Towards swarm calculus: urn models of collective decisions and universal properties of swarm performance

Abstract

Methods of general applicability are searched for in swarm intelligence with the aim of gaining new insights about natural swarms and to develop design methodologies for artificial swarms. An ideal solution could be a ‘swarm calculus’ that allows to calculate key features of swarms such as expected swarm performance and robustness based on only a few parameters. To work towards this ideal, one needs to find methods and models with high degrees of generality. In this paper, we report two models that might be examples of exceptional generality. First, an abstract model is presented that describes swarm performance depending on swarm density based on the dichotomy between cooperation and interference. Typical swarm experiments are given as examples to show how the model fits to several different results. Second, we give an abstract model of collective decision making that is inspired by urn models. The effects of positive-feedback probability, that is increasing over time in a decision making system, are understood by the help of a parameter that controls the feedback based on the swarm’s current consensus. Several applicable methods, such as the description as Markov process, calculation of splitting probabilities, mean first passage times, and measurements of positive feedback, are discussed and applications to artificial and natural swarms are reported.

Appendix: Details on curve fitting

All curve fitting was done with an implementation of the nonlinear least-squares Marquardt–Levenberg algorithm (Levenberg 1944; Marquardt 1963) using gnuplot 4.6 patchlevel 1 (2012-09-26).²

1.1 A.1 Foraging in a group of robots

The data for the curve fitted in Fig. 2(b) are shown in Table 1.

Table 1 Fitting data, foraging in a group of robots, Fig. 2b

1.2 A.2 Collective decision making based on BEECLUST

The data for the curve fitted in Fig. 2(c) are shown in Table 2.

Table 2 Fitting data, collective decision making based on BEECLUST, Fig. 2(c)

1.3 A.3 Aggregation in tree-like structures and reduction to shortest path

The data for the curve fitted in Fig. 2(d) are shown in Table 3. In this case weighted fitting was used (values ρ ₁=0.00524 and ρ ₂=0.00598 were weighted ten times higher than other values) to enforce the limit P<1.

Table 3 Fitting data, aggregation in tree-like structures and reduction to shortest path, Fig. 2(d)

1.4 A.4 Emergent–taxis behavior

The data for the curve fitted in Fig. 3(a) are shown in Table 4. Fitting was done in two steps. First, the interference function I(N) was fitted. Second, the performance function P(N) was fitted while keeping the parameters a ₂ and c fixed.

Table 4 Fitting data, emergent–taxis behavior, Fig. 3(a)

1.5 A.5 Emergent–taxis behavior, narrow fit

The data for the curve fitted in Fig. 3(b) are shown in Table 5. The parameters a ₂ and c of the interference function I(N) as obtained in Sect. A.4 were reused. The performance function P(N) was fitted within the narrow interval of N∈[20,22] while keeping the parameters a ₂ and c fixed.

Table 5 Fitting data, emergent–taxis behavior, narrow fit, Fig. 3(b)

1.6 A.6 Mean first passage times

The data for the curve fitted in Fig. 7 are shown in Table 6. Weighted fitting was applied based on the measured standard deviation and weights scaled by $\sqrt{\tau^{\text{theor}}}$, respectively.

Table 6 Fitting data, mean first passage times, Fig. 7

1.7 A.7 Density classification

The data for the curve fitted in Fig. 8(a) are shown in Table 7. For times t∈{100,200,400}, we set φ=0 as otherwise the fitting would result in φ<0.

Table 7 Fitting data, density classification, Fig. 8(a)

1.8 A.8 Feedback intensities

The data for the curve fitted in Fig. 8(b) are shown in Table 8. Weighted fitting was applied with zero-weight for data points of t<700, which means we ignore the initial values of φ(t)=0. Data points of t≥3000 had double the weight than values of 700≤t<3000.

Table 8 Fitting data, feedback intensities, Fig. 8(b)

1.9 A.9 Positive feedback probability

The data for the curve fitted in Fig. 9(a) are shown in Table 9.

Table 9 Fitting data, positive-feedback probability, Fig. 9(a)

1.10 A.10 Swarm alignment in locusts

The data for the curve fitted in Fig. 10 are shown in Table 10. We set φ=1 as otherwise the fitting would result in φ>1. Weighted fitting was applied values of s<0.17 and s>0.83 had double weight than values of 0.17≤s≤0.83.

Table 10 Fitting data, swarm alignment in locusts, Fig. 10