Does the hα-index reinforce the Matthew effect in science? The introduction of agent-based simulations into scientometrics

Abstract Recently, Hirsch (2019a) proposed a new variant of the h-index called the hα-index. The hα-index was criticized by Leydesdorff, Bornmann, and Opthof (2019). One of their most important points is that the index reinforces the Matthew effect in science. The Matthew effect was defined by Merton (1968) as follows: “the Matthew effect consists in the accruing of greater increments of recognition for particular scientific contributions to scientists of considerable repute and the withholding of such recognition from scientists who have not yet made their mark” (p. 58). We follow up on the point about the Matthew effect in the current study by using a recently developed Stata command (h_index) and R package (hindex), which can be used to simulate h-index and hα-index applications in research evaluation. The user can investigate under which conditions hα reinforces the Matthew effect. The results of our study confirm what Leydesdorff et al. (2019) expected: The hα-index reinforces the Matthew effect. This effect can be intensified if strategic behavior of the publishing scientists and cumulative advantage effects are additionally considered in the simulation.


INTRODUCTION
The h-index, introduced by Hirsch (2005), is one of the most popular bibliometric indicators worldwide. The paper by Hirsch (2005) has been cited more than 3,500 times (date of search in Web of Science (WoS), Clarivate Analytics: March 2019). The h-index has been adopted as one among other indicators in WoS and Scopus (Elsevier). In the bibliometrics literature, however, many critical points have been raised about it: For example, Waltman and van Eck (2012) argued that "for the purpose of measuring the overall scientific impact of a scientist (or some other unit of analysis), the h-index behaves in a counterintuitive way. In certain cases, the mechanism used by the h-index to aggregate publication and citation statistics into a single number leads to inconsistencies in the way in which scientists are ranked" (p. 406). Furthermore, the counting of papers with at least h citations has not been justified by Hirsch (2005); it is equally possible to count papers with at least h 2 citations or h/2 (Egghe, 2006a,b).
Since the introduction of the h-index, many variants have been proposed, targeting one or several disadvantages of the h-index. Bornmann, Mutz, Hug, and Daniel (2011)  concluded, on the basis of a metaevaluation, that most of these variants correlate highly: "depending on the model, the mean correlation coefficient varies between .8 and .9. This means that there is redundancy between most of the h index variants and the h index" (p. 346). Recently, Hirsch (2019a) himself proposed a new variant called the h α -index: "we define the h α index of a scientist as the number of papers in the h-core of the scientist (i.e. the set of papers that contribute to the h-index of the scientist) where this scientist is the α-author" (p. 673). The α-author is defined "as the author of the paper with the highest h-index among all the coauthors" (p. 673). Hirsch (2019a) recommended using the new index in combination with the h-index. The author formulated as follows: "a high h index in conjunction with a high h α ∕h ratio is a hallmark of scientific leadership" (p. 673).
The h α index was criticized by Leydesdorff et al. (2019). One of their most important points of criticism is that the index "adds the normative element of reinforcing the Matthew effect in science" (p. 1163). The Matthew effect was defined by Merton (1968) as follows: "the Matthew effect consists in the accruing of greater increments of recognition for particular scientific contributions to scientists of considerable repute and the withholding of such recognition from scientists who have not yet made their mark" (p. 58). Merton (1968) cited a physicist as follows: "The world is peculiar in this matter of how it gives credit. It tends to give the credit to (already) famous people" (p. 57). The Matthew effect is similar to Price's (1976) "cumulative advantages," which he noted as a core mechanism in the sciences explaining, among other things, the skewed distributions in the indicator values. Barabási (2002) reinvented Price's cumulative advantages and Merton's Matthew effect as "preferential attachment" without any knowledge of or reference to this bibliometric literature.
In his rejoinder, Hirsch (2019b) partly denied that the h α -index reinforces the Matthew effect in science: "Strictly speaking at most half of this is true, the higher h-index author in a collaboration benefits, however the lower h-index author does not get negatively affected, his/her h α remains the same. More importantly, lower h-index authors have the choice to not collaborate with high h-index authors but rather pursue their own independent work, or work with more junior collaborators" (p. 1168).
