g_m-index: a new mentorship index for researchers

Abstract

Advisor–advisee relationships in academic genealogy offer opportunities to understand the contribution of a mentor in shaping the research community. In this paper, we adapt the bibliometric g-index to study the mentorship role of a researcher. We call the new index $g_m$-index. It has some important differences from the mentorship h-index or the $h_m$-index. We compute the values of the $h_m$ and $g_m$ indices for researchers indexed in the Mathematics Genealogy Project and the Academic Family Tree. We observe for the majority of researchers, these index values are zero, but in non-zero cases, sometimes, the $g_m$-index can be significantly higher than the $h_m$-index. Moreover, the $g_m$-index decays less rapidly to zero than the $h_m$-index. It appears the $g_m$-index can be used to discriminate between researchers with the same $h_m$-index. We study how these mentorship indices are correlated with other indicators of academic success, and how they are correlated across generations of researchers.

Appendices

Appendix 1: Distribution of $h_m$ and $g_m$ indices in MGP

Here we analyze the general nature of the plot in Fig. 6 and its underlying reasons. By Corollary 1 in "Some properties of the g_m-index" section, the number of researchers with $h_m=1$ is always same as or greater than the number of researchers with $g_m=1$. In the present case, there are $6.5\%$ more researchers with $h_m=1$ than with $g_m=1$. In the $g_m$-index space, these researchers are distributed among the bins for higher $g_m$-index. Indeed, for index values other than one in Fig. 6, the frequency of researchers at a given $g_m$-index is always observed to be greater than that at the same $h_m$-index (unless both are zero). Let us explore this in greater detail. Denote the number of researchers having $h_m=\beta$ as $M(\beta )$ and the number of researchers having $g_m=\beta$ as $N(\beta )$. For $\beta = 0$, $M(\beta ) = N(\beta )$ by Theorem 2. For $\beta = 1$, $M(\beta ) \ge N(\beta )$ by Corollary 1. For $\beta >12$, $M(\beta ) = 0$ (from Fig. 6), therefore, $N(\beta ) > M(\beta )$ trivially wherever $N(\beta ) \ne 0$. Therefore, let us focus on $2 \le \beta \le 12$. Observe that around $40\%$ of researchers with $h_m=2$ and around $17\%$ of researchers with $h_m=1$ possess $g_m=2$. The frequency of researchers at a given $h_m$-index falls steeply with $h_m$-index. In particular, the frequency of researchers with $h_m=1$ is almost four times the number of researchers with $h_m=2$, i.e., $M(1) \approx 4M(2)$. Thus, $N(2) \ge 0.4M(2) + 0.17M(1) = 1.1 M(2)$. In other words, $N(2) > M(2)$. Similarly, when $\beta =3$, $N(3) >0.18M(3) + 0.36M(2)$. When $\beta =4$, $N(4) > 0.33M(3) + 0.16M(2)$. Since $M(2) \approx 3M(3)$ and $M(3) \approx 2M(4)$, we have $N(3) >M(3)$ and $N(4) > M(4)$. Thus, for $2 \le \beta \le 4$, $N(\beta )> M(\beta )$. To infer the ordering between $N(\beta )$ and $M(\beta )$ for $5 \le \beta \le 12$, it is sufficient to consider only the most frequent $h_m$-index among the researchers with $g_m=\beta$. Let us call the most frequently occurring $h_m$-index among researchers with $g_m=\beta$ as the largest $h_m$-contributor to $g_m=\beta$. (There can be more than one largest $h_m$-contributor to a given $g_m$-index.) As expected, the largest $h_m$-contributor typically has $h_m < \beta$. For example, the largest $h_m$-contributor to $g_m=5$ is $h_m=3$. Similarly, the largest $h_m$-contributor to $g_m=8$ is $h_m=5$. In particular, there are 747 researchers with $g_m=5$ out of which 367 have $h_m=3$, and there are 196 researchers with $g_m=8$ among whom 87 have $h_m=5$, and in both these cases, no other $h_m$-index is more common in the corresponding $g_m$-index set. If the largest $h_m$-contributor (in $g_m=\beta$) has $h_m=\beta -\delta$, and its contribution is a fraction $0 < q \le 1$ of its own population size $M(\beta -\delta )$, then clearly, $N(\beta ) > q M(\beta - \delta )$. As the size $M(\beta )$ falls fast with $\beta$ (e.g., $M(5) \approx M(3)/4$ and $M(7) \approx M(5)/8$), it turns out, for $5 \le \beta \le 12$, that $N(\beta ) > M(\beta )$ even if this largest contributor contributes only a small fraction of its own population size. For example, for $g_m=5$, the largest contributor is $h_m=3$ and it contributes $q = 0.3$ of its frequency M(3). Hence, the number of researchers with $g_m=5$ is $N(5)> q M (3) \approx 0.3 \times 4 M(5) = 1.2 M(5) > M(5)$. Due to this behavior, we find $N(\beta ) > M(\beta )$ for $5 \le \beta \le 12$. The above discussion is a quantitative analysis of the variation in the frequency of researchers with various $h_m$-index and $g_m$-index values, and how researchers with a certain $h_m$-index get distributed among various $g_m$-indices.

Appendix 2: Mathematicians with high mentorship index

Tables 2 and 3 show mathematicians who graduated in the twentieth and nineteenth centuries respectively, in order of their $g_m$-index. We restrict to mathematicians whose $h_m$-index is at least 10. The data are taken from the MGP snapshot in Rossi et al. (2017). The 1st column of each table mentions the identifier of the researcher in MGP, the 2nd column mentions the name of the researcher, the 3rd column mentions the researcher’s $g_m$-index, and the 4th column mentions the researcher’s $h_m$-index. The mentorship fecundity is shown in the 5th column and the total number of descendants in the 6th column. The year in which the researcher was awarded the doctoral degree is shown in column 7, while column 8 mentions the academic institution that awarded the degree. The 9th column, Active mentoring life, mentions the temporal span bounded by the years in which the first and the last PhDs were granted under the researcher’s supervision. The last column mentions the top-3 most prolific DDs of the researcher, i.e., the protégés who produced the highest number of PhDs. It is written in the form Name (id in MGP, number of DDs). For example, in Table 2, the most prolific DD of Jacques-Louis Lions is Roger Temam, whose MGP id is 11498 and who has 118 DDs. A very high $g_m$-index indicates the presence of very prolific DDs among the protégés of a researcher. This can help distinguish mentors with the same $h_m$-index. The highest $g_m$-index and $h_m$-index in the twentieth century are 23 and 12, respectively. They are 17 and 11, respectively in the nineteenth century. In the MGP data for pre-nineteenth century, the highest $g_m$-index is 5 and the highest $h_m$-index is 2.

Table 2 Mathematicians included in MGP who obtained their PhD in the twentieth century and have $h_m \ge 10$, ordered by their $g_m$-index

Table 3 Mathematicians included in MGP who obtained their PhD in the nineteenth century and have $h_m \ge 10$, ordered by their $g_m$-index