial stages of the
algorithm but are later driven to extinction by memes
1A color version of this figure can be accessed at
http://dx.doi.org/10.6084/m9.figshare.1235204.
hitchhiking on high quality genotypes to which they
stuck by chance. The situation is quite different when
genotypes are initially drawn from [0,0.5]: in this case
the algorithm does gradually converge to the upper part
of the meme distribution, with low-potential memes
quickly disappearing from the population. A more
detailed perspective on this is provided by Fig. 2, in
which qualified runtime distributions (QRTDs) (Hoos and
Stützle, 2004) are provided. These indicate the probability
of a certain target (in this case, convergence to a meme
in a desired percentile) being reached as a function of
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0 1000 2000 3000 4000 5000 6000 7000 8000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
iterations
P(success)
100
99
95
90
0 1000 2000 3000 4000 5000 6000 7000 8000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
iterations
P(success)
100
99
95
90
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1
0
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
iterations
P(success)
100
99
95
90
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1
0
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
iterations
P(success)
100
99
95
90
Fig. 2. Qualified runtime distributions for simulations with μ= 625. The upper row corresponds to panmictic connectivity and the
lower row to a von Neumann neighborhood. Similarly, the left column corresponds to genotypes initialized in [0,1] and the
right one to initialization in [0,0.5] (memes are initialized in [0,1] in both cases). The curves indicate the probability that the
population converges to a meme in the i-th initial percentile of the population as a function of the number of iterations. Note
the different scale on the x-axis.
the number of iterations. Observe how the probabilities
are below 10% for memes above the 95% percentile in
the first scenario, whereas this probability is 100% in the
second scenario. In the latter a spurious match between
a very good genotype and a bad meme cannot happen
since these very good solutions do not exist initially.
Furthermore, high-potential memes by initially attaching
themselves to bad genotypes can highly improve the
quality of the latter in the initial steps, thus increasing their
chances of survival.
Let us now turn our attention to the effect of the
spatial structure. The bottom row of Fig. 1 shows the
distribution of memes for the case of a von Neumann
neighborhood. Note how a pattern similar to the panmictic
case is observed with respect to genotype initialization.
A more detailed inspection reveals several differences,
though. Firstly, observe how the convergence is slower
in this case (e.g., the scale in the x-axis is larger). This
is a well-known effect of using the spatial structure and
is commonly exploited in the context of evolutionary
algorithms to promote diversity and thus decrease the
chances of getting stuck in local optima (Dorronsoro and
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0 1000 2000 3000 4000 5000 6000 7000
0
10
20
30
40
50
60
70
80
90
100
iterations
prevalence of best meme (%)
μ=625
μ=400
μ=256
μ=100
0 1000 2000 3000 4000 5000 6000 7000
0
10
20
30
40
50
60
70
80
90
100
iterations
prevalence of best meme (%)
μ=625
μ=400
μ=256
μ=100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1
0
4
0
10
20
30
40
50
60
70
80
90
100
iterations
prevalence of best meme (%)
μ=625
μ=400
μ=256
μ=100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1
0
4
0
10
20
30
40
50
60
70
80
90
100
iterations
prevalence of best meme (%)
μ=625
μ=400
μ=256
μ=100
Fig. 3. Growth curves for different population sizes. The upper row corresponds to panmictic connectivity and the lower row to a
von Neumann neighborhood. Similarly, the left column corresponds to genotypes initialized in [0,1] and the right one to
initialization in [0,0.5] (memes are initialized in [0,1] in both cases). Note the different scale on the x-axis.
Alba, 2008; Tomassini, 2005). In the case of MMAs
this has an additional advantage: a slower convergence
increases the lifespan of individual memes, thus giving
them more chances to improve their hosts if they have the
potential to do so. Hence, the algorithm is more robust
and can better cope with hitchhiking inert memes. This
can be seen in the meme map in the bottom row of Fig. 1
by a larger prevalence of lighter-gray areas, and more
clearly in the QRTDs (bottom row of Fig. 2), e.g., the
95% percentile is reached with nearly 20% probability
in the case of [0,1]-initialization (cf. below 10% in
the panmictic case), and the 99% percentile is reached
with nearly 80% probability for [0,0.5]-initialization (cf.
about 65% in the panmictic case). A sign-rank test
(Wilcoxon, 1945) indicates that the difference in the final
percentile reached is statistically significant in both cases
(α=.05).
