Abstract
To gain a deeper understanding of the Matthew effect in academic recognition among researchers, three social influence models of status hierarchies in the form of agent-based models are presented, where the positivity of the Lyapunov characteristic exponent is the quantitative operationalization of the Matthew effect. We find that the Matthew effect in status attribution is a generic property of the social influence model where skills depend on status (28.1% of all examined parameterizations) twice as often as in the model with constant skills (14.1%) and that the Matthew effect is a generic property of the model where skills depend on previous skills (43.0%) trice as often as in the model with constant skills. Hence, if one argues in favor of a meritocratic system in which research grants are awarded based on researchers’ skills rather than their status in the scientific community, and if grant-providing bodies also adhere to this principle, there is a higher probability of observing the Matthew effect in academic recognition among researchers. Employing the Lyapunov characteristic exponent to investigate the presence of the Matthew effect in a theoretical model is novel in the literature.
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Introduction
A central concept in the sociology of science is the Matthew effect. This is due to Merton’s [29, 30] observation that better-known researchers tend to receive more academic recognition for similar achievements than do lesser-known researchers and that better-known researchers attract more resources at the expense of lesser-known researchers, which widens the gap between the resources and achievements of the two groups of researchers. In the words of Merton [30] p. 609,
“The Matthew effect is the accruing of large increments of peer recognition to scientists of great repute for particular contributions in contrast to the minimizing or withholding of such recognition for scientists who have not yet made their mark. The biblical parable [in the Gospel of Matthew 25:29] generates a corresponding sociological parable” (added text in brackets).
Indeed, the Matthew effect is well documented in the literature with reference to academic careers (e.g., [3]), citation counts of scholarly papers (e.g., [24]), productivity in research (e.g., [2]), and research funding (e.g., [8]) and is an important theme in the study of stratification in science. There is also theoretical literature that aims to explain the Matthew effect in academic careers (e.g., [16]), academic recognition and scientific productivity (e.g., [1]), and citation counts of scholarly papers (e.g., [37]) (see also the review of the empirical and theoretical literature by [14]).
The present paper considers the literature on status hierarchies to develop a theory on the Matthew effect in academic recognition. Specifically, to better understand the inequality-generating mechanisms underlying Merton’s [29, 30] observation, the aim of this paper is to present three social influence models of status hierarchies and examine the conditions under which the Matthew effect in academic recognition, or status attribution, is a generic property of the models in the sense that the Matthew effect is the expected outcome of a model. That is, the Matthew effect should not only be an occasional feature of a proper model of the Matthew effect but also be the expected feature of the model. The three status models developed herein are agent-based models characterized by low-dimensional realism. This means that the agents in the models exhibit empirical realism along a few specific dimensions [9].
In the agent-based models, we model a network of researchers showing appreciation for each other where the amount of deference depends on how skillful a researcher is perceived to be. Specifically, the more skillful a researcher is perceived to be, the more appreciation she receives from other researchers and the higher her status is in the network. However, researchers do not necessarily interact with all other researchers in the network; if the status of two researchers is too dissimilar, they will not interact with each other (see [4] on status-based homophily). Moreover, researchers rely not only on their own evaluations of the skills of other researchers in the network but also on their colleagues’ evaluations of other researchers’ skills because a researcher’s skill is not directly observable and is therefore difficult to evaluate [27]. Finally, it is agonizing to show respect to a peer researcher if the favor is not repaid. In this case, the humiliated researcher will show less appreciation to this particular researcher than if the favor had been repaid [21].
The three status models differ regarding how researchers’ skills are modeled. In the first model, skill is constant, whereas in the other two models, skill is time-varying. Specifically, in the first of the latter two models, a researcher’s skill is a function of her status in the sense that higher status implies a higher probability of receiving skill-improving research funding, whereas in the second model, higher skill of the researcher in the previous period implies a higher probability of receiving skill-improving research funding in the current period. In the absence of research funding, the researcher’s skill deteriorates in both models. The idea behind the mechanism is that a research grant gives the researcher the opportunity to spend more time on her own research and thereby improve her skill as a researcher (cf., learning-by-doing). However, a lack of research funding makes it more difficult for the researcher to conduct research that is close to the research frontier, which means that her skill as a researcher declines.