We agree with Hirsch (2019b) that authors or co-authors with low h-index values cannot be expected to become "poorer," because nothing is taken away from them. The first problematic point in his statement is, however, the implicit demand to search strategically for cooperation in science. According to the norms in the ethos of science (Merton, 1942(Merton, , 1973, cooperating partners should be selected based on the quality of their research or the fit to the needed expertise for a certain research project, but not for nonscientific reasons such as the increase of indicator values. Supervisor-supervised relationships may be another reason for co-authorship. The second problematic point in Hirsch's statement above is the remaining uncertainty about the reinforcement of the Matthew effect by using h α in research evaluations. In this study, we address this point. We use a recently developed Stata command (h_index) and R package (hindex) for the simulation of the h-index and h α -index. Based on simulation results the user can empirically investigate whether h α reinforces the Matthew effect or not.

The Role of Simulations in Scientometrics
Albeit that simulations are not in the focus of the bibliometric literature, both bibliometrics and simulation studies have been used as methods in quantitative science and technology studies (e.g., Ahrweiler, 2001;Edmonds, Gilbert, Ahrweiler, & Scharnhorst, 2011;Scharnhorst, Börner, & van den Besselaar, 2012). Gilbert (1997) set the stage with the first simulations of the structure and dynamics of academic science. He introduced "kenes" as knowledgevariants of genes; the resulting events showed Lotka-type distributions and were interpretable using Simon's (1957) models of social processes. Gilbert (2004, 2011) developed a large innovation model called SKIN: "Simulating Knowledge Dynamics in Innovation Networks." Different from data-oriented studies, simulations enable us to theorize mechanisms and to specify expectations. Not only observable behavior but also coordination and selection mechanisms can be studied. Leydesdorff and van den Besselaar (1998), for example, showed that the Cobb-Douglas production function can be elaborated into a representation of technological trajectories and technological regimes by assuming feedback mechanisms (Leydesdorff & van den Besselaar, 1994). In a similar vein, one can simulate lock-ins and deadlocks in technological innovation (Leydesdorff, 2001;Leydesdorff & van den Besselaar, 1998) and synergy in Triple-Helix models (Ivanova & Leydesdorff, 2014). In the confrontation with data, the insights in mechanisms can be developed into what Bornmann and Marewski (2019) further elaborated into bibliometrics-based heuristics (BBH: see section 2.3).
In a recent study, Backs, Günther, and Stummer (2019) used agent-based modeling as a decision support system when planning measures to encourage academic patenting within universities. The authors suggest "the application of agent-based modeling and simulation, an approach that has been successfully used in other, similar, contexts (e.g., when selecting useful measures for market introduction and diffusion of new products). We have presented herein an agent-based model that is suitable for this purpose, and we have demonstrated its applicability and its potential value for practice [i.e., TTO (technology transfer offices) management drives increased patenting] and subsequently for society (i.e., more academic patents lead to an increase in knowledge transfer between universities and industry and/or provide a basis for spin-off companies) by means of an application example" (p. 454). You, Han, and Hadzibeganovic (2016) used an agent-based simulation model to assess how the impact of scientists' work efficiency and their capability to select important topics for their research affects the h-index (and other measures). In this simulation model, the agents (authors or research teams) try to occupy nodes in a citation network (publications). By providing the citation network a priori, the simulations focus on the process of competing for possible publications, rather than the collaboration or the citation process. The model proposed by You et al. (2016) is an example of how the influence of individuals' actions on macro-level patterns can be analyzed by means of simulations in scientometrics.
