# Scale-Free Features in Collective Robot Foraging
## 3 Results and Discussion
### The Role of Feedback Mechanisms in the Emergence of Scale-Free Features
In short, when the probability of finding a food item is low, with \(i_{r}=0\) and \(s_{e}>0\) enabling the swarm to significantly damp the feedback mechanisms that drive the swarm towards inactivity. Next, we consider the particular contributions of social cues \(s_{e}\) and \(s_{r}\). Social interactions represent a direct form of feedback loops, enabling the swarm to drift towards an absorbing state (e.g., uninterrupted resting or exploring) or maintain a balance between positive and negative feedback. In general, note that high values of \(s_{e}\) often lead to \(p_{r\to e}=0\) due to the high probability of encountering a robot with a failed exploration (due to the low density of food items). With such relatively high values of \(s_{e}\), \(p_{r\to e}\) can be reduced to zero within a few time steps. By contrast, with \(s_{e}=0.01\), \(p_{r\to e}\) does not fluctuate as strongly. Similar considerations hold for \(s_{r}\) and \(p_{e\to r}\). In terms of active exploration, i.e., long exploration times and high number of retrieved items, it is beneficial for the swarm to have robots with \(p_{r\to e}=1\) and \(p_{e\to r}=0\). Indeed, in the present system we observe that the swarm approaches such behavior for \(s_{r}=0.5\) and \(s_{e}=0.01\) (i.e., over the \(LS_{e}\) region). More importantly, such balance of \(s_{r}\) and \(s_{e}\) allows positive and negative feedback loops to coexist with the positive feedback being slightly more dominant. Due to this feedback coexistence, a robot that happened to be surrounded by unsuccessful neighbors will tend to have low \(p_{r\to e}\) and high \(p_{e\to r}\), i.e., its resting time \(\tau_{r}\) will increase (together with its \(d\) or \(d_{c,rec}\)) and vice versa. Over time, such dynamics will result in robots that are increasingly inactive (with increasingly higher \(\tau_{r}\), \(d\) or \(d_{c,rec}\)) and robots that are increasingly active (with increasingly lower \(\tau_{r}\), \(d\) or \(d_{c,rec}\)). When the majority tends towards active exploration, the inactive group of robots experiences negative feedback and while the active group is subject to positive feedback. The prevalence of the positive feedback decreases the number of consistently resting robots significantly below the number of consistently exploring ones. Ultimately, this leads to skewed or heavy tailed distributions, such as the power law and, consequently, to the emergence of scale-free features. Similar considerations apply to the DCM over the \(HS_{e}\) region. The difference is that in DCM each robot can broadcast its exploration result only once. Thus, \(s_{e}\) needs to have high values for dynamics similar to CCM to emerge. To illustrate the above considerations, let us examine the scenario configuration of CCM with \(s_{e}=0.01\), \(i_{e}=0.9\), \(i_{r}=0.0\), and \(s_{r}=0.5\) (see Figure 4a), in which a high performance value of \(N_{coll}\) was observed (i.e., this scenario is similar to the peak in Figure 8). The value of \(s_{r}\) has a high impact on \(p_{e\to r}\) (the probability to transition to resting). For example, if a robot receives at least two'success' messages more than 'failure' messages--i.e., if \(\delta_{\eta}(t_{r})\geq 2\) in Equation (2)--its \(p_{e\to r}\) drops to zero. When \(p_{e\to r}=0\) and \(i_{r}=0.0\), the robot will stop exploring only if it finds an item. During the subsequent resting, this robot is likely to cause one of its neighbors to reach \(p_{e\to r}=0\), which repeats an analogous cycle of events. The corresponding dynamics can be translated in terms of the positive feedback pushing the robots out of the nest and increasing the number of robots in the foraging area (i.e., in the exploring state). In the long term, due to the positive feedback, the swarm drifts towards the absorbing state in which all robots have \(p_{e\to r}=0\) and \(p_{r\to e}=1.0\). In the short term, while most robots is exploring, some robots remain in the nest, e.g., due to crowding at the entrance of the nest.