## 4 Methodology
* Is there any pattern in the scholars' trajectories across topics over time? In this section, we present the different analyses that were performed and the related results to answer these questions. The first one can be related to the concept of mobility volume (Radius of Gyration), computed in the semantic space for each researcher. Moreover, some differences in the behavior of researchers working in different countries are investigated. The second one has been investigated looking for the probability density function (PDF) that could better approximate the scholars' mobility in the semantic space over the analyzed time. The analyzed dataset is composed of about 3000 researchers. For each author, information about articles, country, and H-index is collected, considering only the selected reference period (Section 4.1). We compared the semantic spaces computed with Word2 Vec and SBERT in terms of who is the most performing into revealing latent trajectories for the target knowledge discovery task. This means that we looked for a correlation between the respective Radius of Gyration computed in the semantic spaces and the H-index of the researchers. ### Correlation Analysis and Comparison of the Word Embedding Models

In general, the linear correlation between two variables \(X\) and \(Y\) is measured by the _Pearson correlation coefficient_, \(Corr(X,Y)\), which assumes a value between \(+1\) and \(-1\), where \(+1\) is total positive linear correlation, \(0\) is no linear correlation, and \(-1\) is total negative linear correlation. \[Corr(X,Y)=\frac{\Sigma(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\Sigma(x_{i}-\bar {x})^{2}\Sigma(y_{i}-\bar{y})^{2}}} \tag{2}\]

The considered variables for computing the linear correlations are:

* \(X:\) The Radius of Gyration of each researcher;
* \(Y_{1}:\) The H-index of each researcher;
* \(Y_{2}:\) The H-index of each researcher, normalized for subject. In Equation (2) is shown the Pearson correlation where each \(x_{i}\) is the Radius of Gyration of each researcher for each time unit, the \(\bar{x}\) is the average RoG and the y represents the H-Index. Since the researchers' H-index ranges are related to their research subjects, all H-index values are grouped by subject and normalized (\(Y_{2}\)) to bring all values into the range [0,1]. In this way, it is possible to make a fair comparison among researchers working in different fields. Table 2 shows the correlation values \(Corr(X,Y_{1})\) and \(Corr(X,Y_{2})\) for both Word2 Vec and SBERT. It is evident that the semantic space originates with the pre-trained transformers does not return any correlation since in both cases the correlation value is around 0. On the other hand, the semantic space originates from Word2 Vec is able to return a negative strong correlation in both cases. We further analyzed this result with a significance analysis in the next section. In light of this first result, we performed the other analysis using only the Word2 Vec semantic space. ### Significance Analysis

Statistical hypothesis testing has been used to determine if results are statistically significant. In particular, we evaluated the following significance:

* Correlation, considering the results shown in Table 2.
* Two Regression Models, considering the H-index as the dependent variable for the first regression, and the H-Index normalized for the subject for the second one. Both cases use the Radius of Gyration as the independent variable. For the correlation analysis, the \(p\)-value represents the probability that you would have found the current result if the correlation coefficient were in fact zero (null hypothesis). If this probability is lower than the conventional \(5\%\) (\(p<0.05\)), the correlation coefficient is called statistically significant. Conventionally the \(5\%\), \(1\%\) and \(0.1\%\) (\(p<0.05\), \(0.01\) and \(0.001\)) levels have been used. Most authors refer to the statistically significant as \(p<0.05\) and statistically highly significant as \(p<0.001\) (less than one in a thousand chance of being wrong).