The Compound F²-index and the Compound H-index as Extension of the f² and h-indexes from a Dynamic Perspective^①

Thanks to its simplicity, Hirsch's h-index has become a widely adopted indicator in science (Hirsch, 2005). The h-core of a dataset with an h-index of h represents the h articles that have at least h citations (Rousseau, 2006; Sanz-Casado et al., 2016). Some alternatives to the h-index have been presented, as the h²-index (Kosmulski, 2006) or the g-index (Egghe, 2006), The related h²-index corresponds to the h articles that have at least h² citations (Kosmulski, 2006). The g-index is defined as “the unique largest number such that the top g articles received together at least g² citations” (Egghe, 2006). Just as the h-index, most of those h-related indicators have a static character that do not reflect the dynamics of bibliometrics.

Despite its wide diffusion, the h-index (Hirsch 2005) has received a lot of criticism and presents serious drawbacks (Costas & Bordons, 2007; Waltman & Van Eck, 2012). The lack of differentiation and the lack of acknowledgement of highly-cited papers, have been criticized by many authoritative bibliometric scholars (Bornmann, Mutz, & Daniel, 2008; Bouyssou & Marchant, 2011; Costas & Bordons, 2007). Besides these drawbacks, the difficulty for inter-field comparison has posited the quest for normalization (Bornmann & Leydesdorff, 2018). A more recently proposed alternative, the f²-index, awards different weighted factors to the articles in function of their place in the citation distribution (Fassin, 2018). Other bibliometric scholars underlined the difficulty to grasp a complete oeuvre in a single indicator and suggested additional elements (Bornmann & Daniel, 2009).

While bibliometric citations evolve over time, indicators also vary over the years. The author's h-index continuously increases over the years, which advantages established scholars. In order to cope with this problem, some evaluation panels make use of the indicators applied to a certain time-window such as the last 10 years, or the last 5 years, or the last 2 years. But these “recent” indicators disadvantage older researchers who made impactful contribution in their field in previous years.

In this article, an extension of the f²-index to a compound F²-index is developed to integrate the dynamic aspect in bibliometric research. A similar method is applied to the h-index. An activity index is defined to evaluate the proportion of the recent contribution in a scholar's career performance.

The method is tested and illustrated with a practical application to a specific field of management research, selected for its limited size and its young but mature stage: governance. Corporate governance refers to the system of rules, practices, and processes by which a firm is directed and controlled (OECD, 1999). Corporate governance principles try to balance and align the interests of shareholders and management and other company stakeholders (Aguilera et al., 2008). From an academic perspective, the research agenda has evolved in the last decades to develop corporate governance as a real discipline. Durisin and Puzone (2009) provide a first bibliometric study in the main specialized journal Corporate Governance An International Review (CGIR). They map the intellectual structure of corporate governance research and its evolution since 1993 to 2007 and conclude to the maturation of a specific field with its distinct subfields.

A more comprehensive study of the status and evolution of a field focuses on the main research questions of exploratory nature: Who are the most influential authors in the field? Who are the scholars with increasing impact? And who are the rising stars in the field?

Classic bibliometric studies have focused on number of articles, independently of their number of citations; while many articles do not gain much references, and have little impact. Other studies re-centered their analysis around citations and thus privileges mainly the contribution of the top articles irrespective on the theme that is treated.

2.1

The gh-rating as a innovative bibliometric methodology

To execute a more qualitative view on the impact of academic journals, a more advanced analysis of the distribution of the publications can be executed. It follows a recent methodology based on the ghent-rating or gh-rating that categorizes all articles in categories in function of their citation distribution as presented in Figure 1 (Fassin, 2018).

Figure 1

The various grades (from AAA to E) depend on their relative position in the distribution of citations: all articles situated in the top 50 % (D), top 25 % (C), top 10 % (B), and top 5 % (BB). To have a better insight on the most influential articles, the top 2, 1, and 0.5% are also determined, as well as the h-core (A) and the h²-core (AAA) of the sample, and the g-core (BA) (Egghe, 2006).

The original search query for “corporate governance” in the Web of Science, restricted to the science categories business, management and ethics, leads to a selection of 3,686 articles written till 2018, by 1,235 different authors; together 43,351 citations (average is 11.8 citations per article).

Table 1 gives the number of articles within a specific percentile category, for the field of corporate governance and for the recent distribution, and also for the journal Corporate Governance An International Review (CGIR). The field has an h-index of 83 which corresponds to 2.2%, a h²-index of 16 corresponding to 0.4% while the g-index 208 lies above 5%. The h-index of the journal CGIR is 37 and its h²-index 7, which corresponds to 4.4% respectively 0.4%; its g-index is 89.

Table 2 shows the number of citations needed to be part of the percentages for the corporate governance field, the recent field and the journal CGIR; it also provides the maximum amount of citations in the dataset.

Who are the most influential researchers in corporate governance? Bibliometric studies traditionally start from two predominant methods: the list of the most productive authors in a field in Table 3—those with the most publications (excluding editorials)—and the list in Table 4 of the most influential authors—those authors who have received most citations to their publications.

Those lists illustrate some similarities but also huge differences between those rankings. Half of the top 20 in productivity appear in the top 20 of citations. 5 out of the 7 most cited authors belong to the most productive authors. But 5 of the most productive authors have not yet reached the top 20 in citations.

