A quantitative micro–macro link for collective decisions: the shortest path discovery/selection example

Abstract

In this paper, we study how to obtain a quantitative correspondence between the dynamics of the microscopic implementation of a robot swarm and the dynamics of a macroscopic model of nest-site selection in honeybees. We do so by considering a collective decision-making case study: the shortest path discovery/selection problem. In this case study, obtaining a quantitative correspondence between the microscopic and macroscopic dynamics—the so-called micro–macro link problem—is particularly challenging because the macroscopic model does not take into account the spatial factors inherent to the path discovery/selection problem. We frame this study in the context of a general engineering methodology that prescribes the inclusion of available theoretical knowledge about target macroscopic models into design patterns for the microscopic implementation. The attainment of the micro–macro link presented in this paper represents a necessary step towards the formalisation of a design pattern for collective decision making in distributed systems.

Appendix

1.1 Survival analysis

We estimated the transition rates of the macroscopic model from the multi-agent/swarm robotics simulations through survival analysis. In this appendix, we shortly introduce this methodology and we present the Nelson–Aalen estimator that we employed for our analysis. Finally, we describe how to estimate the transition rates directly from experimental data. We believe that showing the application of this statistical tool for the analysis of a swarm robotics experiment may be of interest to the community.

1.1.1 Survival analysis

Survival analysis is a branch of statistics that offers tools to estimate the change over time of the probability of an event from experimental data. Survival analysis has been initially introduced in medicine to estimate the probability of survival (or death) of an organism under some treatment. Subsequently, these tools generalised to the estimate of any transition probability between populations. Nowadays, survival analysis is employed in several fields, such as economics—e.g. to estimate the probability of a stock market crash—or mechanical engineering—e.g. to estimate the probability of engine failures. In this work, we apply survival analysis to estimate the transition rates of the macroscopic model of collective decision making.

Other works have used this type of analysis to estimate the parameters of multi-agent systems behaviour. Jeanson et al. (2003) use survival analysis to estimate the probability with which cockroaches change their behaviour. Garnier et al. (2008) and Reina et al. (2014) employed survival analysis to compute transition rates of artificial agents behaviour.

1.1.2 Hazard curve

We consider the three populations $\{A,B,U\}$ in accordance with the agent’s commitment state described in Sect. 2. To compute the rate at which agents switch (transit between) their commitment state, we log the number of timesteps $t$ interlaying between two commitment switches (transitions) and the relative type of event causing the switch (e.g. discovery or recruitment). At the end of an experiment, we log the timesteps $t$ from the last commitment switch as censored event, which indicates that after $t$ timesteps no transition happened. We use the Nelson–Aalen estimator (Nelson 1969) to compute the hazard curve $H(t)$ from the collected experimental data. The hazard curve $H(t)$ shows the cumulative probability of events occurring until time $t$ and is computed as follows:

$$\begin{aligned} H(t) = \sum _{t_i\le t}d_i/n_i, \end{aligned}$$

(8)

where $d_i$ is the number of events recorded at $t_i$, and $n_i$ is the number of events occurring (or censored) at time $t \ge t_i$. In a memoryless system, the probability of an event does not change over time, therefore the curve of the cumulative probability as function of time corresponds to a line with a slope equal to the constant event rate (i.e. the transition rate). Assuming our system as memoryless, we compute the transition rate by linear fitting the hazard curve with a line passing through the origin. Additionally, the quality of the fitting can be used to verify the correctness of a memoryless implementation.

1.1.3 Rate estimation

The rate of spontaneous transitions, in this study discovery and abandonment, can be directly estimated by computing the slope of the hazard curve, as detailed above. For instance, Fig. 12a shows the hazard curve of the discovery rate computed from multi-agent experiments with target area at distance $d= 1.5\,\hbox {m}$. Differently, rate estimates of transitions consequent to an interaction—in this work, recruitment and cross-inhibition—include the probability of interaction with an agent of the other population, which we call hereafter the interacting population. This probability changes during the process as the interacting population size changes and must be taken into account to estimate a constant transition rate independently from the interacting population size. Therefore, we first compute one aggregate transition rate for every interacting population size, and then, we normalise the rates for the interacting population fraction. For instance, from Eq. (1), the cross-inhibition rate for the population $B$ is $(-\sigma _A\varPsi _{A})$, which includes the size of the interacting population $A$ that delivers the inhibition signal. Through survival analysis, we compute the aggregate rate ($\sigma _A\varPsi _{A}$) for varying values of $\varPsi _{A}$ in the range $]0,1[$. Then, by linear fitting, we discount from the aggregate rates the interacting population fraction to obtain $\sigma _A$. Figure 12b shows a set of 50 aggregate rate estimates plotted as function of $\varPsi _{A}$, the slope of the fitted line corresponds to the estimate of $\sigma _A$.

Fig. 12

a Cumulative discovery probability over time for target area at distance $d= 1.5\,\hbox {m}$. The slope of the resulting line corresponds to the discovery rate estimate. b Cross-inhibition rate $\sigma _A$ estimates over population fraction $\varPsi _A$ for target area distances $d_A=d_B= 2.5\,\hbox {m}$ and probability $P_{\sigma }=0.1$

To estimate the transition rates with constant population sizes, we run ad hoc experiments where the population sizes are fixed. In these experiments, agents follow the normal behaviour and, in case of commitment state transitions, they only log the event but do not change commitment state. In this way, we can quickly gather a large amount of data for every population size and parameterisation.