Towards a 2-dimensional Self-organized Framework for Structured Population-based Metaheuristics

Carlos M. Fernandes1,2

J.L.J. Laredo3

J.J. Merelo1

Carlos Cotta4

Agostinho C. Rosa2

1Department of Computers Architecture and Technology, University of Granada, Spain

2 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST), Portugal3 University of Luxembourg

4 University of Malaga

2

Objectives and Motivation

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Objective: describe the properties of a swarm of simple entities that interact (communicate) on a 2-dimensional heterogeneous environment.

Motivation: improve the state-of-the-art on dynamic population structures for bio-inspired algorithms.

Local rulesStigmergySelectionStochastic no central coordination

3

Non-panmictic Evolutionary Algorithms (Eas)

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Island Models

Cellular EAs

EAs are based on the selection, recombination and mutation of populations of solutions

Panmictic EAs

Non-Panmictic EAs

4

Particle Swarm Optimization

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Bio-inspired: bird flocks and fish schools.

Topology: ring, star,...

5

The System

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Habitat: 2-dimensional toroidal grid of nodes with size N×N

Swarm: population of n particles (p) with a random fitness value [0,1].

Initialization: the particles are randomly distributed in the grid

Dynamics: particles move to neighboring sites

Communication: particles leave marks (m) with their “status”

Evaporation: marks are erased after one iteration

Rules:check free nodes in Moore neighbourhood.if no free nodes → don’t moveif no marks → random siteif marks → move to more similar

P P P

P P P

P P P

P

P P P

m P P

P m

m

fitness

6

Self-Organization

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Local rules → Global patterns

Dynamic, robust and displays power-laws

Self-Organized Criticality (SOC)

Edge of Chaos (EOC)

Highly Optimized Tolerance (HOT)

Self-Organized complex system?

No central coordination

No order, no chaos

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Dynamic Behaviorone dimension

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n=25

n=50

n=75

n=100

Space-time diagrams of a 1-dimensional habitat with 150 nodes

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Dynamic Behaviortwo dimension

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()

t = 0

t = 1

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

t = 0

t = 9

t = 10

t = 1000t = 500t = 250t = 100

k =4.02 k =4.02k =4.00k =3.77

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Dynamic Behavior2D: clustering degree and distance

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0 75 1502253003754505256006757508259009752.5

3

3.5

4

4.5

iteration t

k

0 75 1502253003754505256006757508259009750

0.05

0.1

0.15

0.2

0.25

iteration t

d

1 10 100 1000 100000.01

0.1

1

10

100

1000f(x) = 2553.98182 x^-1.1750431R² = 0.742468825507372

k

frequency

inte

nsi

ty

1 10 100 1000 100000.01

0.1

1

10

100

1000

f(x) = 77.9227875 x^-0.9867793R² = 0.700408288010165

d

frequency

inte

nsi

ty

k, average clustering degree (number of neighbours)

d , average distance (difference between fitness values) to neighbours

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Dynamic Behavior2D: robustness

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n:nodes→ 1:24 1:12 1:6 1:3 1:2 1:1.5 1:1.2

k1.18(0.76)

1.23(0.76)

1.23(0.76)

1.20(0.76)

1.07(0.70)

0.88(0.60)

0.56(0.60)

d0.82

(0.60)1.00(0.72)

0.97(0.68)

1.01(0.69)

1.00(0.69)

0.93(0.64)

0.42(0.60)

n→ 33 75 147 300 616 1200 2408 4800

k1.15

(0.72)1.29

(0.77)1.18

(0.75)1.22

(0.77)1.18

(0.74)1.20

(0.76)1.17

(0.74)1.18

(0.76)

d0.87

(0.62)1.04

(0.70)1.04

(0.71)1.10

(0.75)1.03

(0.70)1.01

(0.69)1.02

(0.69)0.97

(0.69)

n = 300 n = 600 n = 1200 n = 1800 n = 2400 n = 3000

Slope and r-squared of the power-laws

k =3.18 k =3.88k =3.92 k =4.58 k =6.27k =5.28d =0.038 d =0.076 d =0.243d =0.190d =0.144d =0.085

10000 iterations

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Dynamic Behavior2D: fitness distribution

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t = 0

t = 10000

Distribution of the fitness values on the the habitat (darker → higher fitness)

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Dynamic Behavior

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10481120

1140

1160

1180

1200

k=0k=2

k=4k=6

k=80

50100150200250300350400450

t=0

part

icle

s

k=0k=2

k=4k=6

k=80

50100150200250300350400450

t=1

k=0k=2

k=4k=6

k=80

50100150200250300350400450

t=2

part

icle

s

050

100150200250300350400450

t=1000

number of particles that move in each iteration.

number of articles classified according to the clustering degree

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Conclusions

o Local rules lead to complex behavior

o Particles form highly dynamic clusters connected by paths

o The system’s output variables (degree of clustering and distance between neighboring particles) display a power-law relationship

o The system is robust

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14

Future Research

o Study the model under different fitness distributions (normal, for instance) and under perturbations (remove particles, change fitness values, etc)

o Increase the memory of the system.

o Use the model for designing a cellular GA.

o Use the model as a basis for a dynamic PSO topology.

o Investigate if the model fits into a Self-Organization theory (SOC, EOC, HOT)

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Questions?

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