# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### Presence of Power Law Distributed Features  
The analysis of our simulation data shows that in most scenarios there was only weak or no statistical support for the (truncated) power law distribution (see Figure 2A). In particular, in roughly half of all scenarios no power distributed features were found. This observation suggests that in the present system, scale-free features are rare. Nevertheless, we found 245 + 71 = 316 (truncated) power law distributions with moderate or strong statistical plausibility for different features in various scenario configurations and for both communication modes, DCM and CCM. Thus, our findings are in line with a recent study showing that scale-free networks may occur rarely but across different areas [55]. As the scatter plots in Figure 2 show, most of the distributions with weak or moderate support for power law are concentrated below or close to the average values of swarm performance while the distributions with strong support for power law are associated with above-average performance in terms of \(N_{coll}\) and \(\omega_{ca}\). Swarm performance was measured using (i) the number of items retrieved by the robots,\(N_{coll}\), (ii) the average number of collected items per time spent on collision avoidance \(\omega_{ca}\), and (iii) the average number of collected items per time spent exploring \(\omega_{e}\). Figure 3 shows the values recorded for these three metrics under both continuous and DCMs and for the entire range of 256 scenario (cues) configurations, respectively. Repeating performance patterns can be observed over different sets of configurations. The regions over which these patterns emerge are (from left to right): (i) all scenarios with \(s_{e}=0\) and \(i_{e}=0\), i.e., constant \(p_{r\to e}\) (blue region with a left tilted mesh in Figure 3), (ii) all scenarios with \(s_{e}=0\) and \(i_{e}>0\), i.e., no social and only internal influence on \(p_{r\to e}\). This region is henceforth referred to as \(NS_{e}\) (shown in orange, no mesh), (iii) all scenarios with \(s_{e}=0.01\), i.e., low social impact on \(p_{r\to e}\) (green region with vertical mesh, henceforth denoted as \(LS_{e}\)), and (iv) all scenarios with \(s_{e}=0.5\) or \(s_{e}=0.9\), i.e., high social influence on \(p_{r\to e}\) (red region with right tilted mesh, henceforth denoted as \(HS_{e}\)). Please note that in all four regions \(p_{e\to r}\) is altered in the same way, i.e., for \(s_{r}\) and \(i_{r}\) all values from \(\{0.0,0.01,0.5,0.9\}\) are included. The best swarm performance in terms of \(N_{coll}\) and \(\omega_{ca}\) emerges when the influence of internal cues on the swarm dynamics is negligible compared to social cues, i.e., when \(s_{e}\sum_{t}|\delta_{\eta}(t)|\gg i_{e}\sum_{t}|\delta_{\phi}(t)|\) and \(s_{r}\sum_{t}|\delta_{\eta}(t)|\gg i_{r}\sum_{t}|\delta_{\phi}(t)|\). Figure 2: (**a**) Feature data sets obtained from simulations, sorted by the type of statistical support for a corresponding power law fit. The classifications follow Table 3. (**b–d**) Log-linear scatter plots relating the power law fit quality ratio \(\rho_{q}\) to the swarm performance in terms of \(N_{coll}\), \(\omega_{ca}\) and \(\omega_{e}\), respectively. The vertical dashed lines indicate the mean performance values while the horizontal dashed lines separate the quality categorizations taken from Table 3.  
Figure 3: Swarm performance in terms of (**a,b**) \(N_{coll}\); (**c,d**) \(\omega_{ca}\) and (**e,f**) \(\omega_{e}\), respectively. For each performance measure, 256 scenario configurations were implemented (i.e., with all cue values from Equation (14)), using one of two communication modes: DCM (left) and CCM (right). The x-axis represents the IDs of the scenario configurations. The colors and the mesh patterns highlight regions that display different dynamics. Apart from (**f**), in all plots the red dots mark the scenarios in which the feature mentioned in the inset demonstrated a high value of \(\rho_{q}\), i.e., there was a strong support for the distribution to be power law.