Ilyas Ahmad¹, Abdul Shahid¹, Sana Ullah Khan^1*,  Tariq Hussain², Ahmad Ali Alzubi³,*, Oleksandr Korochkin⁴

¹        Institute of Computing, Kohat University of Science and Technology (KUST), Kohat , 26000, Pakistan; ilyasphd@kust.edu.pk ; ashahid@kust.edu.pk ; sana.ullah@kust.edu.pk

²        School of Computer Science and Technology, Zhejiang Gongshang University, Hangzhou 310018, China; uom.tariq@gmail.com

³        Department of Computer Science, Community College, King Saud University , Riyadh, Saudi Arabia; aalzubi@ksu.edu.sa

⁴        Faculty of Informatics and Software Engineering, National Technical University ,  of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Kyiv , Ukraine; korochkin.oleksandr@lll.kpi.ua

^*        Correspondence: aalzubi@ksu.edu.sa;

Abstract

The evaluation of authors based on their contributions to the scientific community is typically performed through both qualitative and quantitative assessments. Qualitative evaluations depend on subjective assessment and lack uniform criteria, whereas quantitative methods have established more than 50 parameters like publication count, citation count, and variants of the h-index. However, existing author ranking parameters fall short of determining the best parameter that agrees with expert qualitative evaluations. To thoroughly analyze the behavior of each index, it is essential to evaluate these indices using a large dataset. This study focuses on evaluating the h-index, along with several of its variants and extensions divided into seven distinct categories, for their effectiveness in ranking. The evaluation is conducted using a large dataset in the field of Mathematics. The dataset used in this study comprises data from 1,050 individuals, equally distributed between awardees and non-awardees. This balanced composition shall provide for an impartial assessment of the indices concerned. The dataset contains a total of 204,896 publications and 14,370,007 citations. Such a large-scale and diverse collection of data provides a robust foundation for assessing the effectiveness of the h-index and its variants in ranking, particularly within the field of Mathematics. A two-step approach was adopted to evaluate the results obtained from the indices. First, the occurrences of awardees were evaluated across all ranked lists generated by the indices. This step involved identifying the presence of prominent awardees within the top-ranked authors for each index. The analysis was intended to assess the effectiveness of each index in identifying top-performing authors who have received significant recognition in their respective fields. The analysis was carried out in several steps. First, the presence of awardees within the top-ranked authors for each index was identified to assess its performance. The best performing index in each category was selected. Next, a Random Forest approach was applied to calculate the importance scores of the indices, further identifying the top-performing index for each category. The top indices identified by both methods were analyzed using Spearman correlation to measure their similarity. The two indices with the highest correlation values were then selected for further analysis. Finally, various statistical methods were employed to combine these indices and develop a novel author ranking index, providing a more comprehensive evaluation framework.

Keywords: h-index, g-index, awardees, Machine Learning

Received: 15 May 2025; Revised: 18 Jun 2025; Accepted: 23 Jun 2025; Published: 10 July, 2025

1. Introduction

The rapid growth of research output worldwide has led to an overwhelming number of publications being produced each day. However, the actual impact and quality of this research may take time to become clear, as many esteemed authors may wait years for their contributions to receive full recognition. An author’s impact can be measured in several ways; however, different scientific organizations have different criteria they use. Unfortunately, there is no universally accepted way to acknowledge and commend individuals for their outstanding achievements. The literature shows three primary approaches for assessing an author’s contributions. The first approach relies on peer evaluation, wherein experts in the field assess the study’s impact on their expertise and judgment. This approach is informal and subjective because of the absence of defined guidelines; yet, expert opinions enhance it. The second approach entails citation analysis, using computerized technologies to monitor the frequency of an author’s citations, so providing a more objective evaluation and reducing human error. A third approach assesses the importance of an author’s achievements by examining prestigious awards conferred by prestigious scientific organizations, using these awards as a standard of excellence and influence in their field[1].

The scientific community has developed a plethora of quantitative metrics to measure the influence of authors and their contributions after the Hirsch index became widely used. Ever since 2005, when the h-index was first announced [2], Various variants of  h-index,  like g-index [3], A-index [4], w-index [5], m-index [6] and f-index [7] have been introduced. Though novel approaches keep popping up, the majority of them have not yet been conventionally proven. It is difficult to comprehensively assess the effectiveness of these quantitative indicators because each new approach is tested on different datasets. Currently, there are no clear standards in the literature on which indices should be used or minimum requirements that determine an author’s influence within their field. Consequently, explicit guidelines are needed to assist those in the early stages of their careers in achieving recognition as significant contributors [8]. Furthermore, there is a lack of consensus on the criteria for nominating individuals for recognized awards. Recent empirical research has examined h-index and its variants to understand their influence on awardees from international scientific societies. However, only a limited number of parameters and datasets have been considered in this study [9]. These insights could help authors improve their chances of receiving grants and provide scientific societies with more precise criteria for evaluating the quantitative impact of scholarly work [10].

We consolidated the dataset curated by Bilal et al[11] as the foundational source for our analysis. Our research uses a dataset of 1050 individuals (525 awardees and 525 non-awardees) spanning from 1990-2023. Awardee information was obtained from leading mathematical societies that formally recognize and honor outstanding contributions to the discipline. These societies include the International Mathematical Union (IMU), the London Mathematical Society (LMS), and the American Mathematical Society (AMS).

The present study makes the following key contributions:

• Conducted a comprehensive evaluation of author assessment parameters using a well-established mathematics dataset.
• Categorized 64 indices into seven different categories, ensuring a structured evaluation.
• Proposed a novel index derived from the best-performing statistical model.

Our analysis delineates the most significant features correlated with award recipients in discipline, offering vital insights for the scientific community in the allocation of research awards. These recommendations may also be advantageous for early career seeking to elevate their profiles. The subsequent sections of this work are organized as follows: The “Literature Review” section encapsulates current ranking criteria. The Methodology section outlines the systematic approach adopted for author ranking and the formulation of quantitative standards. The Evaluation section presents a detailed assessment of the effectiveness of the proposed index, examining its performance across relevant benchmarks. The Results and Discussion section summarizes the findings and interprets them in light of current research, highlighting points of convergence and divergence. The Conclusion section distills the core insights derived from the study, underscoring its contributions to the field. Lastly, the Future Work section explores prospective avenues for advancing the methodology and enhancing the robustness of the evaluation framework.

