# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### Correlation with Swarm Performance
However, by implementing Equation (16), the correlation coefficient is less sensitive to fluctuations than the standard Kendall's tau, while still being more robust to outliers and non-linearity than the Pearson correlation measure. Therefore, in the following we will use this modified Kendall's tau rank correlation coefficient to investigate the presence of correlations between \(\rho_{q}\) and swarm performance. The correlations are shown for all features in Table 5 between the three measures of the swarm performance and the feature'scale-freeness' quantified by \(\rho_{q}\). We found strong correlations of the scale-free property of various features with the swarm performance. In particular, high correlations exist for \(\tau_{ca},\tau_{r}\), \(\Delta d\) in DCM; and, additionally, for \(d,d_{c,rec}\) in CCM. Remarkably, for those features for which we found moderate or high correlation values (highlighted in blue in Table 5), most high-quality power law fits appear in the same scenarios as the highest swarm performance peaks. The red dots in Figure 3 illustrate this finding by highlighting the scenarios in which the quality ratio is \(\rho_{q}>\frac{0.5}{7}\). Moreover, the swarm tends to demonstrate low performance with respect to \(N_{coll}\) and \(\omega_{ca}\) for those scenarios in which \(\omega_{e}\) is highest, the latter being well correlated with \(\Delta d\). Figure 7: Ranked distribution of \(N_{coll}\) (dark red, left y-axis). For the same cue configurations, the CCM distributions of \(\rho_{q}\) (right y-axis) for (**a**) \(d\) and (**b**) \(d_{c,sent}\) are shown in blue, respectively. The insets depict the corresponding scatter plots with data points representing weak (circles), moderate (triangles) and strong (squares) support for power law distribution; gray lines indicate the onsets of the different support classifications (similar to Figure 2). The correlation coefficients confirm the observation, supported by the data shown in Figures 2 and 3, that most power law distributions with strong support (i.e., high \(\rho_{q}\)) appear in scenarios with peak swarm performance. To further examine this observation, we consider the correlations of the swarm performance with different \(\rho_{q}\) support classifications (based on Table 3). As Table 6 shows, there are moderate and strong positive correlations between features with strong support for power law distribution and swarm performance in terms of \(N_{coll}\) and \(\omega_{ca}\) for both communication modes. This suggests that the observation of scale-free features is more likely in scenarios in which the agents are more successful in retrieving a high number of food items. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Support for pl** & **Correlation with \(N_{coll}\)** & **Correlation with \(\omega_{ca}\)** & **Correlation with \(\omega_{e}\)** \\ \hline \multicolumn{4}{c}{_Discontinuous communication mode (DCM)_} \\ \hline weak & \(0.18\) & \(0.18\) & \(-0.16\) \\ moderate & \(0.10\) & \(0.08\) & \(0.21\) \\ strong & \(0.31\) & \(0.40\) & \(-0.14\) \\ moderate + strong & \(0.50\) & \(0.53\) & \(0.22\) \\ \hline \multicolumn{4}{c}{_Continuous communication mode (CCM)_} \\ \hline weak & \(0.10\) & \(0.11\) & \(0.03\) \\ moderate & \(0.37\) & \(0.33\) & \(0.03\) \\ strong & \(0.66\) & \(0.63\) & \(-0.25\) \\ moderate + strong & \(0.74\) & \(0.76\) & \(-0.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Correlation coefficients between \(\rho_{q}\) and \(N_{coll}\), \(\omega_{ca}\) or \(\omega_{e}\) for different power law support classifications.