We are aware of only a few simulation studies in scientometrics that focus on the h-index. These simulations-as a rule-have dealt with the development of single h-index values without considering collaborations between scientists. Lobet (2016) published an h-index evolution simulator which reveals the development of single h-index values based on various inputs (e.g., starting year of publishing, papers per year). The simulator is able to consider certain behaviors of researchers, for example, to always cite their own papers. Guns and Rousseau (2009) investigated the h-index's growth based on computer simulations of publication and citation processes. They found that "in most simulations the h-index grows linearly in time.
Only occasionally does an S-shape occur, while in our simulations a concave increase is very rare" (p. 410). Ionescu and Chopard (2013) published two agent-based models which refer to performance measurements of single scientists and a group of scientists (see also Żogała-Siudem, Siudem, Cena, & Gagolewski, 2016). They studied, for example, what happens when low h-index researchers are removed from a community. Their results suggest "a stratified structure of the scientific community, in which the lower h levels mostly cite papers from the upper h levels" (p. 426).

Analytical Sociology
This study follows the approach of analytical sociology, which focuses on the mechanisms leading to social phenomena (Hedström, 2005;Hedström & Ylikoski, 2010). Epstein (2006) formulated what he called a "generativist" research program as a manifesto, stating that one cannot explain a social phenomenon until one has "grown" it by simulating the phenomena under study as emerging from the bottom up. It is the goal of analytical sociology to work out the mechanisms (on the micro level) that are the causes of the phenomena (on the macro level; Bornmann, 2010). In this study, we are interested whether the phenomenon Matthew effect can be produced by the mechanism "h α -index." In the simulations, action is an operation at the micro-level, because action (publishing, being cited, collaborating, and performance measuring) is done by a single agent. The possible outcome is on the macro-level-structures in the form of certain h α -index distributions. In order to test the relationship between the mechanism and the phenomenon in this study, several agent-based simulations have been performed using the Stata h_index command. Most of the model parameters are held constant across the simulations. Compared to a baseline simulation, only a single parameter is changed in each of the other three simulations in order to inspect the effect of this parameter. The interested reader of this paper can use the command or package to investigate the effects of further parameter changes.

Bibliometrics-Based Heuristics
The h_index command and the hindex package can be used to define rules for running various simulations. For example, we work with certain distributions of h-index values as starting points and define how the agents in the simulation interact. The simulations are used to obtain an experimental view on the effects of the h α -index use in research evaluation.
Recently, Bornmann and Marewski (2019) introduced BBHs. They discussed the use of bibliometrics in research evaluations against the backdrop of the fast-and-frugal heuristics research program (e.g., Gigerenzer, Todd, & ABC Research Group, 1999). Heuristics are "models for making inferences that (a) rely heavily on core human capacities (such as recognition, recall, or imitation); (b) do not necessarily use all available information and process the information they use by simple computations (such as lexicographic rules or aspiration levels); and (c) are easy to understand, apply, and explain" (Katsikopoulos, 2011, p. 10). In the fast and frugal heuristics framework, "a model of a heuristic specifies (i) process rules such as search, stopping, and decision rules; (ii) the kinds of problems the heuristic can solve, that is, the structures of environments in which it is successful; and (iii) the capacities that the heuristic exploits" (Marewski, Schooler, & Gigerenzer, 2010, p. 77). The application of heuristics in many environments, for instance business, medicine, sports, and crime (Gigerenzer & Gaissmaier, 2011), has shown that they come to judgments of similar quality compared with more complex decision strategies. Bornmann and Marewski (2019) defined BBHs as decision strategies in research evaluation which ignore much data, but use limited information about an entity (i.e., citation and publication data of a researcher) to assess the entity. By following the fast-and-frugal heuristics research program, Bornmann and Marewski (2019) defined for the use of BBHs some search, stopping, and decision rules. These rules help to formulate and apply BBHs for a certain research evaluation environment. For example, the rules can be defined as follows: In economics, publications in top-five journals (American Economic Review, Econometrica, Journal of Political Economy, Quarterly Journal of Economics, and Review of Economic Studies) decide about scientific careers (Bornmann, Butz, & Wohlrabe, 2018); reaching a professorship without having published in these journals is frequently not possible. The search, stopping, and decision rules for filling a professorship can be defined as follows: (a) search for all publications of a group of candidates (economists); (b) stop search when all publications have been identified; (c) select the candidate with the largest number of papers in the top-five journals.
