## 4 Methodology
For example, it could be considered in the evaluation criteria that occur in many different national academic systems or even to evaluate research projects or their management. Before discussing the specific findings, in this section we present the methodology that we have used and the kind of results obtained. This kind of analysis can be seen as borrowed from the Human Mobility field, where it emerges that the length of human travels can be described by a Power Law distribution in general [1; 39] and by a Lognormal distribution if analyzed with a transport modality decomposition (walk, car, bike, train, airplane) [18; 40]. In our analysis the scholars travel among the topics, modeled as the estimated locations. We have studied the mobility for each scholar and for each year, as discussed in Section 4.4. The distance between two consecutive locations is measured with the Euclidean distance. The Probability Density Function (PDF) was computed over the frequency of displacements, in order to estimate which distribution can better describe the authors' mobility across the topics. We have considered the following distributions: Power law, Truncated Power Law, Exponential, and Lognormal. The following methodology was adopted to select which distribution best fits the scholar data:

1. For each distribution, the specific parameters have been selected using the Maximum Likelihood Estimator (MLE) between the candidate distribution and the empirical data [41; 42]. Figure 6: Scatter plot of single scholars (**a**) and their average values when grouped by country (**b**). Radius of Gyration is on the \(x\)-axis. H-index is on the \(y\)-axis. Labels are associated with the corresponding number of instances. 2. To quantify the goodness of each fit, the Likelihood Ratio test [43] has been performed among each couple of distributions. As reported in the literature, this comparative test is preferable to an individual one (e.g., Kolmogorov-Smirnov test) for its robustness to noise in the data [44]. The result of this test is the Log-Likelihood ratio **R** between the two candidate distributions and a p-value. The result is considered acceptable if the \(p\)-value \(<\) 0.1. 3. Once the best distribution has been selected, we have performed an individual test for the resulting distribution with the Kolmogorov-Smirnov (or KS statistic) and bootstrapping [45] anyway. The KS returns the distance between data from the target distribution. Figure 7 presents the PDFs describing the empirical data from scholars trips and the candidate distributions, whose parameters have been fitted with the MLE. It is possible to note that the black curve related to the Lognormal distribution is the one that best approximates our data. This evidence is also shown by the Log-Likelihood ratio test, whose results are presented in Table 4 for each comparison. The Lognormal distribution parameters estimated with the MLE are \(\mu=2.967\) and \(\sigma=0.449\). The goodness of the Lognormal fit has been also evaluated for the sake of completeness with individual Kolmogorov-Smirnov tests. As presented in Table 5, the only distribution fit that is acceptable is the Lognormal one with a \(p\)-value = 0.03. This result is interesting also considering that several patterns related to the human mobility follow the same distribution [18]. Once detected that scholars' travels around research topics are well approximated by a Lognormal, we try to explain what are the implications of this result. First of all, this distribution results to be concordant with the previous results concerning the Radius of Gyration: top scholars tend to obtain better academic results by focusing on specific research areas. Lognormal distribution confirms this result and an interpretation can be resumed in this way: top scholars obtain better academic results by focusing on a specific research area but moving around within the sub topics. Most of the top scholars perform medium flight length travels each year, while it is less probable that they make important shifts in space.