The two status models with time-varying skills proposed herein are novel in the literature. We introduce these models because we believe that people’s skills in various arenas are not constant but change over time. Our specific concern in this paper is to better understand the inequality-generating mechanisms behind the Matthew effect in academic recognition in the scientific community. One mechanism noted by Merton [29, 30] was researchers’ ability to attract resources, and this ability was positively affected by their academic recognition in the scientific community. For this reason, we assume, in one of the models with time-varying skills, that a researcher’s status affects her chances of receiving skill-improving research funding.
The fundamental question we address in this paper is whether research funding from grant-providing bodies should focus on agents’ skills in doing research instead of their academic recognition, or status, in the scientific community when financing research proposals. For this reason, we also assume, in the other model with time-varying skills, that a researcher’s skill affects her chances of receiving skill-improving research funding. When evaluating the two grant-providing schemes, we study the extent to which a specific scheme is associated with the Matthew effect in status attribution (see [36] for a field experiment on how texts in research proposals matter for funding decisions). As a complement to this analysis, we also examine to what extent we observe the Matthew effect in status attribution in a model with constant skills. The latter model is similar to the model developed in Manzo and Baldassarri [27].
To recapitulate, the approach in the paper is straightforward. First, we posit that grant-providing bodies make funding decisions exclusively based on the status of the researchers in the scientific community; then, we examine under which parameterizations the agent-based model is characterized by the Matthew effect. Thereafter, we change the assumption to funding decisions being based exclusively on the researchers’ skills in conducting research and then examine under which parameterizations the agent-based model is characterized by the Matthew effect. Thus, agent-based models can be viewed as “laboratories” in the form of mathematical models in which we explore the consequences of various assumptions about the behavior of grant-providing bodies for the occurrence of the Matthew effect.
Although a large body of literature exists in the social sciences on the Matthew effect in various arenas, a quantitative measure of the Matthew effect did not exist in the literature before Bask and Bask [6] proposed their measure. One reason was the lack of a common and precise understanding of the Matthew effect among scholars [14]. The definition of the Matthew effect proposed by Bask and Bask [6] is that after taking the average of the slopes of all the time series showing how the distance between two researchers’ status evolves over time, restricting attention to the pairs of status trajectories that were initially close to each other, the average slope should be positive when the Matthew effect is in play. Hence, although researchers experience a mix of cumulative advantages and disadvantages in academic recognition, the expected outcome of a particular parameterization of a model that is characterized by the Matthew effect is an interindividual divergence of status trajectories. Employing the proposed measure in Bask and Bask [6], the Lyapunov characteristic exponent, to investigate the presence of the Matthew effect in a theoretical model is novel in the literature.
We find that the Matthew effect in status attribution is a generic property of the status model where skills depend on status (28.1% of all examined parameterizations) twice as often as in the model with constant skills (14.1%) and that the Matthew effect is a generic property of the model where skills depend on previous skills (43.0%) trice as often as in the model with constant skills. These findings are due to the reinforcement effect of the learning-by-doing mechanism. That is, some researchers experience a cycle of more research funding, improved skills, higher status, and even more research funding, whereas other researchers experience a cycle of less research funding, deteriorated skills, lower status, and even less research funding. The former researchers experience a series of cumulative advantages, and the latter researchers experience a series of cumulative disadvantages, resulting in the Matthew effect in academic recognition. The reinforcement effect is weaker when researchers’ skills depend directly on their status because status is affected by unrepaid favors from other researchers.
Consequently, if we return to the fundamental question that we address in this paper, what we learn from the numerical analysis of the status models with time-varying skills is that if one argues in favor of a meritocratic system in which research grants are awarded based on researchers’ skills rather than their status in the scientific community, and if grant-providing bodies also adhere to this principle, there is a higher probability of observing the Matthew effect in academic recognition among researchers. From the numerical analysis, we also learn that in the status model where researchers’ skills depend on their status, if researchers give more consideration to other researchers’ evaluations of colleagues in the network, then the discrepancies in these evaluations diminish, reducing the likelihood of the Matthew effect in status attribution.