Although this BBH can be expected to exist in economics, it is not clear whether it is a favorable BBH leading to valid and reliable suggestions for decisions. One objective of the fast-and-frugal heuristics research program is to identify heuristics used in practice; anothermore important-objective is to develop heuristics leading to reliable and valid decisions. Further questions in this program are the following: Is the application of bibliometric indicators in a given environment reasonable? Does the indicator's use lead to nondesired effects?
In this paper, we follow therefore the program's approach by studying the possible advantages and disadvantages of the use of the h α -index in research evaluation. We especially focus on the assumed sensitivity of the h α -index for the Matthew effect.

IMPLEMENTATION OF OUR SIMULATION MODEL IN STATA AND R
The ado h_index and the hindex package simulate agents who collaborate on publishing papers. In Stata, type net install h_index, from(https://raw.githubusercontent.com/chrgan/h_index/master/) to install the ado. 1 The R package hindex is available on CRAN and can be installed by typing install.packages("hindex"). 2 The simulation procedure is as follows: 1. As a starting point of the simulation, n agents are generated. The user can specify n, the number of agents. The agents have published in the past. The user can choose between a Poisson or negative binomial distribution for the number of previously published papers and set parameters of the distribution (the expected value for both distributions, and an additional dispersion parameter for the negative binomial distribution specifying a factor by which the variance exceeds the expected value). The choice of these distributions reflects that the number of papers is a discrete variable and allows modeling different distributional forms. By using a negative binomial function and setting a low expected value together with a high dispersion, one obtains a distribution with many agents who have published only a few papers and few agents having published many papers; this may roughly approximate a power law (Newman, 2001b). (Note that our approach allows for agents who have not yet published.) By specifying a higher expected value and a lower dispersion, it is possible to approximate a normal distribution as well. It is assumed in the simulation that each paper has been written one to five periods ago (imagine years, for example). For a share of these papers, the agent is the alpha author. The user can specify the expected value of this share, which is the same for all agents. The initial papers have already received citations in the past. The number of citations depends on (a) the citation distribution and (b) the age of a paper: (a) The user can choose between a Poisson and a negative binomial distribution (parameters are specifiable as above). He or she can specify the maximum expected number of citations. (b) The expected number depends on the papers' age following a log-logistic function. It first increases with time (periods), reaches the maximum specified in (a) after a configurable number of periods, and then decreases. The steepness of the log-logistic function can also be specified. Thus, for each given age of the papers, the number of citations follows the distribution specified in (a) with an expected citation number given by its maximum and the age of the paper.
2. For the initial set of n agents, the h-index and h α -index are calculated. 3. Then, the agents start to collaborate according to the following routine. The user can specify how many periods the agents collaborate. In each period, the agents form teams publishing new papers (each agent can publish or co-author at most one paper in each period). The user can set some properties: the average number of co-authors, the share of agents who collaborate or publish in each period, and the correlation between the probability of being active in the current period and the h-index values calculated in step 2 3 . Thus, one can specify that agents with high initial h-index values are more productive than agents with low initial h index values. By default, the collaborating/ publishing agents are assigned to (co-)authorships at random. For example, if the user sets the number of agents to 100 and the average number of co-authors to 10, 100/10 = 10 groups of collaborating authors are created by assigning a random integer from the interval [1, 10] to each agent. Agents with the same number are treated as one group. It is possible to specify that agents with high h-index values avoid co-authorships with agents who have equal or higher h-index values. In this case, the agents with high h-index values strategically select co-authors to improve their h α . Suppose again there are 10 groups of agents because the desired average group size is 10 and there are 100 agents. Then, the 10 agents with the highest h-index values are assigned to groups 1 to 10 before performing the aforementioned random assignment for the rest of the agents. This assures that the agents with the highest h-index values become members of groups where all other agents have equal or lower h-index values.
