Presentation at the 2nd International Conference on the Theory and Practice of Natural Computing, TPNC 2013.

1. An Analysis of a Selecto-Lamarckian Model of
Multimemetic Algorithms with Dynamic
Self-Organized Topology
Rafael Nogueras1
Juan L.J. Laredo3
Carlos Cotta1 Carlos M. Fernandes2
Juan J. Merelo4 Agostinho C. Rosa2
1 Universidad
3 University
de M´laga (Spain), 2 Technical University Lisbon (Portugal),
a
of Luxembourg (Luxembourg), 4 University of Granada (Spain)
TPNC 2013, C´ceres, 3-5 December 2013
a
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

2. What are Memes?
Memes are information pieces that constitute units of imitation.
“Examples of memes are tunes, ideas,
catch-phrases, clothes fashions, ways of
making pots or of building arches. Just as
genes propagate themselves in the gene pool
by leaping from body to body via sperms or
eggs, so memes propagate themselves in the
meme pool by leaping from brain to brain via a
process which, in the broad sense, can be
called imitation.”
The Selfish Gene, Richard Dawkins, 1976
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

3. What is a Memetic Algorithm?
Memetic Algorithms
A Memetic Algorithm is a population of agents
that alternate periods of self-improvement
with periods of cooperation, and competition.
Pablo Moscato, 1989
Memes can be implicitly defined be the choice of local-search (i.e.,
self-improvement) method, or can be explicitly described in the
agent.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

4. Multimemetic Algoritms (and Memetic Computing)
The term “multimemetic” was coined by N. Krasnogor and J.
Smith (2001). In a MMA, each agent carries a solution and the
meme(s) to improve it.
Evolution works at these two levels, cf. Moscato (1999).
Memetic Computing
A paradigm that uses the notion of meme(s) as units of
information encoded in computational representations for the
purpose of problem solving.
Ong, Lim, Chen, 2010
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

5. Scope
Some interesting issues in MMAs:
Memes evolve in MMAs alongside with the solutions they
attach to. It is up to the algorithm to (self-adaptively)
discover good fits between genotypes and memes.
Memes are indirectly assessed via the effect they have on
genotypes.
We consider an analyze an idealized model of MMAs to analyze
meme propagation with dynamic self-organized spatial structures.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

6. Background
Dynamic of meme propagation is more complex than genetic
counterparts.
Genes represent solutions objectively measurable via the
fitness function.
Memes are indirectly evaluated by their effect on solutions.
A first analysis was done by Nogueras and Cotta (2013) with
panmictic and spatially-structured populations. Population
structure is very important to determine the behavior of the
algorithm.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

7. Dynamic Self-Organized Topology
A dynamic model defined by Fernandes et al. (2012) combining
ideas from swarm intelligence and cellular automata is considered
in this study.
Model uses simple rules for movement on a
large 2D-lattice, giving rise to self-organized
clusters of particles.
The clusters evolve and change their shape
with some kind of dynamic order.
We consider how memes propagate in this environment.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

8. Preliminaries
Each agent is a pair g , m ∈ R2 – i.e., gene, meme . The effect
of a meme is captured by a function f : R2 → R, i.e.,
meme application
g , m − − − − − → f (g , m), m
−−−−−
m actually represents the improvement potential of the meme.
lim f n (g , m) = m if g < m
n→∞
f (g , m) = g
if g
m
The population P = [ g1 , m1 , · · · , gµ , mµ ] of the MMA is a
collection of µ such agents.
Agent communication is constrained by a spatial structure,
characterized by a µ × µ Boolean matrix S.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

9. Model Pseudocode
Algorithm 1: Selecto-Lamarckian Model
for i ∈ [1 · · · µ] do
Initialize gi , mi ;
end
while ¬ Converged (P) do
i ← URand(1, µ) // Pick random location
g , m ←Selection(P, S, i);
g ← f (g , m) // Local improvement
P ← Replace(P, S, i, g , m );
end
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

10. Concepts
Let G be a grid of size r × s > µ. Each cell Guv of the grid is a
tuple (ηuv , ζuv ), where:
ηuv ∈ {1, · · · , µ} ∪ {•} and ζuv ∈ (D × N) ∪ {•}.
ηuv indicates the index of the individual that occupies position
u, v in the grid.
ζuv is a mark placed by individuals which occupied that
position in the past, where:
f
ζuv is the fitness value of the individual.
t
ζuv is a time stamp.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

11. Individual Movement
The system combine ideas from swarm intelligence and cellular
automata.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

12. Individual Movement
Algorithm 2: Individual Movement (i, t, G )
u, v ← ρ(i); move←true;
f
if exists u (i) , v (i) ∈ N u, v such that ζuv > gi then
f
f
u , v ← arg min{ζuv | ζuv > gi };
else
f
if exists u (i) , v (i) ∈ N u, v such that ζuv < gi then
f
f
u , v ← arg max{ζuv | ζuv < gi };
else
if N u, v = ∅ then
Pick u , v at random from N u, v ;
else
move ← false;
end
end
end
if move then
f
t
ζuv ← gi ; ζuv ← t; // mark old cell
ηuv = •; ηu v = i; // move to new cell
end
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

13. Setting
Goal
Explore the dynamics of meme propagation and how it is affected
by factors such as the selection probability, the improvement
potential of memes and the spatial structure of the population.
µ = 256.
pLS ∈ {1/256, 0.1, 0.5, 1.0}.
Spatial structure:
1
2
3
Panmictic: full connectivity with static structure.
Von Neumman neighborhood (r = 1) with static structure.
Moore neighborhood (r = 1) with dynamic structure.
Meme application:
f (g , m) =
g
(g + m)/2
R. Nogueras et al.
if g m
if g < m
MMAs with Dynamic Self-Organized Topology

14. Numerical Simulations
Individual Distribution – Evolution
Individuals start from a random distribution and quickly group in
clusters during the run.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

15. Numerical Simulations
Growth Curves
The number of copies of the dominant meme grows until taking
over the population:
the panmictic model is the first to converge.
the dynamic model is closer to von Neumann model
depending on the value of pS .
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

16. Numerical Simulations
Qualified Run-Time Distributions – α = 1
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

17. Numerical Simulations
Qualified Run-Time Distributions – α = 1/2
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

18. Numerical Simulations
Spectral Analysis
The spectrum of the average number of neighbors indicates:
the intensity is proportional to f a for some a < 0.
the spectrum slope is closer to pink noise.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

19. Conclusions
A dynamic model provides promising results in comparison to
unstructured populations and to populations arranged in static
lattices.
By tuning the ratio between self-organization and evolution the
convergence of the algorithm can be adjusted.
Future work:
other topologies and movement policies,
decouple movement for neighborhood and evolutionary
interaction,
full-fledged MMA.
R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology

20. Thank You!
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and in Twitter
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R. Nogueras et al.
MMAs with Dynamic Self-Organized Topology