# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### Presence of Power Law Distributed Features
In (**f**), the red dots mark the scenarios with moderate support. See Supplementary Material Section 3 for combined plots of \(N_{coll}\) and \(d\) distribution in CCM over the complete set of 256 scenario configurations. The best performance levels in terms of \(N_{coll}\) and \(\omega_{ca}\) were reached over the \(LS_{e}\) region. For instance, the maxima of \(N_{coll}\) and \(\omega_{ca}\) correspond to the scenario configurations in which \(s_{e}=0.01\), \(i_{r}=0\) and \(s_{r}\geq 0.5\). For the same configurations, (truncated) power law distributions of space features were found in the CCM (examples shown in Figure 4). Contrary to CCM, in the DCM the robot interactions are interrupted. These interruptions may explain why, in DCM, space features such as communication degree tend to not follow a power law distribution (weak overall support for the presence of a power law behavior). Nevertheless, we found fits with moderate to strong support for (truncated) power law to time features, such as \(\tau_{r}\) and \(\tau_{ca}\), demonstrated in Figure 5. The best power law fits of the DCM correspond to the peaks in swarm performance in terms of \(N_{coll}\) and \(\omega_{ca}\) over the \(HS_{e}\) regions. Figure 4: Log-log scale plots of the degree \(d\) (top) and the critical degree \(d_{c,rec}\) (bottom) distributions 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) and (b) differ by the scenario configurations shown in the insets; similarly for (c) and (d). These scenarios are among the top five swarm performances with respect to \(N_{coll}\). Please note that \(\lambda x_{min}\) is relatively small, i.e., power law is a good fit for \(x\) close to \(x_{min}\). See Supplementary Material Section 3 for plots of \(d\) in CCM over all 256 scenario configurations. The third performance measure, i.e., \(\omega_{e}\), reached its best values over the \(NS_{e}\) region. Its maxima correspond to cases where \(s_{e}=0\) and \(s_{r}=0\). Interestingly, for these scenario configurations we found fits with moderate to strong support for (truncated) power law to the data of \(\Delta d\), i.e., the change of the average communication degree of the robot (examples shown in Figure 6). This is an interesting finding because it indicates that a communication feature may be power law distributed also in those scenarios in which the swarm tries to minimize the number of foraging robots and maximize the number of resting ones. Moreover, in most \(\Delta d\) distributions with strong or moderate support for the power law, the fit includes only 10-20% of data points. The reason for the relatively low ratio of power law fitted data is that the tail of the distribution is likely to represent by the fraction of robots that rest or move close to the border between the nest and the foraging area. In general, the findings suggest that internal cues (in the absence of social cues) keep robots at the edge of minimal activity while social cues (in the absence of internal cues) drive the robots towards maximal activity. Figure 5: Log-log scale plots of the resting time \(\tau_{r}\) (top) and the collision avoidance time \(\tau_{ca}\) (bottom) distributions in DCM. 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; similarly for (**c**,**d**). These scenarios belong to a subset of the best swarm performance with respect to \(\omega_{ca}\). Please note that \(\lambda=0\) for the fits of \(\tau_{ca}\), indicating better support for the power law fit than for truncated power law.