3 If the user specifies the correlation, an auxiliary variable av is calculated as av where c is the desired correlation, μ(h 0 ) is the mean of the initial h-index and σ (h 0 ) its standard deviation (therefore, the term h0 − μ h0 ð Þ σ h0 ð Þ is the z-standardized distribution of h 0 ), and rnorm is a standard normal random variable. Only those agents with the s% highest values on this auxiliary variable engage in publishing, where s is the desired share of publishing agents as specified by the user. 4. All papers can receive citations each period. The number of citations is determined by the procedure outlined in step 1. To reflect the possibility of self-citations, the user can specify an option leading to one additional citation for each paper (published at least one period ago) where at least one of its authors has an h-index value which exceeds the number of previous citations of the paper by one or two. This reflects agents strategically citing their own papers that have citations just below their h-index value. This accelerates the growth of the agents' h-index values. Finally, a "boost" effect can be specified: Papers of agents with higher h-index values are cited more frequently than papers of agents with lower h-index values. The citations of a paper are increased by a user-specified number for every additional h point of the agent with the highest h-index among the paper's co-authors, rounded to the next integer. For example, suppose a single paper where the highest h-index of its agents is 11. If one specifies the boost effect to be 0.5, this paper receives additional round (11×0.5) = 6 citations. 5. For each period, the new h-index and h α -index values are calculated. The alpha author of a paper can be determined at the time of its publication (without changing later on) or the alpha author of a paper is determined after each period of action based on the current h-index values of the authors (see Tietze, Galam, & Hofmann, 2019). 6. To ensure the robustness of the results, steps 1 to 5 are repeated r times. Figure 1 visualizes the single steps of the simulation procedure.

RESULTS
The Matthew effect implies that the more reputable scientist receives more credit than the less reputable scientist for a scientific contribution, although the contribution is of the same scientific quality. Thus, the credit is not attributed fairly on the basis of the performed contribution, but (unfairly) on the basis of previous contributions. If we compare this definition of the Matthew effect with the definition of the h α -index, the similarities are obviously observable. In case of the h α -index, the credit for a paper is assigned to the co-author with the highest h-index. Although all authors conributed to the co-authored paper in question, only one author receives the full credit. Furthermore, the credit is assigned to the co-author who is most reputable in terms of h-index values. These similarities between the definitions of the Matthew effect and h α already point out that the simulations that are presented in the following can be expected to reveal the appearance of the Matthew effect by using the h α -index in performance measurement.

First Agent-Based Simulation with 200 Agents (Baseline Simulation)
Similar to the BBHs program with search, stopping, and decision rules (see above), the first agent-based simulation has three phases: initial setting, acting (collaborating) several periods, and final data set for further analysis (visualization of the results). Whereas the initial setting and the final data set are on the macro level (certain distributions are set or analyzed), acting is on the micro level (see section 2.2). It is the goal of the first agent-based simulation-the baseline simulation, compared to which one parameter is changed in each of the simulations presented in the following sections-to compare the mean h α -index values of agents with initial low or high h-index values after several periods of action (e.g., collaboration with other agents). The Stata command for the first agent-based simulation is h_index, r(50) n(200) per(20) co(3) dp(poisson, mean(10)) dc(poisson, mean (5) Acting: Agents act (publish, collaborate, receive citations) across 20 periods [per(20)]. Each collaborating group of agents has three agents on average [co (3)]. The citations that the co-authored papers published by the agent groups receive follow a Poisson distribution with a specified time-dependent expected value [dc(poisson, mean (5))]. The timedependent expected value follows a log-logistic distribution reaching its maximal value of 5 after 3 years (following the general guideline by Glänzel & Schöpflin, 1995) [p(3)]. The agentbased simulation is repeated 50 times [r(50)] to ensure the robustness of the simulation. After each simulation, new h-index and h α -index values are calculated for all agents.