Finally, it should be emphasized that the social influence models of status hierarchies that are presented herein are applicable not only to research networks. On the contrary, the status models are also applicable in other contexts in which agents’ skills are not constant but change over time. For example, the agents in the models could be workers in the labor market with skills that are determined by their human capital, where either their status in the labor market or their task-specific skills determine whether they are given the opportunity to invest in their human capital via task-specific training. In addition, the status models could be incorporated into a suitable labor market model for further analysis (see [12] for a review of the literature on agent-based modeling in macroeconomics).
The rest of this paper is organized as follows. In Sect. "Literature review", we present a review of the relevant theoretical research on the Matthew effect in academic careers and academic recognition. The three social influence models of status hierarchies are developed in Sect. "Three status models". Cumulative advantage (disadvantage) and the Matthew effect are reviewed in Sect. "Cumulative advantage (disadvantage) and the Matthew effect". In Sect. "The Matthew effect in the status models: a numerical analysis", we scrutinize the three social influence models of status hierarchies numerically for the Matthew effect in status attribution across a large number of parameterizations. Finally, Sect. "Discussion" concludes the paper with a discussion.
Literature review
Allison et al. [1] were among the first to formulate a formal theoretical model characterized by the Matthew effect in science, first and foremost in terms of the number of publications by researchers. The authors extended the work of Allison and Stewart [2] and de Solla Price [37] by presenting a model based on a contagious Poisson process [11] and noting with reference to previous research that a “model of cumulative advantage [or the Matthew effect] does not [necessarily] imply increasing inequality. When the model is modified to allow for heterogeneity in the rate of cumulative advantage, however, increasing inequality is implied” (p. 615 in [1], added text in brackets). However, the model in Allison et al. [1] is not a behavioral model in the sense that it describes how agents choose to act. That is, there are no agents who choose to publish in their model. Instead, they formulated a stochastic process describing how the number of publications by researchers progresses over time. Moreover, it is not a model for academic recognition in the scientific community, which is the focus of our models.
A behavioral model of hierarchies in academic recognition, or status attribution, was presented in Manzo and Baldassarri [27] when they developed an agent-based model with agents who base their decisions on heuristics when interacting with other agents in a network. As mentioned in the introductory section, the model in Manzo and Baldassarri [27] is close to one of the models presented in this paper. (The differences between the two models are explained in Sect. "Three status models".) Their contribution to the literature was twofold. First, as already noted, they incorporated agents with heuristic-based decision making into the model. Second, they were able to demonstrate the importance of social influence for the status-quality gaps among agents in the presence of a cumulative advantage process in status attribution. That is, “in science, it has been observed that status hierarchies display a sizable disconnect between actors’ quality [or agents’ skills] and rank and that they become increasingly asymmetric over time, without, however, turning into winner-take-all structures” (p. 329 in [27]; added text in brackets).
Manzo and Baldassarri [27] built their research on Gould [21] and Lynn et al. [26], who presented models of the emergence of status hierarchies in small groups. Gould [21] noted that social hierarchies “emerge and persist spontaneously rather than by conscious creation, but at the same time without ensuring that rewards exactly reflect differences in individual qualities” (p. 1146 in [21]). In his model, Gould [21] assumed that two opposing forces were at work to create a status hierarchy with sizeable status-quality gaps among agents but without having a winner-take-all structure. First, the repelling force, which widens the status-quality gap, is a social-influence force in which agents’ deferential gestures are influenced by other agents’ deference attributions. Second, the attracting force, which narrows the status-quality gap, is a force that reflects agents’ aversion to unrepaid deferential gestures.
Lynn et al. [26] extended the model in Gould [21] in two ways. First, they assumed that agents’ qualities could not be observed perfectly by other agents, which means that agents’ qualities can be under- or overestimated by other agents. Second, they also assumed that agents were involved in a sequence of deference exchanges, where agents’ perceptions of other agents’ qualities are affected by the agents’ previous behavior. Hence, Lynn et al. [26] adopted a dynamic approach to the emergence of status hierarchies rather than a static approach, as Gould [21] did. In contrast with the model in Manzo and Baldassarri [27] and our models, neither Gould [21] nor Lynn et al. [26] assumed that agents have a preference for status-based homophily. That is, agents prefer not to interact with status-dissimilar others. Finally, the analysis in Lynn et al. [26] differs from the analyses in Gould [21] and Manzo and Baldassarri [27] in the sense that they focused on status reordering instead of status dispersion.