• Literature Review

The practice of ranking authors was initially introduced to assist students in selecting the most suitable supervisors for their Ph.D. research [12]. Liang and Jiang assert that an author rankings are crucial in multiple decision-making processes, including promotions, research opportunities, award nominations, service role allocations, peer-reviewer identification, and professional contract awards [13]. Jiang et al. observed that the ranking of  authors has been recognized as a significant concern among the scientific community [14]. Different measurements, like  publications , citation count, collaborative authorship, and integrated approaches like the h-index, have been employed to evaluate authors [8].

Standard ranking approaches frequently depend on fundamental arithmetic parameters, like the overall count of publications or citations. In 2005, Sidiropoulos carried out a thorough study into these parameters [15]. However, these indicators are not generally relevant and offer issues when comparing different  types of  authors, such as those who publish frequently or those who focus on fewer, high-impact research [16]. The number of published papers and their citation counts are two well-known indices [2]. Bibliometric indicators serve to statistically assess an individual’s productivity, the quality of their publications, and their influence, along with the relationships between various universities and organizations [7, 17]. However, these indices have also faced criticism. Each index has unique pros and cons, as addressed by Bogers and Van den Bosch [17].

A significant issue with publication count is that the quantity of publications does not accurately represent the importance or impact of research. It just denotes the quantity of work [18]. An author with a higher volume of publications may be regarded exceptionally prolific [19], However, publication count alone fails to present a comprehensive overview, as certain publications may appear in substandard venues. Moreover, citation counts can be deceptive, as they may be distorted by extensively review research papers or augmented through self-citation [20]. Furthermore, variations in citations can be quite pronounced among different fields, complicating the process of making across fields comparisons [21]. Smolinsky and Lercher emphasized the differences in citation patterns across mathematics, pointing out that certain subfields gain from associations with well-cited areas, whereas others are less frequently cited [22]. Behrens and Luksch also noted that citation indices in mathematics frequently neglect numerous subfields, privileging only a limited selection [23].

The h-index was proposed by Hirsch in 2005 to address the shortcomings of publication and citation counts. The h-index’s straightforwardness and utility have made it favoured throughout the scientific community, as it succinctly represents an author’s output and impact through a single index [2]. The h-index provides a more comprehensive evaluation of scholarly output by simultaneously accounting for both the total number of publications and associated citation counts. However, it remains subject to several methodological constraints. For instance, the h-index may be disproportionately affected by a small number of highly cited papers, particularly in cases where citation counts do not align with the overall volume of publications [24]. Furthermore, the h-index lacks discriminatory capacity between authors who share identical values, as it does not reflect differences in individual citation distributions. The use of self-citation further complicates such comparisons, potentially inflating h-index values and obscuring true scholarly impact[18]. Calculating an h-index has its own set of issues because it necessitates a meticulous and precise compilation of an author’s work. Furthermore, the h-index generally disadvantages early-career academics, who typically have fewer publications and citations than their more experienced competitors.

Teixeira and Mota feel that the h-index allows scientists to utilise their previous successes and accolades, as it can rise owing to citations even when there are no new publications [25]. The h-index may be advantageous for comparing the contributions of top scientists; however, it is inadequate for evaluating the contributions of average individuals in the field. To address the limitations of h-index, many other methods have been developed to offer a more thorough evaluation of an individual’s scientific work [26]. Sidiropoulos et al. identified multiple inefficiencies in the conventional h-index system, leading them to propose  extensions: the Standardised h-index, Trend h-index, and Contemporary h-index [1]. The alternative versions seek to reveal more profound insights within various citation networks and serve as valuable tools for evaluating publications in journals and conferences. The findings indicate that these extensions provide enhanced performance relative to the original h-index. Bornmann et al. assessed h-index along with its thirty-seven variants, analysing the interrelationships among them [26]. Despite these modifications, the h-index possesses intrinsic limitations, primarily attributable to its dependence on data over time.

Wu stated in 2010 that the average h-index is four times greater than the overall w-index, it was designed as a more significant index for evaluating the impact of an author, especially for highly cited publications [5]. The w-index is utilised for assessing, research groups, organizations, and scientists. Aoun et al. conducted an evaluation of indices that assess authors’ scientific productivity, highlighting the h-index’s robustness against self-citation biases. However, they also acknowledged that the h-index might place junior  at a disadvantage because it relies on the length of an individual’s academic career [16]. Mazloumian et al. in 2013 introduced a network-based index aimed at assessing scientific productivity, enabling an extensive global assessment of scholarly knowledge shaped by geographic influences [27]. Fukuzawa evaluated the distribution of patented research publications and citations, estimating their relationship through a sample of more than 4,000 works from prominent authors in Japan, uncovering a U-shaped correlation [28].

Ayaz et al. performed an evaluation of h-index and its variants, like g-index and complete-h, employing mathematical techniques on data collected from award-winning mathematicians [29]. The findings suggested that the complete-h index not only enhances the h-index but also incorporates a community aspect, surpassing both the g-index and h-index in ranking awardees on elite lists. The literature review indicates that several often employ the same dataset while adjusting different parameters. For example, Schreiber, two studies by Liu and Rousseau and Liu et al. evaluated same dataset [30-32]. On the other hand, Bornmann et al.’s research  used various datasets but concentrated on comparable parameters [33]. These indices seek to evaluate publishing productivity and identify prominent ; however, they frequently depend on restricted datasets and lack standardized evaluation benchmarks [8].

• Methodology

The comprehensive evaluation of the current advancements in author ranking has encouraged us to investigate various scholarly indices for evaluation. To provide a clear picture of our approach, Figure 1presents overall framework of the proposed methodology. This framework serves as a roadmap for our study, highlighting the interconnected features that contribute to our evaluations. Each segment of the methodology will be elaborated upon in the subsequent section, where we will detail the specific processes and approaches employed.

