# Scale-Free Features in Collective Robot Foraging
## 2 Methods
### Data Analysis
#### 2.4.2 Quality Ratio \(\rho_{q}\)
This equation approaches the power law distribution asymptotically for \(\lambda x\to 0\) and the exponential distribution for \(x\lambda\gg 1\), respectively. Thus, accepting that our systems are significantly constrained within physical boundaries we can slightly soften the criteria given by Equation (7) in the following way:  
If the truncated power law passes more likelihood-ratio tests than the power law fit, i.e., if \(n_{lrt,lpl}>n_{lrt,pl}\), we consider the success-ratio of the former. In short:  
\[\rho_{q,lrt}= \begin{cases}0,&\text{if at least one likelihood-ratio test yields }R<0\\ \frac{n_{lrt,lpl}+1}{7},&\text{if }n_{lrt,lpl}>n_{lrt,pl}\\ \frac{n_{lrt,lpl}+1}{7},&\text{otherwise.}\end{cases}  \tag{9}\]  
Finally, including all the above criteria, we define the quality ratio:  
\[\rho_{q}=\rho_{q,p}\cdot\rho_{q,data}\cdot\rho_{q,lrt}. \tag{10}\]  
Consequently, we obtain \(\rho_{q}=0\) if \(p\leq 0.1\) or \(R<\)0. Conversely, \(\rho_{q}=1\) in the case of \(p>0.1\), \(N_{data,pl}=N_{data,tot}\) and \(n_{lrt}=6\), which is an unlikely but nevertheless possible scenario. Using this ranking, we can link the quality of a fit to a quantifiable value and describe the support for the (truncated) power law as illustrated in Table 3.  
The denominator value represents the total number of considered distributions (i.e., the power law and the six alternative distributions we compare it to). The lower limit of the'moderate' classification corresponds to the case with \(\rho_{q,p}=1\), \(\rho_{q,data}=0.1\) and \(\rho_{q,lrt}=\frac{1}{7}\)--i.e., at least 10% of the data is included in the fit and none of the alternative distributions is a statistically better fit than the power law. The upper limit considers the case with \(\rho_{q,p}=1\) and \(\rho_{q,data}\cdot\rho_{q,lrt}=\frac{0.5}{7}\)--i.e., either the fit includes a high number of data or the power law is statistically a better fit than other distributions. Please note that \(\rho_{q}\) multiplicatively combines standard power law fitting techniques [53; 54; 55; 56] into a quantitative estimate of the quality of the (truncated) power law fit. It is important to emphasize that even if the hypothesis of the data following the power law distribution is found to be plausible using the above statistical analysis, care needs to be taken when interpreting this observation. Firstly, there is still no guarantee that the data is in fact power law distributed and although our rigorous analysis includes several common distributions, other non-obvious distributions may prove to be a better fit. Secondly, the power law fit may be valid only for a small fraction of data. However, as the power law behavior is commonly found for a subset of data, namely at the tail of the distribution, the group that displays power law (i.e., scale-free) behavior includes individuals that stand out from the rest of the swarm by having features with values that are significantly above average. The way in which such individuals impact the global swarm performance remains an open question worth investigating. \begin{table}
\begin{tabular}{c c} \hline
**Support for the (Truncated) Power Law** & **Numerical Value of \(\rho_{q}\)** \\ \hline No support & \(\rho_{q}=0\) \\ Weak & \(0<\rho_{q}\leq\frac{0.1}{7}\approx 0.0143\) \\ Moderate & \(\frac{0.1}{7}<\rho_{q}\leq\frac{0.5}{7}\approx 0.0714\) \\ Strong & \(\frac{0.5}{7}<\rho_{q}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Classification of the power law fit quality with respect to the quality ratio \(\rho_{q}\) used in our study.