Feichtinger et al. [16] presented a model of academic careers in which a researcher’s scientific reputation—or, in our terminology, a researcher’s status—is positively affected by the particular researcher’s amount of scientific effort (doing research, contributing to scientific events such as conferences and seminars, networking in the scientific community, etc.). However, a researcher’s reputation is negatively affected by forgetfulness in the sense that the reputation decreases over time if the researcher does not make any efforts to maintain her reputation in the scientific community. Moreover, Feichtinger et al. [16] added a term that accounts for the Matthew effect in reputation in the sense that equal increases in the amount of effort do not necessarily cause equal increases in reputation. Specifically, a researcher with a high reputation gains more in reputation than a researcher with a low reputation, whose reputation could actually decrease. In other words, in Feichtinger et al. [16], a researcher’s reputation is described by a standard capital accumulation equation augmented with a function for the Matthew effect.
Under the assumption that a researcher’s capacity puts an upper limit on the amount of scientific effort she can make, that the same researcher’s utility is a concave function of her scientific reputation but that her efforts are costly, Feichtinger et al. [16] solved the researcher’s optimal control problem to find the optimal amount of effort that the researcher should make to maximize the lifetime utility of being a researcher. The authors make several interesting findings. First, if the researcher’s reputation is low from the beginning (e.g., due to lack of talent or because her university has a low reputation), her reputation is doomed to decrease over time until her scientific career finally ends. Second, if the researcher’s reputation is high enough to start with (e.g., due to talent or because her university has a high reputation), the efforts made by the researcher might either result in a career that prematurely ends at some point in time or, instead, result in a successful research career.
A main difference between the findings of Feichtinger et al. [16] and those of our paper is that they explicitly include the Matthew effect in their model and, thereafter, study how this effect affects the researcher’s problem of finding the optimal amount of scientific effort that maximizes lifetime utility; however, we study under which conditions the Matthew effect is a generic property of our models. Specifically, we examine under which conditions ‘networking in the scientific community’—an example of scientific effort according to Feichtinger et al. [16]—is associated with the Matthew effect in academic recognition. Hence, the models in Feichtinger et al. [16] and our paper complement each other from more than one perspective.
We now develop the status models and, thereafter, scrutinize numerically for the Matthew effect in status attribution for a large number of parameterizations.
Three status models
The status models are initialized in Sect. "The status models are initialized", and the dynamics of the models are described in Sect. "The dynamics of the status models", where four of the five parts are the same in the models. Specifically, the sirens heuristic in partner selection, the imitation heuristic in skill perception, the sour grapes heuristic in deference attribution, and the averaging heuristic in status update are modeled identically in the models. What differs between them is how skill is modeled in the learning-by-doing heuristic in skill updating.
In one of the status models, agents’ skills—or, in our context, researchers’ skills—do not change over time. This model is similar to the model in Manzo and Baldassarri [27]. The models differ from each other regarding how agents rely on their colleagues’ evaluations of other agents’ skills in the network, coined the imitation heuristic by these authors. We make use of a rule that resembles how Gould [21] formalized quality perception under social influence, whereas they impose a cumulative advantage rule when modeling the same heuristic.
In the other two status models, agents’ skills are not constant, as in the first model. Here, skill is either a function of status or a function of the previous period’s skill. Specifically, in the first model, skill is a function of status in the sense that higher status for an agent implies a higher probability of receiving research funding, which improves the agent’s skill, whereas in the second model, higher skill in the previous period implies a higher probability of receiving research funding in the current period, which improves the agent’s skill. In the absence of skill-improving funding, the agent’s skill deteriorates in both models.
The status models are initialized in period \(t=0\) (see Sect. "The status models are initialized", where it is emphasized in the mathematical expressions that they are valid for \(t=0\)) and the dynamics of the models start in period \(t=1\) (see Sect. "The dynamics of the status models").
