## 3 Influence Metrics
### Effectiveness Analysis of the Proposed Metrics
#### 4.2.3 Correlation Analysis between Local and Global Metrics  
Our goal is to study the probable association between the ranking of global influence and the ranking of local impact through time and to address the consequences of anticipating global influence by looking at the consistency and predictability of human social qualities. A statistical metric used to quantify the strength of the association between the relative changes of two variables is the correlation coefficient. It includes values between \(-1\) and \(1\). A complete negative correlation is represented by a correlation coefficient of \(-1\), and a perfect positive correlation is represented by a correlation coefficient of \(1\). The absence of a linear link between the changes in the two variables is shown by a correlation value of \(0\). A statistical metric used to quantify the degree of linear association between two variables is the Pearson correlation coefficient. This coefficient reveals the nature and strength of the relationship, given that a change in one variable leads to a proportionate change in the other. When there is no apparent association, the Pearson coefficient returns a value of \(0\). Another correlation statistic used to assess rank correlation, which indicates the statistical dependence between the ranks of two variables, is Spearman's rank correlation coefficient. The Spearman coefficient measures the extent to which the relationship between variables can be characterized by a monotonic function, in contrast to a linear relationship where the rate of increase or decrease is constant. Without necessarily adhering to a consistent rate of change, it assesses how well the data align monotonically. The Spearman correlation evaluates the monotonic relationship between two variables, whether they are continuous or ordinal in nature, by considering the ranked values of each variable rather than the raw data. In a monotonic relationship, one of the following is true:  
* As one variable increases, the value of the other variable decreases; or
* Conversely, as one variable increases, the value of the other variable also increases. The methods employed by the two correlation coefficients are fundamentally distinct. While the Spearman coefficient considers both linear and monotonic correlations, the Pearson coefficient focuses exclusively on linear relationships between variables. Furthermore, Spearman utilizes rank-ordered variables, whereas Pearson employs the raw data values of the variables. It is advisable to employ the Spearman coefficient instead of the Pearson coefficient when a scatterplot reveals a potential link that could be either monotonic or linear. Using the Spearman coefficient does not cause any harm, even if the data eventually demonstrate a perfect linear relationship. Nevertheless, choosing Pearson's coefficient might lead to missing crucial insights that Spearman could provide in cases where the connection is not  
Figure 8: Illustration of Complementary Cumulative Distribution Function of proposed global centrality metrics. exactly linear. Therefore, as illustrated in Figure 9, we utilize the Spearman correlation coefficient to analyze the relationship between the ranks of Closeness and Degree throughout their dynamics. By establishing a consistent relative order of observations within each variable (e.g., first, second, third, and so on), it is intuitively understandable that the Spearman correlation between two variables becomes strong when observations possess similar (or identical, resulting in a correlation of 1) ranks. Conversely, the correlation is low when observations exhibit disparate ranks across the two variables (or entirely opposite rankings, leading to a correlation of \(-1\)). Let \((x_{1},y_{1})\), \((x_{2},y_{2})\),..., \((x_{n},y_{n})\) represent a collection of composite rankings from two distinct ranking lists, \(X\) and \(Y\). The \(n\) raw scores (\(x_{i}\), \(y_{i}\) are converted to ranks (R(\(x_{i}\)), R(\(y_{i}\))), and \(r_{s}\) is derived as follows:  
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