# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### Correlation with Swarm Performance  
In the previous section we have illustrated that swarm performance is likely to reach its peaks over cue configurations that include asymptotically scale-free space or time features (see Figure 3). In this section, we analyze this observation statistically, using correlation measures such as the Pearson and Kendall's tau rank correlation coefficients introduced in Section 2.4.3. However, note that both correlation measures have strengths and shortcomings. On the one hand, while the Pearson correlation coefficient is widely used and has an elegant mathematical form, it is sensible to outliers and may not be appropriate for non-linear distributions. On the other hand, Kendall's tau is suitable for non-linear distributions as well as robust to outliers. Nevertheless, reducing the values to ranks may disregard the significance of the variable's value being far from the average. In particular, replacing the real value of the quality ratio \(\rho_{q}\) by its rank leads to loss of information about the extent to which \(\rho_{q}\) represents the quality of the power law distribution. Moreover, following the definition of Kendall's tau in Equations (11)-(15), each difference between data point pairs is assigned the same weight which may not always be appropriate. For instance, consider the ranked swarm performance in terms of \(N_{coll}\) and the corresponding distribution of \(\rho_{q}\) in CCM for \(d\) in Figure 7a and for \(d_{c,sent}\) in Figure 7b, respectively. In both cases, Kendall's tau defined by Equations (11)-(15) returns values indicating no correlation (i.e., \(\tau_{Kendall}=0.02\) and \(\tau_{Kendall}=-0.04\), respectively). However, as evident in Figure 7, both cases show different dynamics, with \(\rho_{q}\) for \(d\) following \(N_{coll}\) more closely than for \(d_{c,sent}\). The main reason is that the dominant fluctuations of \(\rho_{q}\) close to zero are assigned the same weight (i.e., rank step 1) as the more permanent increase of \(\rho_{q}\) for high values of \(N_{coll}\). Similar considerations hold for the other features and the DCM. To account for this type of behavior, we use a generalization of Equation (15) that weights the ranking steps by a parameter \(\kappa\), which is relative to the average change, such that:  
\[ \begin{split}\delta_{i,j}^{(c)}&=\begin{cases} \kappa,&\text{if }sgn(x_{i}-x_{j})sgn(y_{i}-y_{j})>0\\ 0,&\text{else}\end{cases}\\ \delta_{i,j}^{(x)}&=\begin{cases}\kappa,&\text{if }x_{i}=x_{j}\\ 0,&\text{else}\end{cases}\end{split}  \tag{15}\]  
Figure 6: Log-log scale plots of \(\Delta d\) per 100 ts in CCM. The black lines represent the corresponding truncated power law fits. The insets show the fit parameters as well as the scenario configurations. The plots (**a**,**b**) differ by the scenario configurations shown in the insets. These scenarios are among the best swarm performances with respect to \(\omega_{e}\). and similarly, for \(\delta_{i,j}^{(d)}\) and \(\delta_{i,j}^{(y)}\). The weight parameter \(\kappa\) is given by  
\[\kappa=\frac{1}{2}\left(\frac{|x_{i}-x_{j}|}{\mu_{x}}+\frac{|y_{i}-y_{j}|}{\mu_{ y}}\right), \tag{16}\]  
where \(\mu_{x}\) and \(\mu_{y}\) are averages over all \(|x_{i}-x_{j}|\) and \(|y_{i}-y_{j}|\), respectively. For each \(i,j\) pair, Equation (16) considers the data distances of both distributions, normalized by their respective mean distances. Consequently, \(\kappa\) does not favor any distribution and weights each ranking step relative to other distances. Please note that in general, there is no correlation metric that is perfectly adequate for all types of studies and data distributions; it is thus common to consider appropriate modifications [67; 68; 69; 70]. In the present case, for \(\kappa=1\) we obtain the standard Kendall's tau rank correlation coefficient described in Section 2.4.3.