The status models are initialized
There are \(N\) agents in the network with various qualities; in our case, skills as researchers. Exactly how skill is measured is not the focus here. Instead, we make the point that skill can be measured. Nevertheless, the agents in the network cannot perfectly observe other agents’ skills and must therefore form opinions about their skills. Specifically, agent \(i\)’s perception of agent \(j\)’s skill, \({q}_{i,j,t=0}\), equals agent \(j\)’s actual skill, \({Q}_{j,t=0}\), plus agent \(i\)’s misperception of agent \(j\)’s skill, \(e_{i,j,t = 0}\):
where \(Q\sim N\left(0,{\sigma }_{Q}\right)\) and \(e\sim N\left(0,{\sigma }_{e}\right)\); \(N\left(0,\sigma \right)\) is the normal distribution with mean \(0\) and standard deviation \(\sigma\). Note that the agents in the network have perfect perceptions of other agents’ skills when \({\sigma }_{e}=0\). Moreover, as noted above, agents’ skills are constant in one of the status models.
The amount of deference that agent \(i\) assigns to agent \(j\), \(a_{i,j,t = 0}\), equals agent \(i\)’s perception of agent \(j\)’s actual skill:
Then, agent \(j\)’s status, \(S_{j,t = 0}\), equals the average of all deference attributions received from the other agents in the network:
and the status models are initialized.
What conclusions can be drawn from the equations that formalize the initialization of the status models? To answer this question, we substitute (1)-(2) into (3):
Hence, agent \(j\)’s status equals agent \(j\)’s actual skill plus the average of all other agents’ misperceptions of agent \(j\)’s skill. In particular,
Thus, when there is a large number of agents in the network, agent \(j\)’s status equals agent \(j\)’s actual skill. This conclusion holds for all distributions of \(e\) as long as the mean of the distribution equals zero.
In addition, according to (4), there is a tradeoff between the number of agents in the network, \(N\), and the standard deviation of agents’ misperceptions of other agents’ skills, \({\sigma }_{e}\), in the sense that the agents’ status-quality gaps, \({S}_{\forall j,t=0}-{Q}_{\forall j,t=0}\), become more pronounced when there are fewer agents in the network (\(N\) smaller) or when agents’ misperceptions of other agents’ skills are more severe (\({\sigma }_{e}\) higher). Conversely, agents’ status-quality gaps become less pronounced when there are more agents in the network (\(N\) larger) or when agents’ misperceptions of other agents’ skills are less severe (\({\sigma }_{e}\) lower).
The dynamics of the status models
Four heuristics describe the common dynamics of the status models: the sirens heuristic in partner selection (see Sect. "The sirens heuristic in partner selection"), the imitation heuristic in skill perception (see Sect. "The imitation heuristic in skill perception"), the sour grapes heuristic in deference attribution (see Sect. "The sour grapes heuristic in deference attribution"), and the averaging heuristic in status update (see Sect. "The averaging heuristic in status update"). The difference between the models is revealed when skill is modeled either as constant or as time-varying due to a learning-by-doing heuristic in skill updating (see Sect. "The learning-by-doing heuristic in skill updating").
Here is an overview of the timing in the status models. The agents’ actions follow a sequential order within each period in the three models. First, each agent decides which agents she will interact with in the network (determined by \(h\)). Second, each agent evaluates the skills of all the other agents she will interact with in the network (determined by \(w\)). Third, each agent decides the amount of deference she will assign to the agents she will interact with in the network (determined by \(s\)). Then, after all deferential gestures have been exchanged in the network, the status of each agent is updated. Finally, in two of the three models, the skill of each agent is updated. In one of the models, higher status implies a higher probability of receiving skill-improving research funding; in the other model, higher skill in the previous period implies a higher probability of receiving skill-improving research funding (in the current period). In the absence of skill-improving funding, the agent’s skill deteriorates. In the third model, agents’ skills are constant.
We will now proceed to describe the five heuristics in the status models in detail.
The sirens heuristic in partner selection
The first step in the dynamics of the status models is that each agent decides which agents she will interact with in the network. Here, we follow Manzo and Baldassarri [27] when we assume that an agent does not necessarily interact with all other agents in the network. This is in contrast with Gould [21] and Lynn et al. [26], who assumed that all agents interact with everyone else in the network.