Figure 1:Struture of Methodology

• Domain Selection

The dataset was enlarged by Bilal et al. with the acquisition of current awardee data till 2023. Table 1 presents an extensive overview of the dataset’s statistics. They visited several society websites in the previous three decades of mathematics to collect information on awardees, compiling names and matching award years. They used the Publish or Perish platform to obtain an author data, employing a “hold on” strategy that enabled us to get data prior to the years in which they received their awards. We adopt a novel approach by examining a substantial dataset in the mathematical domain to evaluate different categories of indices. Since mathematics is fundamental to science and plays a key role in it, our group finds it to be an interesting subject to study. By focusing on this vital area, we hope to improve understanding of h-index and its variants This will demonstrate their value and importance in assessing scientific output in a vital field of research.

Table 1: Dataset Description

Descripton	Number
Total no.of authors	1050
Total no.of authors(awardees)	525
Total no.of authors(non-awardees)	525
Total no.of publications	204,896
Total no.of citations	14,370,007

• Search Engine

This study utilised a Google Scholar database to gather research papers, citations, and co-author networks pertinent to specific fields within the field of mathematics. Web of Science (WoS), Scopus, and other databases provide author-related information; however, they do not consistently offer the same degree of coverage or accessibility as Google Scholar [29]. As a result, it was decided to collect mathematics dataset from Google Scholar due to its free availability and vast coverage in the mathematics field. Google Scholar is favoured for the availability of publicly accessible full-text publications, a quality that many other databases lack [34]. Numerous studies have advocated for Google Scholar due to its comprehensive data coverage and ongoing enhancements in the accessibility of publications and citation metrics. Its substantial expansion within the mathematical domain has further elevated its utility for research evaluation and bibliometric analyses[35]. Comparative evaluations between Google Scholar and the Web of Science indicate that Google Scholar demonstrates superior dynamism, with weekly updates that ensure the provision of the most current and relevant data[36].

Google Scholar effectively identifies top-ranked, highly cited publications relevant to targeted search queries, rendering it a valuable resource for bibliometric research. Accordingly, it served as the primary data source for this study. A dedicated web crawler was developed to interface with Google Scholar using domain-specific categorization phrases tailored to mathematics. This automated approach facilitated the extraction of essential bibliometric information, including publication titles, author names, publication years, total citation counts, publication URLs, and publishing venues.

Table 2: Major Awards Conferred by Leading Mathematical Societies

Society	Award Name	Description
International Mathematical Union (IMU)	Fields Medal	Honors significant mathematical contributions and demonstrates commitment to supporting future advancements.
	Carl Friedrich Gauss Prize	Honors mathematical research with impact outside mathematics.
	IMU Abacus Medal	Recognized for exceptional achievements in the mathematical facets of information sciences.
	Leelavati Prize	Recognizes contributions to increasing public awareness of mathematics.
London Mathematical Society (LMS)	De Morgan Medal	Premier award for outstanding contributions to mathematics.
	Pólya Prize	Recognizes outstanding creativity or exposition in mathematics.
	Fröhlich Prize	Awarded for original and innovative mathematical work.
	Shephard Prize	Given for mathematical contributions with strong intuitive or visual components.
	Senior Berwick Prize	Awarded biennially for outstanding research published by LMS.
	Senior Whitehead Prize	Recognizes influential work and service to mathematics in the UK.
	Senior Anne Bennett Prize	Awarded for outstanding contributions to mathematics, particularly in the UK.
	Naylor Prize and Lectureship	Recognizes work in applied mathematics and mathematical lecturing.
	Berwick Prize	Given for outstanding mathematical research published by LMS.
	Whitehead Prize	Recognizes significant mathematical contributions, particularly in the UK.
	Anne Bennett Prize	Recognizes outstanding contributions to mathematics in the UK.
	Hirst Prize and Lectureship	Awarded for contributions to the history of mathematics.
American Mathematical Society (AMS)	Leroy P. Steele Prizes	Recognize outstanding research contributions in mathematics.
	Bôcher Memorial Prize	Given for notable research in analysis.
	Cole Prizes	Recognize outstanding contributions to algebra and number theory.
	Veblen Prize	Awarded for notable research in geometry or topology.
	Chevalley Prize	Recognizes notable work in Lie Theory.
	Fulkerson Prize	Given for outstanding papers in discrete mathematics.
	Moore Prize	Recognizes outstanding research in algebra or number theory.
	Satter Prize	Recognizes an outstanding mathematical contribution by a woman.
	Doob Prize	Awarded for an outstanding research book making a seminal contribution.
	Conant Prize	Given for an outstanding expository paper published by AMS.

• Crawling metadata of authors

A specialized web crawler was developed to systematically extract relevant data from Google Scholar with the objective of constructing a comprehensive dataset of mathematical publications. Initially, the crawler retrieved papers linked to each key term entered into Google Scholar and extracted pertinent bibliographic details. However, preliminary results revealed that several query phrases yielded outputs unrelated to the field of mathematics. To address this issue, a team of domain experts was convened to refine and validate the set of search terms, ensuring their disciplinary relevance. With expert input, a carefully curated list of categories and subcategories was constructed to optimize the precision of data retrieval. These revised subcategories were embedded into the PHP-based crawler, which was programmed to extract essential metadata, including publication title, author name(s), journal or conference name, citation count, year of publication, and the corresponding paper URL. To maintain dataset integrity and prevent noise from marginally relevant entries, the crawler was configured to collect only the top 600 results for each search query. This conclusion was derived from actual evidence gathered from human evaluations of random requests, which indicated that results beyond the 600th entry often contained records irrelevant to the original query. Consequently, restricting the results to the top 600 records produced a more concentrated and pertinent dataset for evaluation.

