Fi-index

The Fi-index (or Fiorillo index, from the Greek letter φ, phi) is a bibliometric index introduced in 2022 by Italian researcher Luca Fiorillo, published in Publishing Research Quarterly (Springer). The index was designed to assess the extent to which an author's h-index has been influenced by self-citations.

Background

The h-index, proposed by Jorge E. Hirsch in 2005, is one of the most widely used bibliometric indicators for evaluating the productivity and scientific impact of researchers. However, the h-index has several well-documented limitations, including its inability to distinguish between citations received from third parties and self-citations, which can allow manipulation of the index value through strategic self-citation practices.

Over the years, numerous corrective or complementary indices to the h-index have been proposed, including the g-index, the a-index, r-index, and ar-index, the h(2)-index, the b-index for self-citation exclusion, and the q-index for detecting strategic self-citation placement. The Fi-index differs from previous approaches in that it does not merely remove self-citations but quantifies the effect of self-citations on the h-index.

Formula

The Fi-index is calculated using the following formula:

${\text{Fi-index}}=h{\text{-index}}-\left[{\frac {(100-\%{\text{selfcit}})}{100}}\times h{\text{-index}}\right]$

where:

• $h{\text{-index}}$ is the author's Hirsch index;
• $\%{\text{selfcit}}$ is the percentage of self-citations out of total citations.

The percentage of self-citations is obtained as:

$\%{\text{selfcit}}=100-{\frac {\text{citations without self-citations}}{k}}$

with $k={\frac {\text{total citations}}{100}}$.

The resulting value can range between 0 and the author's h-index value. A value of 0 indicates that self-citations have not influenced the h-index, while higher values indicate a greater influence of self-citations on the bibliometric parameter.

Subsequent developments

Fi-index tool

In 2022, the same author proposed an extension called the Fi-index tool, also published in Publishing Research Quarterly, to apply the method not to an author's entire career but to a single manuscript during the publication process. The Fi-index tool score for a given manuscript is obtained by calculating the difference between the author's Fi-index before and after the presumed publication of the article. This variant was designed to provide certification of the quality of a manuscript's reference list with respect to self-citations.

The tool provides an interpretive scale:

• 0: reference list free of self-citations;
• < 1: normal self-citing;
• 1–2: moderate self-citing;
• > 2: h-index variations and aggressive self-citing.

Fi-score

In 2024, the Fi-score was proposed as a further development, published in Publishing Research Quarterly, with a simplified formula:

${\text{Fi-score}}={\frac {h{\text{-index}}^{2}}{\text{total citations}}}\times 100$

The Fi-score evaluates the reliability of citation counts independently of self-citations, by comparing the ratio of the squared h-index to the total number of citations. The index was calculated on a sample of 194,983 researchers drawn from the database of the world's top 2% scientists published by Stanford University. Results indicate a mean Fi-score value of 25.03 with a standard deviation of 6.85; the maximum admissible value was set at approximately 32 (mean + one standard deviation).

Reception and applications

Discussion in the scientific community

The index has been the subject of analysis and discussion in the international academic community. Mechanician Michele Ciavarella analysed the Fi-score on iMechanica, an academic portal affiliated with the Harvard School of Engineering and Applied Sciences, highlighting the correlation between high Fi-score values and high self-citation percentages, as well as the compatibility of the results with the Stanford–Ioannidis ranking. Ciavarella further observed that the Fi-score constitutes a quick tool for identifying anomalous bibliometric profiles, regardless of discipline, with a calculation far simpler than the multidimensional composite indicator of the Stanford database.

Convergence with the work of Ioannidis (Stanford)

In June 2024, Evdaimon, Ioannidis et al. proposed, as part of a large-scale study on citation orchestration conducted with Scopus data, a set of metrics to detect citation manipulation patterns. Among the indicators presented, the authors adopted the ratio of total citations to the square of the h-index — corresponding to the inverse of the Fi-score introduced by Fiorillo two years earlier — as a marker of possible small-scale orchestration. The independent convergence toward a mathematically equivalent indicator was noted by Ciavarella on iMechanica, who observed that the Stanford group proposed, subsequently to Fiorillo, an index substantially analogous to the inverse of the Fi-score.

Empirical applications

The Fi-index has been applied empirically by independent researchers. A 2025 study by Saputri Nur et al., published in Khizanah al-Hikmah: Jurnal Ilmu Perpustakaan, Informasi, dan Kearsipan, employed the method to analyse the self-citation practices of the ten most productive lecturers at Halu Oleo University in Indonesia, concluding that the Fi-index values fell within normal limits for all subjects examined.

The Fi-score has also been cited in the broader context of anomalous citation detection research. Ebadulla et al. (2025) referenced the Fi-score in a study on detecting anomalous self-citations using citation network analysis and large language models, published as a Springer book chapter.

Limitations

The index has several limitations acknowledged in the literature:

• For authors with very low citation and publication counts (e.g., a single article cited once), the formula produces mathematically extreme values that are not meaningful;
• The calculation for manuscripts with many authors requires individual analysis of each co-author, making it more complex;
• The index does not detect reciprocal citation practices between different authors that do not constitute self-citation in the strict sense;
• The Fi-score can vary significantly across disciplines with different citation conventions;
• The applicability of the index depends on the availability and consistency of data in the bibliometric databases used (Scopus, Web of Science, Google Scholar);
• Ciavarella suggested that the index should not be applied except for statistical or preliminary studies, to be supplemented with in-depth qualitative assessments of suspected cases.

See also

• h-index
• Bibliometrics
• Self-citation
• Impact factor
• Scientometrics
• Citation impact

External links

• Fiorillo, Luca (2022). "Fi-Index: A New Method to Evaluate Authors Hirsch-Index Reliability". Publishing Research Quarterly. 38 (3): 465–474. doi:10.1007/s12109-022-09892-3.
• Fiorillo, Luca; Cicciù, Marco (2022). "The Use of Fi-Index Tool to Assess Per-manuscript Self-citations". Publishing Research Quarterly. 38 (4): 684–692. doi:10.1007/s12109-022-09920-2.
• Fiorillo, Luca (2024). "Detecting the Impact of Academics Self-Citations: Fi-Score". Publishing Research Quarterly. 40: 70–79. doi:10.1007/s12109-024-09976-2.