Final data set: Two groups of agents are defined with low (<7) and high (>7) initial h-index values (7 is the median initial h-index value). For each period with actions (20 in total), the mean h α -index values are computed (across 50 repetitions of the simulation to have robust results). The results are shown in Figure 2. For each period with actions, the advantage of the agents with high initial h-index values is clearly visible: They not only start with higher mean h α -index values (which is as expected), but these values also increase with additional periods-by considering further cooperation, publications, and additional citations. The mean h α -index values of the agents with low initial h-index values also increase over time. However, the difference between the two groups becomes larger with onward periods-as the dotted 4 The equivalent function call to produce the simulated data in R is simulate_hindex(runs = 50, n = 200, periods = 20, coauthors = 3, distr_initial_papers = 'poisson', dpapers_pois_lambda = 10, distr_ citations = 'poisson', dcitations_mean = 5, dcitations_peak = 3, alpha_ share = .33) line in Figure 2 demonstrates. Increasing differences between both groups can be interpreted as a Matthew effect in operation.

Second Agent-based Simulation with an Additional Element Leading to More Citations for Prolific Agents
The second simulation was run using the Stata command h_index, r(50) n(200) per(20) co(3) dp(poisson, mean(10)) dc(poisson, mean(5)) p(3) sh(.33) boost(size(.5)). 5 It is the same command as in the first agent-based simulation (the baseline simulation), but we introduce a new element with boost(size(.5)) (which is printed in boldface). This option means that papers published by agents with higher h-index values are cited more frequently than papers published by agents with lower h-index values.
The number of citations in the second simulation is increased based on the value specified with [size(.5)]. For example, suppose agents with a maximal h-index value of 11 have published a certain paper. The value 0.5 as option means that this paper receives round (11 × 0.5) = 6 additional citations.
Final data set: In the second agent-based simulation, the median of the initial h-index values (median = 7) is the same as in the first simulation. Thus, two groups of agents are defined with low (<7) and high (>7) initial h-index values. Figure 3 presents the results. The results are similar to Figure 2, but the differences between both groups are more pronounced: Whereas the h α -index values of the group with high initial h-index values increase more steeply, the h α -index values of the group with low initial h-index values increase similarly to those in Figure 2. This leads to larger mean h α -index values differences between both groups 5 The equivalent function call to produce the simulated data in R is simulate_hindex(runs = 50, n = 200, periods = 20, coauthors = 3, distr_ initial_papers = 'poisson', dpapers_pois_lambda = 10, distr_citations = 'poisson', dcitations_mean = 5, dcitations_peak = 3, alpha_share = .33, boost = TRUE, boost_size = .5) (as the dotted line reveals). In other words, the Matthew effect is reinforced by letting the papers published by agents with higher h-index values be cited more frequently than the agents with lower h-index values.

Third Agent-based Simulation Considering the Correlation of New Citations with h-index Values: Agents with High h-index Values Receive Disproportional Citations
For the third simulation, the following Stata command was used: h_index, r(50) n(200) per(20) co(3) dp(poisson, mean(10)) dc(poisson, mean(5)) p(3) sh(.33) dil(correlation(.8) share(.6)). 6 Similar to the second simulation, only one option has been changed (which is printed in boldface) in comparison to the first baseline simulation. The new options [dil(correlation(.8) share(.6))] focus on the probability of publishing new papers depending on initial h-index values. The option [correlation(.8)] means that agents with high initial h-index values are more productive than agents with low initial h-index values: the correlation between the probability of publishing new papers and initial h-index values has been set to 0.8. The option [share(.6)] means that 60% of the agents publish. The use of this option can be reasoned, for instance, by the "sacred spark" theory (Cole & Cole, 1973) which claims "that there are substantial, predetermined differences among scientists in their ability and motivation to do creative scientific research" (Allison & Stewart, 1974, p. 596).