Table 3: Categories wise indices

Category	Indices
Complement of h-index	h-index,g-index, h(2)-index, w-index, h.core.citation, m-index , t-index , Gh-index, rm index ,weighted h-index, Woginger index, x-index, V-index,. A-index ,R-index, h-dash index, hi-index
Based on Publication Age	Cite.Year, Total.Years, m-quotient.index, Ar-index, AWCR
Based on Total Number of Authors	normalized.h-index, fractional.g-index, fractional.h-index, HI-index, hc-index, ha-index,Author.papers,cites.author,paper.author,hf.index,hi.index,pure.h.index,P.index,hi,index
Combination of Two or More Indices	hm_index, gf.index, gF.index, hg_index, q2.index, h_m_index, AW_index, gm.index, w.norm.index
Based on Excess Citation Count	e.index, pi.index, Total.Citation, Cite.Paper, wu.index
Based on Total Number of Publications	Total.Publication, i10.index, platinium_h_index, tappered_h_index
Based on Other Variants	f.index, Maxprod, h2upper_index, h2.center_index, h2.lower_index, k.index, Rational.h.index, Real.h.index, k.norm.index, k.dash.index

• Data preprocessing

After data collection, it is imperative to verify and cleanse the data to guarantee its accuracy and utility. The data analysis and cleansing phase focused on two primary aspects. Evaluating the obtained results’ applicability to the field of mathematics was the initial step. Results with invalid characters in the title were first removed. Domain experts then confirmed the journal and conference names that were taken from every publication. The second step of the cleaning procedure focused on finding and removing duplicate author entries and clearing up any confusion caused by similar an author names. Duplicate cases were identified when different authors shared the same first and last names. This study required a comprehensive human verification process for each an author to ensure that the dataset was fine-tuned, which meant reviewing their profiles on their respective home pages. This rigorous approach verified adherence to the requisite standards for further study by verifying the accuracy and integrity of the collected data.

• Taxonomy Building

We obtained data from Bilal et al. [11] and applied various parameters for ranking to evaluate the academic achievement of authors in mathematics. This domain encompasses various branches, one of which is categorized within Mathematics Subject Classification (MSC) system. The most recent version of MSC2010 consists of 64 primary categories, with 45 pertaining to pure mathematics and 19 to applied mathematics. This classification system is employed by prestigious mathematics journals, including the Journal of the American Mathematical Society, Mathematics of Computation, and Conformal Geometry and Dynamics, to organize scientific papers.

We employed a variety of sources, including MathSciNet, ZbMath, WoS, and Google Scholar, to collect data regarding the research activities of authors, including publications, citations, and co-author networks. Nevertheless, the other sources, with the exception of Google Scholar, encounter obstacles in terms of open access and restricted coverage. Although alternatives such as Scopus and WoS exist, Google Scholar’s coverage surpasses that of WoS by approximately 13%, as evidenced by comparisons. Google Scholar offers a more comprehensive selection of indexed publications [34]. The open-access feature, particularly in light of the fact that numerous other databases are not publicly accessible and WoS has its own constraints, further enhances its value. According to recent study, expansion of Google Scholar is approximately 13% greater than that of WoS [37].

• Benchmark dataset

To evaluate the different ranking indices, the lists of award recipients from multiple prestigious mathematical associations. The Table 2 include globally recognised awards granted by esteemed organisations like the American Mathematical Society (AMS), the International Mathematical Union (IMU), and the London Mathematical Society (LMS).The main reason for choosing authors acknowledged by these awarding organisations is the lack of other standards for assessing such research. These societies are highly esteemed in the field of mathematics, and their awards possess considerable significance internationally. The awards function as a reliable indicator of academic distinction and influence within the mathematical community.

• Evaluation
  • Standard Ranking

We start the simple ranking method by classifying the 64 indices into seven separate categories as shown in Table 3. For each category, we generate a ranked list of authors based on the index values of all indices in that category. The number of awardees appearing within the top 50 records of each ranked list, as generated by the respective indices, is then quantified. This step serves to illustrate the relative effectiveness of each index in identifying distinguished authors within its corresponding evaluative category. Subsequently, the top-performing index from each category is selected based on its ability to capture the greatest number of awardees within the top 50 positions. This guarantees that the chosen indices are representative of its categories. Algorithm 1 demonstrates the process of selecting the top-performing index from each of the seven categories based on their capacity to rank the highest number of awardees.

• Machine learning based ranking

Feature ranking in machine learning is vital, significantly assisting in dimensionality reduction, enhancing interpretability, reducing overfitting, improving predictions, and enabling feature selection. This study utilises a Random Forest classifier in conjunction with the Boruta feature selection approach to ascertain the relevance scores of features. Random Forest is a robust ensemble learning technique that constructs a multitude of decision trees during the training phase. For classification tasks, it aggregates the predictions of individual trees and determines the final output based on the mode (i.e., majority vote) of these predictions. This method enhances model stability and accuracy by reducing variance and mitigating the risk of overfitting commonly associated with single decision trees. This model excels at managing high-dimensional datasets and offers intrinsic feature importance indices. To alleviate overfitting, we apply measures such restricting tree depth and utilising an appropriate number of trees to ensure stability and generalization. The approach employs bootstrap sampling, utilises the Gini impurity criterion for decision splitting, and incorporates an early stopping technique to mitigate overfitting, hence maintaining its generalization capability.

The Boruta feature selection approach is an effective wrapper-based methodology that operates in conjunction with Random Forest to identify the most important features in a dataset. Boruta produces shadow features, which are randomly permuted reproductions of the original features, and assesses the relevance of true features relative to their shadow equivalents. Features that exceed shadow features are considered significant whilst those that do not are deemed irrelevant and eliminated. Through recurrent comparisons, Boruta guarantees a solid and dependable selection of features. This study involved the initial division of the dataset into two subsets in an 80:20 ratio, allocating 80% for training and 20% for testing. The training set was divided into 80% for model training and 20% for validation purposes. The allocation of instances among these sets was equitable, guaranteeing uniform representation of various classes. The Random Forest classifier was trained for classification using the training dataset. The trained model was then shown a validation and test sample, and it was able to predict class label for each sample. With all features included, accuracy obtained during this prediction phase served as baseline accuracy.

