[J12][.pdf
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Heiko Hamann Towards Swarm Calculus: Urn Models of Collective Decisions and Universal Properties of Swarm PerformanceSwarm Intelligence, 7(2-3):145-172, 2013 |

[C26][.pdf
.bib
cited by] Heiko Hamann Towards Swarm Calculus: Universal Properties of Swarm Performance and Collective DecisionsANTS 2012, Brussels, Belgium, Marco Dorigo et al. (eds.), LNCS 7461, pp. 168-179, Springer-Verlag, 2012 |

The idea is to define a simple but general model of collective decisions in swarms. We define an urn model:

Say r is the ratio of red marbles in the urn.

1) With probability r we draw a red marble and put it back,

3) we define the probability of positive feedback by

\(P(s)=\varphi\sin(\pi s)\) (1st plot in the applet below),

for having drawn a red marble in step 1) we have P(r).

4a) in case of positive feedback we replace a blue marble from the urn with a red one

4b) in case of negative feedback we replace another red marble from the urn with a blue one

Based on P(s) we can calculate the expected change of the fraction of - say - red marbles s by

\(\Delta s(s)=4(P(s)-0.5)(s-0.5)\) (2nd plot in the applet below).

Based on this expected change we can define a Markov process, define a transition matrix, and then determine the eigenvectors of this matrix to obtain the expected steady state of the probability density of s (3rd plot in applet). powered by NetLogo

view/download model file: matrixInteractive.nlogo

This is a Java applet (iphone and ipad users: You have my apologies). So you need to have Java activated in your browser (for firefox see here).

Math support by MathJax.