The third agent-based simulation is intended to check whether the higher productivity of prolific agents has an effect on the h α -index values development of the groups with high and low initial h-index values. 6 The equivalent function call to produce the simulated data in R is: simulate_hindex(runs = 50, n = 200, periods = 20, coauthors = 3, distr_ initial_papers = 'poisson', dpapers_pois_lambda = 10, distr_citations = 'poisson', dcitations_mean = 5, dcitations_peak = 3, alpha_share = .33, diligence_corr = .8, diligence_share = .6) Final data set: The results of the third simulation are presented in Figure 4. Whereas this third simulation considers a positive correlation between productivity and h-index values, the second simulation includes the positive relationship between citations and h-index values (see Figure 3). The findings in Figure 4 vary (significantly) from the results in Figure 2 and Figure 3. As the dotted line for the differences between the mean h α -index values reveals, the differences in Figure 4 increase less strongly over time than in the previous figures.

Fourth Simulation Considering Strategically Selecting Co-authors
For the fourth simulation, we used the Stata command h_index, r(50) n(200) per(20) co(3) dp(poisson, mean(10)) dc(poisson, mean(5)) p(3) sh(.33) clear st. 7 Compared to the baseline simulation, we considered a strategic element [st], which focuses on the possible tendency of agents to select other agents as co-authors with lower h-index values. Such a strategical element (with another focus) has been mentioned by Hirsch (2019b): "lower h-index authors have the choice to not collaborate with high h-index authors but rather pursue their own independent work, or work with more junior collaborators" (p. 1168). The strategic option of the h_index command means that first the agents with the highest h-index values are assigned to the groups of collaborating agents as described in step 3 of the simulation procedure in section 3. Then, all other agents in the simulation are randomly allocated to the collaborating groups. Thus, the strategic option seizes the idea of collaborating with lower h-index agents. The strategic option gives much weight to the effect of strategic collaboration decisions in our simulations, because the agents with the highest h-index values never collaborate with each other, so that their h α -index values increase after every collaboration. The results of this simulation reveal the potential effect of strategic collaboration decisions on the outcome distribution. The strategic option follows closely Coleman's (1990) classic macro-micro-macro model (i.e., "Coleman's boat"). "The general thrust of this model is that proper explanations of macro-level change and variation entail showing how macro-states at one point in time influence the behavior of individual actors, and how these actions add up to new macro-states at a later time" (Hedström & Swedberg, 1996, p. 296). The model assumes that individual action results from the social context in a social network. Coleman's model for the fourth agent-based simulation (see Figure 5) starts with the possible influence of a social context on the attitudes of agents (A). The current situation in science is characterized by performance-based evaluations: "Especially in universities, government funding of scientific research is increasingly based upon performance criteria. As research institutions operate more and more in a global market, international comparisons of institutions are published on a regular basis" (Moed, 2018). This situation puts pressure on agents doing science in the system.
The second (B) and third (C) steps are characterized by the core components of Hedström's (2005) desires, beliefs, and opportunities (DBO) theory. The second step (B) in the macro-micromacro model is that the social context (here: increasing focus on performance criteria) influences the attitudes of single agents: The agents believe (given the pressure in the system) that they should increase their h α -index values. As acting agents in the system they desire to perform as well as possible in terms of bibliometric indicators. In step C the agents have several opportunities to act: They can collaborate with other agents without considering their h-index values or they can consider that in their reflections (among other alternatives). Because the h α -index can only be improved when agents publish papers with co-authors having lower h-index values, the strategic option simulates this possible tendency of agents. In step D the empirical analyses of the development of h α -index values for agents with low and high initial h-index values across several periods of action reveal how single actions of agents lead to the social phenomenon on the macro level: the reinforcement of the Matthew effect.
The result of the fourth agent-based simulation is shown in Figure 6. It is clearly visible that the strategic element significantly reinforces the Matthew effect, which is already visible in the previous simulations: Agents with low h-index values not only have lower initial h α -index values than agents with high h-index values, the h α -index values also increase on a significantly lower level across the periods of evaluation. Across the periods of actions, the h α index value differences between both h-index groups become larger and larger.