Boruta determines feature importance scores by calculating Z-scores for each feature, quantifying their contribution to the model’s performance in relation to their shadow equivalents. The traits with the highest Z-scores are recognised as the most significant. The equation for the Feature Importance Score (FIS) can be written in a more readable form as:

Where,BLA_i is the baseline accuracy for the ith phase.

WOPA_i is the accuracy without the parameter during the ith phase.

This equation computes the average difference between the baseline accuracy and the accuracy excluding a particular parameter over five epochs, multiplied by 100 to represent it as a percentage.

The Boruta algorithm produces two lists upon analysing all features: one with feature names and another with their respective relevance scores. The parameters are ranked according to these scores. The most significant parameter from each of the seven different categories, determined by their importance scores, is chosen for further research.

• Correlation between indices

The primary goal of evaluating the correlation between these ranked lists is to ascertain the nature of their linkages and the extent of their resemblance.

Algorithm 1: Author Ranking (Count number of awardees)

Index values ← Data_for_ An author_labels ← Label_ (Awardees Non-Awardees) Index-Values_List ← [] Label-List ← [] Awardee counts ← [] Index names ← [] x ← 1 for x < 64     for y from 1 to total_number_of_         Index-Values List.append(Index values[y])         Label-List.append(Author labels[y])     end for     Index-Values_List, Label-List ← Sort(Index-Values_List, Label-List) in descending order     c ← 0     for z from 1 to 50 do         if Reserchers labels[z] == 1 then             c ← c + 1         else             Continue,         end if     end     Index-names.append(Index-values[0])     Awardee counts.append(count)     x ← x + 1 end for Index-names, Awardee-counts ← Sort(Index-names, Awardee-counts) in descending order Return

To achieve this, we applied the Spearman correlation coefficient, as described by Corder and Foreman. The formal formulation of the Spearman correlation is presented in Equation 2 below.

                                                          (2)

Where             r_s  = Spearman’s Correlation Coefficient

                              d_i  = difference between two indices,             n = number of iterations

We employed the Spearman rank correlation coefficient, a non-parametric statistical measure utilized to evaluate the strength and direction of the association between two variables. Specifically, it assesses how effectively the relationship between the variables can be described by a monotonic function, without assuming linearity or normally distributed data. This method is particularly suitable for ordinal data and scenarios where the integrity of ranked relationships is of analytical interest. It is a measure of rank concordance, and can be used to compare multiple indices and assess the concordance or agreement between authors. Using Spearman correlation helps to better comprehend similarities between the ranked lists of rank values within the perspective of differentiating between the indices. The degree of similarity between the ranked lists of indices is found in this part by means of their relationship. This work aims to determine the correlation coefficients between every ranked list in respect to the other lists as well as itself. We computed the correlation values across the different sorted lists in the present study, therefore generating three possible scenarios about their relationships: Positive correlation means that the ranking lists relate to one another. Negative correlation indicates a weak or inverse relationship between the lists. Zero correlation suggests no meaningful relationship between the ranking lists.

• Mathematical models

The results of our correlation evaluation enabled us to select and prioritise relationships with the highest correlation coefficients. A strong correlation between these pairings signifies a substantial association between their respective attributes in the awardee records. This finding indicates each index encompasses complimentary facets of the dataset,  impacting awardee judgements. Utilizing this knowledge, we may leverage the interrelated attributes of these variables to create a novel index that amalgamates their strengths, so enhancing our comprehension of underlying data. To accomplish it, we employed mathematical models to derive the values of these relationships. Mathematical methods are crucial for data analysis in diverse domains. In order to obtain significant insights analyse data in a methodical manner, find patterns, quantify relationships, and draw conclusions based on the findings; this study used advanced statistical methods.

We systematically incorporated the top-ranked pairs utilising various mathematical methods, including the Weighted Arithmetic Mean (WAM), Geometric-Harmonic Mean (GHM), Symmetric Mean (SymM), Log-Euclidean Mean (LEM), Root Mean Square (RMS), Contra-Harmonic Mean (CHM), Hölder Mean (p-Norm Mean), Logarithmic Weighted Mean (LWM), Lehmer Weighted Mean (LWM), and Jensen Mean (JM). These strategies provide an extensive perspective on author ranks and facilitate our analysis of their significance within the dataset. The comprehensive computations for these methods are outlined below.

4.4.1.     Mathematical Means and Their Formulas

4.4.1.1.Weighted Arithmetic Mean (WAM)

The WAM is a modification of the Arithmetic Mean (AM) that employs weights to signify the relative significance of various values within a dataset. It is particularly advantageous when disparate values contribute unevenly to the overall mean.

Formula:

                                                                        (3)

where  x_i are the data points,
w_i are the corresponding weights,
 for equal weights( w=1)

                                                            (4)

4.4.1.2.Geometric-Harmonic Mean (GHM)

The GHM) is a composite mean that integrates Geometric Mean (GM) and Harmonic Mean (HM). It achieves equilibrium between these two methods, rendering it advantageous in contexts where both proportional and reciprocal interactions are essential.

Formula:

                                                             (5)

The mathematical representations of GM and HM are shown below:-

                                                          (6)

                                                              (7)

4.4.1.3.Symmetric Mean (SymM)

The SymM is a statistical measure that balances the AM and HM while maintaining symmetry in its computation. It ensures the evaluation of both direct and inverse relationships among variables, making it advantageous for citation analysis, ranking systems, and financial modelling.

Formula:

                                                        (8)

4.4.1.4.Log-Euclidean Mean (LEM)

The LEM is a statistical measure that use logarithmic transformation to compute a mean that is more robust against scale variations. It is particularly advantageous in situations when data follows a multiplicative distribution rather than an additive one, as shown in citation analysis, signal processing, and machine learning.

Formula:

                                            (9)

4.4.1.5.Contra-Harmonic Mean (CHM)

The CHM is a mean that prioritises larger values, this makes it particularly advantageous in scenarios where higher values carry greater significance. It is defined as the ratio of the sum of the squares of the values to the sum of the values.

