This paper investigates the performance and the scalability of dynamic and partially connected 2-dimensional topologies for Particle Swarms, using von Neumann and Moore neighborhoods. The particles are positioned on 2-dimensional grids of nodes, where they move randomly. The von Neumann or Moore neighborhood is used to decide which particles influence each individual. Structures with growing size are tested on a classical benchmark and compared to the lbest, gbest and the standard von Neumann and Moore configurations. The results show that the partially connected grids with von Neumann neighborhood structure perform more consistently than the other strategies, while the Moore partially connected structure performs similarly to the standard Moore configuration. Furthermore, the proposed structure scales similarly or better than the standard configuration when the problem size grows.

1. Carlos M. Fernandes1,2
J.L.J. Laredo3
J.J. Merelo2
Carlos Cotta4
Agostinho C. Rosa1
1LaSEEB-ISR-IST,
University of Lisbon (IST), Portugal
Departamento de Arquitectura y Tecnología de Computadores, University of Granada, Spain
3 Faculty of Sciences, Technology and Communications, University of Luxembourg, Luxembourg
4 Departamento de Lenguages y Ciencias de la Computación, University of Malaga, Spain
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2. Motivation
Information flow
Speed
(exploitation)
Robustness (exploration)
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3. Bio-inspired: bird
flock and fish
school.
Cultural and social
interaction: cognitive,
social and random
factors.
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4. X i – position of particle i (vector)
V i – velocity of particle i (vector)
X i(t) = Xi(t-1)+Vi(t)
Vi(t) = ω Vi(t-1)+c1 r1(pi-xi(t-1))+c2 r2(pg-xi(t-1))
Social factor
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5. gbest
lbest
k=n
k=2
von Neumann
k=5
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Moore
k=9
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6. grid with size XxY
Rule: The particles move randomly
to adjacent nodes (defined by
Moore neighborhood).
PSO topology: pg is updated
according to the local neighborhood.
a. Maintain local interactions
b. Room for improvements
c. Easily model other topologies
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7. o
Objective: evaluate performance of partially connected grid topologies
with von Neumann and Moore neighborhood.
o
5 functions (d = 30; except f5, with d = 2)
o
von Neumann, Moore, lbest and gbest topologies.
o
Population size: n = 40; c1=c2=1.494; ω= 0.729
o
Grids with different size (7x7, 8x8, 9x9, 10x10)
o
Three performance metrics:
best fitness
iterations to a solution
success rate
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8. lbest
Fitness values: the proposed structure is able to improve lbest in f1, f2, f3
and f5; in f1 and f3 the differences are statistically significant.
Iterations to a solution: improves lbest in every function, with statistical
differences between the results.
gbest
Fitness values: is able to improve gbest in f1, f3, f4 and f5; the differences
are statistically significant.
Iterations to a solution: in general, gbest is faster, but it fails to meet the
criteria in more than 50% of the runs.
standard von Neumann
Fitness values: 9x9 structure improves
significantly in f1; equivalent in the remaining.
Iterations to a solution: 9x9 structure improves
significantly in f1, f3, f4 and f5.
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9. Rank by success rates
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Rank by overall performance
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10. lbest and gbest: similar conclusions.
Standard Moore configuration: a 7x7 structure is able to
clearly improve the standard configuration in f1, f2 and f3
(is worst in f5); however, the 9x9 structure does not
improve the performance in any function (and it is worst
in f1 and f5).
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11. o Tested f1, f2, f3 and f4 with d=15, d=30 and d=60
o von Neumann configurations.
o the proposed partially connected topology scales
better than the standard von Neumann topology in
f3 and f4; similar in f1 and f2.
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12. o
The proposed structure with von Neumann neighborhood
performs consistently throughout the test set.
o
Improves the performance of other topologies in the majority of
the scenarios and under different evaluation criteria.
o
A sparse connectivity degrades Moore neighborhood
performance.
o
o
The structure is robust to the ratio between the grid size and the
swarm size. A fixed size with ratio 1:2 performs well on every
function.
The proposed topology scales similarly to the standard von
Neumann topology in two functions, and better in the other two
functions.
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13. o
Increase the test set size.
o
Non-random movement strategies.
o
Information flow.
o
Islands, multi-swarms.
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