Finally, we consider the takeover time, namely,
the time required for a meme (not necessarily the best
one as shown previously) to completely dominate the
population. Figure 3 shows the growth curves, depicting
the percentage of the meme pool occupied by the most
repeated meme (note that the most repeated meme need
not be the same throughout a run; we simply count the
number of copies of the most repeated one at each time
step). These curves exhibit the typical shape of the
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Table 1. Fitting growth curves to a logistic function. For each algorithm configuration, the scale parameter αand the mean squared
error are shown.
population size
μ= 100 μ= 256 μ= 400 μ= 625
topology G α mse αmse αmse αmse
panmictic 0.5 81.63 0.000042 210.79 0.000069 324.65 0.000064 521.93 0.000017
1.0 77.71 0.000008 187.90 0.000030 297.90 0.000027 462.87 0.000010
von Neumann 0.5 186.82 0.000366 578.91 0.000554 1046.43 0.000361 1975.00 0.000318
1.0 168.57 0.000322 585.80 0.000532 1057.52 0.000537 1945.63 0.000705
well-known logistic model f(t)=1/(1 + Ke−t/α).
Indeed, such a model was proposed early on in the
literature by Sarma and De Jong (1997) in the context
of spatially structured evolutionary algorithms. While by
no means the only alternative (see, e.g., Giacobini et al.,
2003) it serves as a good starting point for quantifying
the growth of the dominant meme. Qualitatively, we
observe, as expected, the well-known pattern of slower
convergence for increasing population sizes and for
the von Neumann topology (Giacobini et al., 2005) as
opposed to the panmictic population. From a quantitative
point of view, we have fitted the growth data to a logistic
curve to identify the scale factor αthat renders the number
of iterations dimensionless. The resulting data are shown
in Table 1. As can be seen, the fit is quite good, yielding
very low mean squared errors. The scale parameters are
quite similar for variants with the same topology, and are
about 2–5 times larger for the von Neumann topology
than for the panmictic case, corresponding to the relative
takeover time which can be seen in Fig. 3. With respect
to the population size, the increase in the scale parameter
admits a linear interpolation α=a+bμ, yielding values
of b=0.84 and b=0.74 for the panmictic case and
b=3.43 and b=3.40 for grid topology with the von
Neumann connectivity.
5. Experimental results
Having analyzed the idealized model of MMAs, we
now turn our attention to an operative algorithmic
incarnation of MMAs so as to validate the previous
findings. The MMA considered will be described next;
subsequently, we present the problem benchmark used
in the experimentation; finally, we report on and analyze
experimental results.
5.1. Multimemetic algorithm. We have considered
an MMA following the framework defined by Smith
(2008; 2012). To be more precise, we have considered a
population of individuals carrying binary genotypes and
single memes, which represent rewriting rules. Such
rules are of the kind A→C, where both Aand Care
patterns of a certain length. These patterns are expressed
as strings composed of symbols from a ternary alphabet
Σ={0,1,#},where‘#’ represents a wildcard symbol.
Given a certain genotype b1b2···bnaruleA→C
could be potentially applied in any position iin which the
antecedent A=a1···arof the rule matches the contents
of the genotype (i.e., bibi+1 ···bi+r−1=a1···ar). The
action of the rule would be to rewrite that portion of the
genotype, implanting the consequent C=c1···crof the
rule (i.e., setting bibi+1 ···bi+r−1←c1···cr). In this
process, the wildcard ‘#’ is interpreted as follows:
•In the antecedent of the rule it denotes a ‘don’t-care’
symbol, meaning that it can match both ‘0’ and ‘1’.
•In the consequent of the rule it denotes a
‘don’t-change’ symbol, meaning that the
corresponding position of the genotype is left
unchanged.
For instance, let a genotypebe 11101100, and let a rule be
10# →0#1. A possible application of the rule could be
the following:
11
A
101 100 rule
−−−−−−→11 001
C
100.