Formula:

                                                         (10)

4.4.1.6.Hölder Mean (p-Norm Mean)

The p-Norm Mean integrates arithmetic, geometric, and harmonic means. This is particularly advantageous when numerous weights are necessary for various values. The Hölder Mean is affected by the parameter p, which dictates the importance of higher or lower values.

Formula:

                                     (11)

4.4.1.7.Logarithmic Weighted Mean (LWM)

The LWM is a statistical average that utilizes logarithmic transformations to equilibrate the impact of varying values. It is especially advantageous for data that encompasses numerous orders of magnitude, as it mitigates the influence of outliers relative to arithmetic means.

Formula:

                                                    (12)

4.4.1.8 Root Square Mean (RMS)

The Root Mean Square (RMS) is a statistical measure that computes the square root of the average of the squares of a set of variables. It is especially handy when both positive and negative numbers exist and we want to determine the total magnitude regardless of sign. RMS offers more weight to larger values due to the squaring process, making it suited for datasets where large values are more important.

Formula:

                                                (13)

4.4.1.9  Lehmer Weighted Mean (LeWM)

The LeWM is a mean that assigns more weight to larger values, making it beneficial in circumstances where higher values should have a greater impact on the conclusion. It is defined as the ratio of the sum of weighted squares to the total of the original values, multiplied by their weights.

Formula:

                                                         (14)

4.4.1.10Jensen Mean (JM)

The JM is derived from Jensen’s inequality, a key notion in convex analysis and probability theory. It describes how to generalise means using convex or concave functions. The Jensen Mean is especially important when a function’s curvature impacts value aggregation, making it crucial in statistics, information theory, and citation indices.

Formula:

                                                         (15)

In addition, comparable methods were used to analyse the top-ranked pairs. Ten different lists were generated by calculating the comparable values for each pair, with each list representing a specific methodology. The lists were evaluated to identify most impactful statistical method for top-ranked pair. The study resulted in the identification of a viable model for efficiently merging the qualities of the top-ranked pair.

• Result and Discussion

This section provides a detailed discussion of our research.

5.1       Awardees in top ranking list

In this section, we examined how different indices contribute to identify awards at the top of the ranking list and how awardees are distributed among the top-ranking categories. To answer this question, we first examined the distribution of awards across each parameter in the top 1-50 of the generated ranked list. The first step enabled us to determine which indices were most efficient in positioning awards at the highest ranks.

Figure 2: Complement of h-index(% of Awardees)

Next, we performed a more comprehensive analysis by categorizing the awards based on their position on the overall ranked list. We specifically looked at how many awardees at top 50 ranked list. The findings of this analysis indicate the distribution of awards across several ranking lists. These evaluates provide critical insights into each category’s success in promoting awardee recognition, as well as the overall number of awardees in the top ranks of the ranking system. The evaluation of indices across different categories reveals different trends in how academic indices relate to an author awards. The Complement of h-index category expands traditional h-index to include extra dimensions of academic influence, such as an author productivity and citation effect, as shown in Figure 2. The indices, like ‘h.index’, ‘g.index’, and ‘weighted.h.index’, have a wide range of values, with ‘weighted.h.index’ and ‘woginger_index’ having the highest values. This variance emphasises the significance of strengthening the basic h-index statistic to more accurately represent an author’s effect, meaning that awards frequently recognize minor differences in the h-index that indicate more impact.

Figure 3: Based on Publication Age(%  of Awardees)

Figure 4: Based on total number  of authors(% of Awardees)

Figure 3 depicts the indices specified by Publication Age, which include ‘Cite.Year’, ‘Total.Years’,’m_qoutient_index’, and ‘Ar.index’. They analyse the duration and consistency of citations over time. These indices show moderate to high values, emphasizing that rewards tend to preferentially recognise individuals with long-term contributions to their professions. This demonstrates the importance of analysing an author’s effect across time, with a focus on career-long impact as a critical aspect in award consideration. In the Total Number of author’s category, indices such as ‘normalized_h_index’, ‘fractional.g.index’, and ‘P.index’ evaluate individual contributions in collaborative efforts as shown in Figure 4.  Numerous indices have higher values, particularly the ‘P.index’, indicating that awards are more likely to recognise measures that reflect individual successes in collaborative contexts. This emphasis reflects a preference for recognising authors who contribute significantly when working in larger groups.

Figure 5: Combination of two or more indices (% of Awardees)

The Combination of Two or More Indices category includes indices like ‘hm_index’ and ‘gf.index’, which combine indices to create a comprehensive view of an author’s effect as shown in Figure 5. Values in this category are generally moderate, with fewer peaks than in other categories.

This suggests that, while aggregated measurements provide a comprehensive picture, they may miss specific features that are highly valued in the reward process, maybe due to dilution of different influential aspects. As shown in Figure 6 higher values for Excess Citation Count, such as ‘e.index’, ‘Total.Citation’, and ‘Cite.Paper’, suggest a strong relationship between citation excess and rewards. Increased values for ‘Cite.Paper’ indicate that measures showing surplus citations over the h-core threshold, emphasizing the relevance of long-term effect in academia.

Figure 6: Based on Excess Citation (% of Awardees)

In the  Based on Total Number of Publications  category, indices such as `Total.Publication` and `i10.index` solely measure production volume, leading to relatively low outcomes as shown in Figure 7. This tendency suggests that although quantity is important, awards emphasize the quality and impact of contributions over mere productivity. The  Other Variants  category includes unique indices, such as `f.index` and `Maxprod`, which demonstrate moderate to high values for certain indices like `k.index`as shown in Figure 8. This balanced presence suggests that while certain alternative indices represent novel aspects of impact and are valued, they may not always hold the same importance as traditional or widely acknowledged indices.