DISCUSSION
The agent-based simulations presented in this paper follow a recent discussion in Scientometrics about the newly introduced h α -index by Hirsch (2019a). Leydesdorff et al. (2019) assumed that the use of the new index reinforces the Matthew effect in research evaluations. Scientists with initial high h-index values will profit disproportionally from the use of the h α -index. Thus, the fear is that the use of the index enlarges a problem that is already prevalent in the science system. According to Merton (1968) the problem of the Matthew effect in science is so great that "we are tempted to turn again to the Scriptures to designate the statusenhancement and status-suppression components of the Matthew effect. We can describe it as 'the Ecclesiasticus component,' from the familiar injunction 'Let us now praise famous men,' in the non-canonical book of that name" (p. 58).
Based on the definition of the h α -index, Leydesdorff et al. (2019) conjectured that the disproportional attribution of credit by the h α -index-the co-author with the highest h-index receives the full credit-reflects the operation of the Matthew effect. Thus, the h α -index is already the Matthew effect in operation. One cannot assume that the co-author with the highest h-index contributes so much to the paper that the other co-authors can be completely discarded in performance measurement. In this study, we abstained from the single case and tested with various simulations whether the Matthew effect is visible on the macro level-when reasonable parameters are set. The results of our study confirm what we expected from the single case: The h α -index reinforces the Matthew effect. This effect can be intensified if strategic behavior of the publishing scientists and accumulative advantage effects are considered in the simulation.
We situated this study in the tradition of analytical sociology which seeks for mechanismbased explanations. These explanations try to focus on the crucial elements of a given process and to abstain from the detailed view (Hedström & Ylikoski, 2010). Agent-based modeling is a way of connecting the individual to the social level (Hedström, 2005). For studying a phenomenon on the macro level, the environment is defined in which the action takes place. Then, the action is run following predefined rules (Hanneman, 1988). The result is a data set that includes the interactions between actions and initial parameters functioning as constraints. This data set can be used to investigate whether the social phenomenon of interest is observable on the macro Figure 6. Results of the fourth simulation: mean h α -index values for two groups of agents with low (<7) and high (>7) initial h-index values (with standard deviations bars).
level. By varying the parameters of an agent-based model used as a baseline, the effect of various situational elements from publishing, being cited, and collaborating on the development of the distribution of h α -index values can be tested.
This study is rooted not only in analytical sociology but also in Bornmann and Marewski's (2019) BBH program. The program demands that indicators are empirically studied to see whether they can be used in specific evaluation environments (and if so, how they can be used). The h_index command and hindex package, which we introduced in this paper, can be used to simulate the use of the h-index and h α -index in certain predefined environments. Using different specifications of the command (package functions), the simulation can be adapted to the environment for investigating where the h α -index is intended to be used. In this study, we used the Stata command to test whether the Matthew effect becomes apparent when the h α -index is calculated for a group of agents who collaborate, publish, and receive citations across several periods.
The R package and Stata command allow consideration of some strategic elements in the agent-based simulations: If the h α -index is used in research evaluation processes, scientists might try to cooperate strategically with co-authors having lower h-index values. The findings of our simulations reveal that the consideration of this element leads to a significant reinforcement of the Matthew effect. By using different options of the h_index command or different parameters for the hindex package functions, the agent-based simulations can consider not only strategic behavior but also information from the literature on the usual behavior of scientists and distributions of publications and citations in different fields and institutions (e.g., Perianes-Rodrigueza & Ruiz-Castillo, 2014). For example, we considered in our agent-based simulations that agents with higher h-index values will publish more frequently than agents with lower h-index values. Many studies have shown that future performance depends on previous performance (Abramo, D'Angelo, & Soldatenkova, 2017;Allison, Long, & Krauze, 1982;Kwiek, 2015). We also included another element in the simulations that can be derived from the literature: that authors might tend to cite highly cited papers.
Because the R package and Stata command are freely available, we encourage their use. We plan to add further functionality to them in the near future.