An index to quantify the disparity between coauthorship and individual’s scientific impact

Technical Summary: An Index to Quantify the Disparity Between Coauthorship and Individual's Scientific Impact

Problem Statement
Research collaboration is widely recognized as a driver of scientific advancement, often correlating with higher quality and quantity of output. However, the current academic incentive structure—relying heavily on publication counts for promotion, hiring, and grant allocation—has fostered the abuse of coauthorship. This manifests as "gift" and "guest" authorship, where individuals are listed as authors without making significant contributions. While conventional bibliometric metrics like the $h$-index and total citation counts measure productivity and impact, they fail to distinguish between single-authorship and multiauthorship, granting full credit to all listed authors regardless of their actual input. Existing attempts to address this, such as fractional counting or contribution statements, are often discipline-dependent or too time-consuming for broad application. Consequently, there is a need for a metric that can identify authors whose citation impact is disproportionately high relative to the number of coauthors on their key papers, potentially indicating inflated author lists.

Methodology
The paper proposes a new author-level metric called the Coauthorship Disparity Index (CDI). The methodology involves the following steps:

1. Data Collection: The study utilized citation data from 48 authors across diverse disciplines (Chemistry, Physics, Mathematics, Biological sciences, Biomedical engineering, and Petroleum engineering), with $h$-indices ranging from 6 to 35. Data sources included published papers and Google Scholar.
2. Definition of Impact Number ($I_h$): The author introduces an "impact number" that couples the $h$-index ($h$) with the total number of citations of the $h$-core papers ($C_h$). This is modeled via regression analysis as:
   $$I_h = \alpha h C_h^\beta$$
   where $\alpha$ and $\beta$ are constants.
3. Derivation of Expected Coauthors ($n'_{k,h}$): The study analyzes the relationship between the number of distinct coauthors on $h$-core papers ($n_{k,h}$) and the citation impact. By normalizing the number of coauthors against citation impact, the author identifies a trend where the expected number of coauthors for a given impact level can be estimated. This relationship is expressed as:
   $$n'_{k,h} = 1.73 h C_h^{0.082}$$
   Here, $n'_{k,h}$ represents the number of distinct coauthors an author should have given their specific $h$-index and citation count.
4. Calculation of CDI: The Coauthorship Disparity Index is defined as the ratio of the actual number of distinct coauthors on an author's $h$-core papers ($n_{k,h}$) to the expected number of coauthors ($n'_{k,h}$):
   $$CDI = \frac{n_{k,h}}{n'_{k,h}}$$
   A CDI of 1 implies the author's coauthorship count aligns perfectly with their citation impact. A CDI > 1 suggests the author has more coauthors than expected for their impact level, while a CDI < 1 suggests fewer.

Key Results

• Correlations: The study confirmed positive correlations between the $h$-index and both the number of citations and the number of coauthors. Similarly, a positive trend was observed between the number of coauthors and the number of citations/publications. These trends suggest that, generally, collaboration leads to higher performance output.
• Outlier Detection: While raw correlations showed no clear outliers, normalizing the number of coauthors by citation impact (as done in the CDI calculation) revealed distinct outliers. These outliers represent authors whose coauthorship counts are disproportionately high relative to their citation impact.
• CDI Thresholds: Analysis of over 500 authors suggested that a CDI below 2 indicates a "good balance" between coauthorship and citation impact. Specifically, a high $h$-index combined with a CDI of less than 1 is considered an impressive score. Conversely, authors with a CDI greater than 2 may warrant reflection on their coauthorship patterns relative to their scientific impact.
• Disciplinary Variation: The results highlighted that CDI scores vary significantly across disciplines (e.g., Physics vs. Mathematics vs. Biology), indicating that cross-disciplinary comparisons of CDI are not appropriate without normalization.

Significance and Claims
The paper claims that the CDI serves as a tool to "decipher" authors with and without outliers, thereby quantifying the disparity between coauthorship and a researcher's citation impact. The author draws an analogy between the CDI and the "Kardashian index," noting that both measure a discrepancy between a scientist's actual research impact and an external metric (coauthors in this case, Twitter followers in the Kardashian index).

The significance of the CDI lies in its ability to provide a quantitative check on coauthorship integrity. The author posits that while collaboration should be encouraged, the metric can help identify cases where the number of coauthors is inflated relative to the quality and impact of the work. The paper concludes that if databases provide the necessary data ($n_{k,h}$ and $C_h$), the CDI can be easily calculated to help ascertain the influence of coauthorship on research output.

Caveats
The author explicitly acknowledges several limitations:

• The metric does not account for "sleeping beauties" (papers that gain citations long after publication).
• It may not fairly reflect early or mid-career researchers who collaborate with large groups of senior researchers.
• Disciplinary differences in coauthorship norms mean CDI scores should not be compared across different fields.
• The metric is intended as a heuristic to prompt reflection rather than a definitive judgment on individual integrity.