In this example, there is another potential application
point, namely, the sixth position, resulting in the genotype
11101001. In general, the order in which genotype bits
are scanned is randomized to avoid positional bias. When
a match is found, the resulting neighboring genotype
is generated and evaluated according to the objective
function under consideration. In order to keep the total
cost of the process under control, a parameter wis used
to determine the maximal number of rule applications.
The best neighbor generated (if better that the current
genotype) is kept. Let us finally note in passing that, by
virtue of the pattern definition used, the whole process
of genotype rewriting by means of rules can be regarded
as a heuristic way of manipulating schemata, injecting
instances of new schemata in the place of extant ones
whenever the fitness function dictates the former are
better.
Apart from the use of memes embedded within
individuals, our MMA otherwise resembles a steady-state
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memetic algorithm in which parents are selected using
binary tournament, recombination, mutation and local
search (conducted using the meme carried out by the
individual as illustrated before). Then they are used to
generate the offspring, which replaces the worst parent
(following our model presented in Section 3).
5.2. Test suite. In order to test the MMA, we
have considered three different problems defined on
binary strings, namely, Deb’s trap function (Deb and
Goldberg, 1993), the massively multimodal deceptive
problem (MMDP) (Goldberg et al., 1992), and Watson et
al.’s (1998) hierarchical-if-and-only-if (H-IFF) function.
Let us describe each of them separately in the following.
Deb’s 4-bit fully deceptive function (TRAP
henceforth) is defined so as to have a single global
optimum in a fairly isolated location and a local optimum
surrounded by increasingly fit individuals (thus deceiving
gradient-based methods to follow the path towards this
local optimum). More precisely, TRAP is defined as a
function of unitation as
f(b1···b4)
=0.6−0.2·u(b1···b4)if u(b1···b4)<4,
1if u(b1···b4)=4,
(16)
where u(s1···si)=jsjis the unitation (number of
1s in a binary string) function. The function above is
used as the basic block upon which to build a higher-order
problem by concatenating ksuch blocks, and defining the
fitness of a 4k-bit string as the sum of the function value
for all blocks/subproblems. In our experiments we have
considered k=32subproblems (and hence opt = 32).
The next function is the MMDP. This is a bipolar
deceptive function with two global optima far apart from
each other (at the extreme unitation values), and a local
deceptive attractor at the halfway point. Placing the
deceptive attractor here implies that there are massively
more local optima than global optima (L
L/2local vs. 2
global). The precise definition of the basic MMDP block
(for 6 bits) is as follows:
f(b1···b6)=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1,u(b1···b6)∈{0,6},
0,u(b1···b6)∈{1,5},
0.360384,u(b1···b6)∈{2,4},
0.640576,u(b1···b6)=3.
(17)
Again, we concatenate kcopies of this basic block to
create a harder problem. We have considered k=24
(thus, opt = 24).
Lastly, the H-IFF function is a recursive epistatic
problem whose hierarchical structure forces the search
algorithm to move from searching combinations of bits
to searching combinations of increasingly higher-order
schemata (and thus challenging the abilities of the
algorithm to identify and combine good building blocks).
This function is defined for binary strings of 2kbits by
means of two auxiliary functions:
f:{0,1,•} → {0,1}.(18)
t:{0,1,•} → {0,1,•},(19)
where ‘•’isusedasanull or NaN value. More precisely,
let us define these two latter functions as follows:
f(a, b)=1,a=b=•,
0,otherwise,(20)
t(a, b)=⎧
⎪
⎨
⎪
⎩
0,a=b=0,
1,a=b=1,
•,otherwise.
(21)
Intuitively, fis the function used to score the contribution
of building blocks and tis a function used to capture their
interaction. They are used as follows:
H-IFFk(b1···bn)=
n/2
i=1
f(b2i−1,b
2i)
+2H-IFFk−1(b
1,··· ,b
n/2),(22)
where
b
i=t(b2i−1,b
2i)(23)
and
H-IFF0(·)=1.(24)
This recursive definition leads to a hierarchical tree-like
dependency structure: increasingly large building blocks
are considered as we move upwards in the tree, and the
weight of their interaction is correspondingly greater. We
have considered k=7(i.e., 128-bit strings), which, using
our definition above of the HIFF function, implies opt =
576.