Figure 7: Based on total number of publication(% of Awardees)

Overall, rewards tend to go to indices that place a premium on a mix of factors, like number of citations, the strength of collaborations, the impact on careers over time, and the unique contributions made inside academic networks. Putting an emphasis on long-term, high-quality impact rather than simply publishing a lot demonstrates that the author has a holistic perspective. The chosen indices represent the best-performing indices from each category of h-index variants, ensuring a thorough assessment of scholarly impact. The woginger_index was recognised as the top performer index in the Complement of h-index category. The m_qoutient_index emerged as the top index in the category Based on Publication Age, whereas the normalized_h_index was selected for the category Based on Total Number of Authors. In the Combination of Two or More Indices category, the w-norm index outperformed all others. Similarly, the Cite.Paper index was the most important statistic in the category of Excess Citation Count, whereas the platinium_h_index led the category of Total Number of Publications. Finally, the h2 lower_index was chosen from the “Based on Other Variants” category.

Figure 8: Based on other Variants (% of Awardees)

These indices provide a fair representation of the most significant measures across various dimensions of author evaluation. Below is a brief description of each selected index, as well as its mathematical form:

5.1.1.   Woginger-Index

This index improves the traditional h-index by incorporating excess citations in a weighted manner, ensuring that highly cited papers contribute proportionally to an author’s impact. It aims to fix the shortcomings of the h-index, which fails to consider citations over the threshold. The following is the mathematical representation:

                                                 (16)

where C_i denotes the citations of the i-th paper, and h signifies the h-index.

 5.1.2.  m-Quotient Index

The m-Quotient index evaluates the sustainability of an author’s impact over time by normalizing the h-index, which is achieved by dividing it by the number of years after an author’s first publication. It is especially beneficial for comparing at various career stages. The mathematical representation of this index is as follows:

                                                       (17)

where h represents the h-index and n is the number of years from the auhtor’s first publication.

5.1.3.   Normalized h-Index

This index modifies the h-index to account for co-authorship impacts by changing it according to the average number of authors per publication. It distinguishes between individual and collaborative contributions, providing a more equitable assessment of scholarly effect. The mathematical representation of this index is as follows:

                                               (18)

where h denotes the h-index and A represents the average number of authors per publication.

5.1.4.   w-norm Index

A composite index that amalgamates various citation-based indices by allocating weights to publications according to their relative significance. This index offers a more refined assessment by accounting for disparities in citation impact instead of treating all papers uniformly. The formula for computing the index is as follows:

                                                  (19)

where w_i is a weight assigned to each paper based on its significance.

  5.1.5. Cite.Paper Index

This index quantifies an author’s mean citation influence per paper. This statistic is straightforward yet efficacious for identifying authors whose papers consistently garner attention within the academic community, balancing productivity and impact. The index is mathematically represented as follows:

                                                           (20)

where C is the total number of citations and P is the total number of publications.

5.1.6.   Platinum h-Index

A modified version of the h-index that considers the total number of publications in its computation. It provides a more fair assessment of an author’s impact, taking into account both the number of highly cited publications and overall publishing productivity. The index is mathematically defined as follows:

                                         (21)

where h represents the h-index and P is the total number of publications

5.1.7.   h² Lower Index

This index calculates the square root of an author’s h-index to determine the lower bound of their impact. It focuses on the minimal number of citations that an author’s most significant publications have received, making it beneficial for evaluating authors with highly consistent but not necessarily high citation counts. The index can be expressed with the following formula:

                                            (22)

where h represents the h-index.

5.2       Ranking with random forest

In this section, we grouped the parameters by their relevance scores, which were computed using a Random Forest classifier and a brute-force feature selection technique, as explained in the methods section. The significance score of a parameter is an attempt to assess its overall contribution to the model’s ability to correctly discriminate between awardees and non-awardees. This technique allowed a thorough assessment of how each parameter affects classification accuracy.The order of all factors is determined by their importance scores.

The analysis revealed that among the ranked parameters, some demonstrated significantly higher importance scores, reflecting their stronger influence on the classification process. Building upon this, we adopted a systematic approach to select seven indices, each representing the top-performing index from the distinct categories. The categories and their respective selected indices are as follows: hi-index from the Complement of h-index category as shown in Figure 9, m_qoutient_index from the Based on Publication Age category in Figure 10, normalized_h_index from the Based on Total Number of Authors category in Figure 11, gf-index from the Combination of Two or More Indices category as shown in Figure 12, pi-index from the Based on Excess Citation Count category as illustrated in Figure 13, platinum_h_index from the Based on Total Number of Publications category as shown in Figure 14, and Maxprod-index from the Based on Other Variants category as shown in Figure 15. These indices provide a comprehensive and balanced evaluation framework for assessing academic impact by displaying the top-performing indicators in their respective categories. The integration of Random Forest-based ranking and category-specific selection guarantees that the selected indices are both statistically significant and pertinent to various aspects of an author performance. This method shows that evaluating authors is multidimensional and that these measures are very important for correctly finding and ranking scholarly contributions. There were two different ways to rank the candidates simple ranking and ranking with Random Forest. Notably, three indices were used by both methods, which shows that they both work well for judging the effect of an author. The m-Quotient Index looks at the age of publications, the Normalised h-Index looks at the total number of authors, and Platinum h-Index looks at total number of publications.  Here is a short explanation of each selected index, along with its mathematical form:

5.2.1.   hi-Index

The hi-index is an addition to the h-index that looks at the extra citation distribution beyond the h-core. It improves the h-index by adding more citation information, which makes it a better reflection of an author’s overall scholarly importance.

                                                     (23)

Where h denotes h-index and Avga is the average number of authors per publication.

Figure 9: Complement of h-index (important score)

Figure 10: Based on Publication age (important score)

5.2.2.   Gf-Index

The Gf-index is a composite statistic that synthesizes two distinct indices to equilibrate productivity and citation impact. It is designed to provide a comprehensive evaluation of an author’s influence by employing various citation-based metrics. This index is calculated using the following formula:

                                             (24)

Y(i) shows the citation count, while φ(i) is the average number of authors per publication.

Figure 11: Based on total number of Authors (important score)

Figure 12: Combination of two or more indices (important score)

5.2.3.   pi-index

The Gf-index is a composite statistic that combines two distinct indices to equilibrate productivity and citation effect. It aims to provide a comprehensive assessment of author influence through citation-based indices. This index is calculated using the following formula:

                                                         (25)

where C represents the total number of citations and h is the h-index.

