This document summarizes research on the effect of topology on diversity in spatially-structured evolutionary algorithms (SSEAs). The researchers modeled SSEAs as spreading processes and investigated how network topology influences diversity. They found that lattice networks maintained more diversity than random networks, leading to finding more optima. Rewiring the lattice to have small-world properties showed how network structure can control the spreading of solutions and thus the algorithm's dynamics. This research was a first step toward understanding how to design network topologies to optimize problem-solving in SSEAs.

1. Effect of Topology on Diversity of
Spatially-Structured Evolutionary
Algorithms
M. De Felice - Energy and Environment Modelling Unit@ENEA, Rome, Italy
S. Meloni - Institute for Biocomputation and Physics of Complex
Systems@University of Zaragoza, Zaragoza, Spain
S. Panzieri - Dept. Informatica e Automazione@ROMA TRE University,
Rome, Italy

2. Outline
What are Spatially-Structured EAs (SSEAs)
Why studying SSEAs?
Experimentations
And now?

3. SSEAs
EA where ‘interaction’ is graph-based
Cellular Genetic Algorithms are SSEAs
Classic EA SSEA
}
Individual 1 Individual 1
Individual 2
Individual 3 Selection
Individual 2 Individual 3
...
Individual N
Individual 4

4. Original Idea
Panzieri et al., A Spatially Structured Genetic Algorithm over
Complex Networks for Mobile Robot Localisation, IEEE Int. Conf.
on Robotics and Automation (ICRA), 2007
Adding a ‘structure’ seemed to improve the diversity
of hypothesis

5. -2 2
2
Robotic Localization
-4
(a)
2 4
-6 6
6 6
1
4 4
2 1 2
0 0
2
-2 -2
2
-4 -4
(b)
-6 -6
6 6

6. Epidemic Spreading
Compartmental models [1920s] used to model
epidemic spreading with differential equations
µ
S I R

7. Epidemic Spreading
Compartmental models [1920s] used to model
epidemic spreading with differential equations
µ
S I R
Epidemic Spreading on Networks (see S.Meloni
et al., traffic-driven epidemic spreading in finite-size
scale-free networks, PNAS, 2009)

8. Main Questions

9. Main Questions
1. Can we model EAs as Spreading
Processes?

10. Main Questions
1. Can we model EAs as Spreading
Processes?
2. How graph topology influences
diversity?

11. Main Questions
1. Can we model EAs as Spreading
Processes?
2. How graph topology influences
diversity?
3. Can we use analytic tools used in
Epidemic Spreading to investigate
EAs dynamics?

12. SSEA as Spreading
Process
Analogy between SI (Susceptible-Infectious) model
and EA
γ
S I S
Non-Optimal Optimal Elitism?
J.L. Payne & M.J. Eppstein, Pair Approximations of
Takeover Dynamics in Regular Population Structures,
Evolutionary Computation, 2009

13. Our Algorithm
1. start with random solutions
in nodes
while (!terminate)
for each individual i
2. select uniformly a random
neighbour
3. mutate it
4. if it’s better or equal than i use it
to replace i
end
end
No Diversity Maintenance Mechanisms!

14. Proposed problem
NMAX: Combinatorial problem 8
7
6
5
fitness
4
Composition of L TWOMAX functions of
3
2
1
0
0 1 2 3 4 5 6 7 8
length b Ones
10010100|00011000|... |11100101
}
}
first TWOMAX of length b k-th TWOMAX of length b
2L optima

15. Experimentations
10000 individuals (i.e. 10000 nodes)
Measuring First Hitting Time (FHT), generation of fitness
convergence (FCT) and n. of optima found (N.OPT.)
[average on 100 runs]
Panmictic (traditional), Random Graph (Erdös-Rényi) and
Lattice 1-D (2-neighbours)

16. Experimentations - 2

17. Entropies
Genotypic Entropy Phenotypic Entropy
1. Random and Panmictic go All the topologies converge at
quickly to the same solution the optimal fitness value
2. Lattice 1D ‘converges’ to
several optima

18. Some numbers
Larger genotype leads to...
...slower convergence
...less diversity

19. Some numbers
Larger genotype leads to...
...slower convergence
...less diversity
What happens in-between?

20. Watts-Strogatz
Small-World model
Landmark paper: D.J. Watts & S.H. Strogatz,
Collective dynamics of ‘small-world’ networks,
Nature, 1998
Rewiring probability parameter r
0 -> Regular Graph (lattice)
1 -> Random Network

21. Rewiring and APL
Average Path Length (APL) is the average
length of all the shortest paths
APL measures the spreading of information on
a network

22. Rewiring Factor

23. Rewiring Factor

24. Rewiring Factor - 2

25. Conclusions

26. Conclusions
We investigated the relationship between
network topology and SSEA dynamics

27. Conclusions
We investigated the relationship between
network topology and SSEA dynamics
This is a first step...

28. Conclusions
We investigated the relationship between
network topology and SSEA dynamics
This is a first step...
...to study how to design an ad-hoc network
for a specific problem

29. Conclusions
We investigated the relationship between
network topology and SSEA dynamics
This is a first step...
...to study how to design an ad-hoc network
for a specific problem
...to apply Epidemic Spreading formalisms
to SSEAs

30. Thank you
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www.matteodefelice.name/research/resources/