Another categorization following the h-index of the researchers is presented in Table 5. That list combines scholars from both previous rankings. The next Table 6 presents the ranking based on the career f²-index. In the field of corporate governance, the career f²-index of the top 50 ranked researcher reaches 6. Following the constraints, the bonus for the f²’-index is thus limited to 50% for researchers with a f²-index higher than 6, and to 3 for lower ranked authors.

Table 7 provides key data and indicators for a number of selected authors, the total citations amount, the h-, h²- and h’ 5 year -index in this dataset, the total number of articles and the repartition of their publications, according the different thresholds, number of articles within the 25%- and 10%- decile, articles in the g-core, h-core and h²-core of the corporate governance selection; further, the number of HCP—highly cited papers in the Web of Science and data on the last five recent years: the number of publications, the number of articles in the 10%-decile and in the h’-core of the recent dataset; and finally, the f², f²’ and compound F²-indexes.

The rankings according to various indicators are presented in Table 8 with the Af activity ratio.

Three researchers, Peng, Westphal, and Aguilera who were already present in most rankings lead the f²-ranking. The recent f²’-index is led by Peng, Aguilera, and Filatotchev; then follow a few active scholars as Zattoni or newcomers as Van Essen. Some researchers as Daily did not publish in the last 5 years.

The dynamic aspect is illustrated with the compound F²-index, that adds the effect of promising recent publications. Aguilera and Filatotchev follow Peng and surpass Westphal and Zajac who have been less productive in recent years. A few younger active scholars like Zattoni catch up with Dalton and Daily and others as van Essen make progress. The activity index highlights the contribution of younger researchers as Van Essen.

The comparison between the career f² and the compound F²-classifications shows the evolution and the progress of the individual researchers compared to their peers. The data reveal the future trends. Even if the top 10 remains stable (the f²-classification advantages the well-known researchers), some ranks vary; especially the writers of HCP (less than 10 years old) and with a high recent f²’-index improve their position.

So, the F²-compound index illustrates the dynamism of a research field. It recognizes past achievements but also values new research work with potential. Those scholars who did not publish much in the last five years continue to benefit from their past publications, but will not substantially increase the gh-rating of their older publications (especially in categories A to AAA), while those researchers who published more in the last years benefit temporarily from the recent f²’-index and will gradually increase the gh-rating of their upcoming publications that will contribute more to the future f²-count.

The quantum approach used in this bibliometric method is quite selective and works at a double level. First, it allows an incremental climb in the rating of the individual articles, and secondly, it also grants an incremental rise of the number of articles that are taken into account, as these are limited by the scholar's h²-index in this dataset. As a result, the f²-methodology will select mainly papers within the h-core or g-core in the selected dataset (A and BA-category), and to a lower extend those in the 10% or 25%-percentiles (B and C-category). Younger scholars will start with a few papers in the C or D-category, as their h²-index will be ranged between 2 and 3. To increase the number of papers to be taken into account necessitates to increase the h²-index; which signifies that a higher number of papers need to reach the higher classification, and means that papers in the lower category will not be taken into consideration anymore. To increase the h²-index from 6 to 7, needs an increase of citations of 6 papers from 36 to 49 and an additional 7^th paper with 49 citations, which will generally imply that all papers are within the 10%-percentile or minimum B-category. In the compound F²-classification, this quantum mechanism works at both the level of the f²-index and the recent the f²’-index.

The compound F²-index just as all f²-indexes need a careful calculation as the Web of Science nor Scopus does not yet provide the percentile categories. It is, therefore, worthwhile to search for a simplified alternative. Following similar reasoning, one could define a compound h-index, that better takes into account the recent contribution of academic articles. Indeed, the h-index is ever increasing over time, and thus privileges senior researchers. The career h-index benefits from an increasing number of publications and time further feeds higher citations for all publications, so that additional articles can reach the h-core and further increase the researcher's h-index. Partial time-restricted h-indexes over the last 10 years (or 5 years of 2 years) on the contrary disadvantage senior researchers who do not publish anymore, while they privilege younger active authors.

If h is the career h-index of the scholar, and h’ their recent h’-index over the last 5 years, the compound H-index is defined as the sum of the career h-index and the recent h’-index (4)H=h+h'{\rm{H}} = {\rm{h}}+{\rm{h}'}

The younger scholar whose main cited publications are published in the last 5 years can double their h-index, as all articles are taken into account, with equal weight. The senior professor who is not active in the last years will remain with their h-index; their compound H-index will be equal to their h-index.

The present study permits to illustrate the advantages of a new bibliometric methodology with a practical application. This implementation of the f²-index to this particular field illustrates the prudent selectivity and discriminative power of the method. The classification following the f²-index mitigates between the classical ranking following the number of papers and the ranking following the total number of citations of those papers, and also between the h-index the h²-index rankings. Thanks to its quantum approach, the f²-methodology increases selectivity and turns out one-time contributions. The results presented in the rankings or classification in tiers show more consistency than other rankings.

With its extension of the f²-index to the compound F²-index that integrates the recent contribution and the application to form the compound H-index, the paper presents an innovation in bibliometrics as a tool to examine dynamism in citation analysis. Especially, the difference in ranking between the F² and f²-ranking offers a view of the dynamic evolution in the field at study. The classification on the basis of the compound F²-index can be applied to estimate the future f²-classification and to identify the rising stars. The study contributes to developing new methods to generalize the h-index to a more comprehensive indicator. Especially the activity index offers an interesting complement to the h-index as a good estimate that highlights the contribution of a researcher's recent work. These new indicators illustrate the underestimated dynamic capacity of bibliometric research.