Figure 13: Based on Excess Citation ( important score)

5.2.4.   Maxprod-index

The Maxprod-index emphasises an author’s optimal citation-productivity potential by maximizing the product of citation counts and publication numbers, hence assuring proper representation of both aspects. The proper representation for this index is:

                                                 (26)

In this case, Ci represents i-th paper’s citation count and Pi stands for the period’s total number of publications.

5.3       Correlation among the selected Indices

The correlation matrix heatmap clearly depicts the Pearson correlation coefficients among different author ranking indicators as shown in Figure 16. Each cell denotes the magnitude and orientation of the association between two indices, with values spanning from -1 (strong negative correlation) to +1 (strong positive correlation). The diagonal of the matrix indicates perfect self-correlation (1.00), but the off-diagonal entries reveal the degree of correlation among the indices. The colour gradient emphasises these links, with red hues representing strong positive correlations, blue hues indicating strong negative correlations, and intermediate hues reflecting weak or no correlation. The heatmap shows that the w-norm index and hi-index have the strongest correlation (0.91), indicating a significant overlap in the aspects of an author impact that they assess.

Figure 14: Based on Total publication ( important score)

Figure 15: Based on other Variants (important score)

Figure 16: Correlation among selected top indices

This strong association suggests potential effectiveness or redundancy in their techniques, analyzing the components that contribute to this relationship may provide important insights into its design and use in an author evaluation.

5.4       Formulation of index

This section focused on identifying the indices with the highest correlation, specifically the w-norm index and the hi-index, across ten different statistical models. The primary goal was to identify the statistical model that best captures and preserves shared features of this highly correlated pair. This method provided substantial insights into the interaction between this pair and the statistical methods designs, improving our understanding of their collective behaviour and impact. For this particular pair of indices, values of ten statistical methods were calculated.

Figure 17: Comparison of Statistical Methods for Author Ranking

The data were systematically sorted by method following the computation of relevant values to establish a clear ranking of results. Subsequently, the top 50 records for each method were evaluated to assess their award-identification capabilities with high precision. The effectiveness and reliability of each statistical approach were determined by the number of awards identified within these top-ranked entries. The findings shed light on the comparative performance of various statistical methods when applied to the most strongly correlated parameters. As illustrated in Figure 17, among the ten statistical methods examined, the Lehmer Weighted Mean (LWM) yielded the highest value of 17.65. Building on this insight, the hwi-Index an author impact index is proposed as a novel ranking index. It leverages the Lehmer Weighted Mean to integrate the strengths of multiple, highly correlated citation-based indices.

This technique is selected for its statistical robustness, preserving the common characteristics of the most interconnected indices while ensuring equitable contribution from each. Notably, the Lehmer Weighted Mean (LWM) yielded a higher number of awards within the top 50 records, reinforcing its reliability in author ranking. Accordingly, the derivation of the hwi-Index from this method ensures its empirical validation and transparency by anchoring it to statistically the most effective approach.

                                        (27)

• Conclusion

This study evaluates the challenges associated with evaluating authors within the scientific community by comprehensively investigating an extensive range of features and providing a novel author ranking index. Despite several efforts in the literature, with over seventy parameters proposed to evaluate academic influence through both quantitative and qualitative dimensions, no single index has yet achieved universal acceptance that can be used for ranking. The ongoing debate underlines the difficulty of establishing an effective and impartial evaluation methodology.

To address this challenge, we employ a systematic methodology that classifies an author ranking indices into seven different categories. Our methodology utilizes two complementing methodologies: (1) a straightforward ranking strategy that orders authors according to calculated index values, and (2) the Random Forest technique, which discerns the most significant factors in assessing an author effect. By utilizing these dual techniques, we want to establish a rigorous and careful evaluation framework. Once the rankings are generated, we use a correlation analysis to determine which indices have the strongest relationships. From this study, we select the top two indices with the highest correlation, ensuring that the selected indices capture a consistent and meaningful scholarly ranking pattern. This phase is critical in improving the index formulation by focusing on parameters with significant interdependence and predictive potential.

To improve the dependability of our proposed index, we rigorously evaluated the selected indices using 10 different statistical methods, each of which provided a new perspective on combining and interpreting the selected indices. Our statistical methods guarantee a theoretically grounded integration of the selected indices, eliminating biases and improving the accuracy of our final ranking index. Based on this systematic approach, we present the hwi- index, a new composite index that combines the top two strongly linked indices using the most effective statistical method discovered throughout our search. This formula provides a balanced and representative ranking index, taking into account both individual strengths and interdependence. While the proposed index contributes to improving author ranking approaches, some limitations remain. First, the hwi- index is currently validated in the mathematics domain, and its application to other fields may necessitate domain-specific recalibrations. Future research should look into how this approach can be used in different fields.

Second, our validation method concentrates on authors who have earned prominent awards, which, while a standard benchmark, may leave out extremely influential authors who have not been formally recognized. The inclusion and generalizability of the ranking index could be improved by including other validation criteria. Our study not only contributes to ongoing discussions about author ranking, but it also provides a scalable, data-driven framework for future research. We propose a systematic methodology that might inform and modify future developments in academic impact evaluation.

The research community has proposed several other indices in addition to those previously mentioned. It is vital to assess these indices utilizing extensive datasets to reveal their genuine efficacy and relevance. Our forthcoming research intends to evaluate these alternative indexes using an extensive dataset that encompasses other fields of study. Furthermore, we advocate for the development of a uniform dataset that delineate awards according to standardized criteria. For example, two separate standards could be established: one targeting juniors and the other seniors. By analyzing the performance of these parameters between the two groups, we can acquire a more profound understanding of how various indices operate at different career stages. This comparison analysis will elucidate the strengths and shortcomings of each parameter, thereby enhancing the understanding of an author assessment across fields.

Competing Interests: The author declares no competing interests.

Acknowledgments: This work was supported by the Ongoing Research Funding program, (ORF-2025-395), King Saud University, Riyadh, Saudi Arabia.

Availability of Data and Materials: NA

Ethical Approval: Not applicable.

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