# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### The Role of Feedback Mechanisms in the Emergence of Scale-Free Features  
An attribute of complex systems that is widely known to support the emergence of scale-free characteristics is the presence of (positive and negative) feedback loops [1; 53; 71]. We specify the feedback effect to be positive or negative based on the individual response to the information input from its neighborhood. Hence, we refer to the feedback mechanism as _positive_ feedback if it pushes the individuals to the same state as the state of the majority, whereas _negative_ feedback pushes them away from it. Most scale-free features were found in scenarios in which (i) the robot behavior was dominated by social interactions, (ii) the swarm attempted to balance positive and negative feedback and (iii) the swarm displayed a tendency towards active exploration. In particular, in CCM, the first 17 scenarios sorted by \(\rho_{q}\) in descending order were found over the \(LS_{e}\) region and with \(i_{r}=0\). Similarly, in DCM, the first 28 scenarios were found over the \(HS_{e}\) region and with \(i_{r}=0\). To understand this, it is necessary to consider in more detail the impact of each cue on swarm dynamics and the feedback mechanisms. For conciseness, we focus our system analysis on CCM and its most relevant set of parameter configurations. Similar conclusions hold for DCM. In particular, we can simplify our analysis based on the repeating patterns of swarm performance (see Figure 3) and the following observations: (i) The swarm performance is qualitatively very similar between the cue values 0.5 and 0.9 (for all four cues). Thus, we focus, in the following, on \(\{a:s_{e},i_{e},i_{r},s_{r}\in\{0.0,0.01,0.5\}\}\). (ii) The cue \(i_{e}\) has a negligible impact on the foraging dynamics when \(s_{e}>0\). By neglecting scenarios in which \(i_{e}\
eq 0\), except those with \(s_{e}=0\), we can further shorten the set of relevant scenarios. Finally, (iii) there are significant differences in the dynamics between scenario configurations with \(i_{r}=0\) and those with \(i_{r}>0\) but negligible differences between \(i_{r}=0.01\) and \(i_{r}=0.5,0.9\). Thus, we focus, in the following, only on scenarios with either \(i_{r}=0\) or \(i_{r}=0.01\). Figure 8 shows the final set of 24 scenarios relevant to the discussion below. Please note that (ii) and (iii) are consequences of the internal cues \(i_{e}\) and \(i_{r}\) acting only on _exploring_ robots. In the exploring state, the crucial parameter is \(p_{e\to r}\) because it defines the probability to stop exploring and change to resting. A non-zero value of the internal cue \(i_{r}\) has a substantial impact on dynamics as it alters \(p_{e\to r}\) after each exploration attempt. As the likelihood of finding and retrieving a food item is low, \(i_{r}\) mostly reduces \(p_{e\to r}\). The more \(p_{e\to r}\) is lowered by \(i_{r}\), the less likely the robot is to find a food item during the next exploration attempt. Thus, \(i_{r}\) has a strong inhibitory influence on the swarm's exploration activity. Consequently, there is a significant difference in swarm performance between the scenario configurations with \(i_{r}=0\) and those with \(i_{r}>0\). As the swarm actively attempts to explore the environment and collect food items, the influence of \(i_{r}>0\) can be considered an important driver of negative feedback. In contrast, \(p_{r\to e}\) acts only on _resting_ robots. Consequently, any change of \(p_{r\to e}\) through \(i_{e}\) is easily distorted by \(s_{e}\), i.e., the social interactions with the neighborhood of the resting robot. Hence,  
Figure 8: Number of collected items for a selected set of 24 scenario configurations \(a\) in CCM. The data labels show the corresponding cue values of \(a=(s_{e},i_{e},i_{r},s_{r})\). only when \(s_{e}=0\) there is an inhibitory impact of \(i_{e}\) on swarm dynamics (similar to \(i_{r}\), due to the scarcity of food items), otherwise \(i_{e}\) is negligible.