5.3. Results. The experiments have considered the
MMA described in Section 5.1 using a population size
of μ= 256, binary tournament selection, recombination
probability pX=1.0and mutation probability pM=
1/,whereis the length of genotypes. Genotype
recombination is done using single-point crossover. As for
the memes, with probability pXthe offspring will inherit
the meme carried by the best of the parents (otherwise, it
inherits the meme of its first parent). Memes are defined
as rules of length r=3and are applied with parameter
w=5. We consider a maximum number of 25,000
evaluations for 128-bit TRAP and H-IFF and 50,000
evaluations for a 144-bit MMDP. In all cases the cost of
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R. Nogueras and C. Cotta
0.5 1 1.5 2 2.5
x 1
0
4
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16
18
20
22
24
26
28
30
32
evaluations
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VN
PANM VN
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26
27
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29
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best fitness
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 1
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16
18
20
22
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evaluations
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VN
PANM VN
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20
21
22
23
24
best fitness
0.5 1 1.5 2 2.5
x 1
0
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250
300
350
400
450
500
550
evaluations
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PANM
VN
PANM VN
350
400
450
500
550
600
best fitness
Fig. 4. Best fitness (averaged for 100 runs) of MMAs with panmictic and von Neumann connectivity. The inset shows the distribution
of fitness values attained. From left to right: TRAP, the MMDP and H-IFF.
0 0.5 1 1.5 2 2.5
x 1
0
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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1
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100
95
90
85
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 1
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50
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x 104
0
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 104
0
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60
50
Fig. 5. Qualified runtime distributions for MMA variants. The top row corresponds to panmictic connectivity and the bottom one to
a von Neumann neighborhood. Similarly, the leftmost column corresponds to TRAP, the middle one to the MMDP and the
rightmost one to H-IFF. The curves indicate the probability that the population converges to a solution whose quality is i%of
the optimum (i.e., pindicates an approximation of (100 −p)%), as a function of the number of iterations. These approximation
percentages are different in the case of H-IFF due to the particularities of the distribution of values in this function (bigger gaps
between the optimal solution and suboptimal ones).
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Table 2. Results (averaged for 100 runs) of MMAs with panmictic and von Neumann connectivity. The number of times the optimum
is found (nopt ), the median (˜x), mean ( ¯x) and standard error of the mean (σ¯x)areshown.
TRAP MMDP H-IFF
nopt ˜x¯x±σ¯xnopt ˜x¯x±σ¯xnopt ˜x¯x±σ¯x
panmictic 49 31.6 30.8 ±0.2 60 24.0 23.0 ±0.1 54 576.0 506.1 ±8.1
von Neumann 91 32.0 31.9 ±0.1 97 24.0 24.0 ±0.0 99 576.0 574.8 ±1.2
meme application is accounted as a partial evaluation (as
the fraction of subproblems that need being re-evaluated
after modifying the genotype), thus effectively measuring
the overhead related to using them. As in Section 4,
two algorithmic variants are considered: an MMA with
panmictic population and with a spatially structured
(toroidal square grid with the von Neumann topology)
population. All experiments with either algorithmand test
problem are replicated one hundred times.
Let us firstly observe the evolution of the best fitness
in Fig. 4. The panmictic MMA (PANM) exhibits a
faster initial convergence but it is quickly surpassed by
the von Neumann MMA (VN), which manages to keep
steadily improving for a longer time, finally attaining
notably better solutions—see the boxplots in the insets
of Fig. 4, showing the distribution of fitness values
achieved by each algorithm, and the detailed numerical
data in Table 2. This improvement is consistent across
all the problems in the test suite and can be shown to
be statistically significant to a level of α=0.05 using
a Wilcoxon ranksum test. A complementary perspective
on this improved convergence of the VN is shown in
Fig. 5. By inspecting the QRTDs we can observe how
there is a marked improvement on the success probability
in the case of the VN with respect to PANM in all three
problems. Not only is the VN capable of finding the
optimal solution more often (91%–99% success rate for
the VN vs. 49%–60% for PANM), but it is also capable
of approaching a high quality solution on a more frequent
basis. Consistent with the findings of the idealized model,
this can be due to the higher diversity of memes which
are capable of sustaining a fruitful Lamarckian search
for a longer time. The evolution of diversity (measured
in terms of the entropy of memes in the population) is
depicted in Fig. 6 (top). Memetic diversity decreases
more gently in the case of the VN, thus giving potentially
good memes more opportunities to express themselves
(ultimately resulting in the improved quality of the results
as mentioned before). In fact, as shown in Fig. 6 (bottom),
the success rate of meme application is eventually higher
in the case of the non-panmictic MMA, which helps to
explain the higher fitness performance of the latter.
6. Conclusions
We have presented some initial steps in analyzing meme
propagation in MMAs. Using an idealized model
of genotypes and memes, we have shown that the
selection intensity plays a very important role in allowing
high-potential memes to proliferate. In a panmictic model,
good memes will dominate the final population when the
starting solutions have a substantial improvement margin
on average. When this margin is smaller, average memes
can hitchhike their way to the final stages of the evolution
and make other comparatively better memes become
extinct. In the presence of a spatial structure inducing
longer takeover times (in our case, a toroidal square grid
with the von Neumann topology), this hitchhiking effect
is somewhat mitigated, allowing good memes to express
themselves and increasing their chances for making it to
the final population. These findings have been empirically
corroborated by deploying an actual MMA on three
different pseudo-Booleanfunctions, namely, concatenated
trap functions (both unimodal and massively multimodal),
and Watson et al.’s (1998) hierarchical-if-and-only-if
function. The MMA with a spatially structured population
has been shown to maintain a higher memetic diversity
and to provide notably improved results with respect to its
panmictic counterpart.
An interesting line of future research will consider
other topologies and study their effect on meme
propagation. We are particularly interested in the use
of the island model (Cantu-Paz, 2000; Schaefer et al.,
2012), which can be readily deployed on distributed
computational environments (Alba, 2005). Work is
already in progress in this area. Looking beyond,
another topic for further research is the extension of this
analysis to coevolutionary memetic algorithms (Smith,
2007; 2012) in which memes are detached from genotypes
and co-evolve alongside the latter in a separate population.
Acknowledgment
This work is an extended version of our previous
results (Nogueras and Cotta, 2013). We acknowledge
the support of Spanish MICINN under the project
ANYSELF (TIN2011-28627-C04-01)2, by Junta de
Andalucía under the project DNEMESIS (P10-TIC-6083)3
and Universidad de Málaga, Campus de Excelencia
Internacional Andalucía Tech.
2http://anyself.wordpress.com/.
3http://dnemesis.lcc.uma.es/wordpress/.
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0.5 1 1.5 2 2.5
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2
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VN
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x 1
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0
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VN
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 1
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0
0.1
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1
evaluations
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PANM
VN
0.5 1 1.5 2 2.5
x 1
0
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
evaluations
individual improvement rate
PANM
VN
Fig. 6. Evolution of diversity in memes (top), evolution of meme success (fraction of local search stages that result in an improved
individual) (bottom). From left to right: TRAP, the MMDP and H-IFF.
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methods, Biometrics 1(6): 80–83.
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512
R. Nogueras and C. Cotta
Rafael Nogueras obtained an M.Sc. degree in
computer science in 1998 from the University of
Málaga, Spain. He is currently working on his
Ph.D. at that university, under the supervision of
Carlos Cotta, in the area of memetic algorithms.
Carlos Cotta obtained his M.Sc. and Ph.D. in
computer science from the University of Málaga,
Spain, in 1994 and 1998, respectively. He
has held a tenured professorship at that univer-
sity since 2001. His main research areas in-
volve metaheuristic optimization, in particular
hybrid and memetic approaches with the focus
on both algorithmic and applied aspects (partic-
ularly combinatorial optimization) and complex
systems.
Received: 6 June 2014
Revised: 26 November 2014
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