The reason why an agent prefers not to interact with a status-dissimilar other could be that she would like to minimize her exposure to unpleasant psychological experiences. This is known as a preference for status-based homophily [4]. For instance, a high-status agent may react with scorn [18, 19] or worry about status leaking [31] when interacting with a low-status agent (e.g., frustrated by poor reasoning), and a low-status agent may react with envy [18, 19] when interacting with a high-status agent (e.g., feeling of inadequacy).
We follow Manzo and Baldassarri [27] when we assume that agent \(i\) interacts with agent \(j\) if their status was similar enough in the previous period. Specifically, if
equals one, agent \(i\) will interact with agent \(j\) in the current period, where \(H\left[\bullet \right]\) is the Heaviside step function and \(0\le {h}_{i}\le 1\) is agent \(i\)’s propensity to interact with status-dissimilar others, with the extremes that agent \(i\) will not interact with any other agent in the network (\({h}_{i}=0\)) and that agent \(i\) will interact with everyone else in the network (\({h}_{i}=1\)). If the sirens heuristic in (6) equals zero, agent \(i\) will not interact with agent \(j\) in the current period.
In the parameterized status models scrutinized below (see Sect. "The Matthew effect in the status models: a numerical analysis"), we allow for heterogeneous behavior among agents. That is, some agents are more inclined to interact with status-dissimilar agents than other agents in the network are.
The imitation heuristic in skill perception
The second step in the dynamics of the status models is that each agent evaluates the skills of all the other agents she will interact with in the network. Since this is a difficult task to accomplish, the agent relies on a cognitive shortcut when performing the evaluations; namely, she weighs other agents’ skill assessments when forming her own assessment of a particular agent. However, because other agents’ assessments are not directly observable, she uses the agent’s status, determined in the previous period, as a proxy for the other agents’ skill assessments (since the agent’s status in the current period is yet to be determined).
Specifically, agent \(i\)’s perception of agent \(j\)’s skill—if agent \(i\) will interact with agent \(j\) in the current period—equals a weighted average of agent \(i\)’s perception of agent \(j\)’s skill when agent \(i\) does not account for other agents’ skill assessments of agent \(j\) and other agents’ skill assessments of agent \(j\) (proxied with agent \(j\)’s status in the previous period):
where \(0\le {w}_{i}\le 1\) is agent \(i\)’s propensity to imitate other agents’ deferential gestures. The imitation heuristic in (7) resembles how Gould [21] formalized quality perception under social influence.
Essentially, there are two models in the literature on learning in social networks: Bayesian learning and DeGroot learning (e.g., [23]). In Bayesian learning, agents update their beliefs about other agents’ opinions using Bayes’ rule. However, Bayesian learning is quite sophisticated and imposes a cognitive load on agents that might not be realistic for human beings [10]. For this reason, DeGroot learning has been proposed as a more realistic model for social learning. In DeGroot learning, agents update their beliefs about other agents’ opinions by calculating a weighted average of those opinions [13]. The imitation heuristic in (7) is an example of DeGroot learning.
There are two extremes in the imitation heuristic in (7). The first extreme is that the agent does not imitate other agents’ deferential gestures (\({w}_{i}=0\)), and the other extreme is that the agent relies only on other agents’ deferential gestures (\({w}_{i}=1\)) in the skill assessment.
The imitation heuristic in (7) differs from the corresponding heuristic in Manzo and Baldassarri [27]. To understand our modification of the imitation heuristic, take the time-\(t\) expectations of both sides of (7):
Then, take the time-\(t\) expectations of both sides of the corresponding equation in Manzo and Baldassarri (p. 342 in [27]):
Hence, agent \(i\)’s expected perception of agent \(j\)’s skill is biased with magnitude \({w}_{i}{S}_{j,t-1}\) in (9). Specifically, when agent \(j\)’s status in the previous period is positive (negative), \({S}_{j,t-1}>0\) (\({S}_{j,t-1}<0\)),Footnote 1 then agent \(i\)’s expected perception of agent \(j\)’s skill is upwardly (downwardly) biased. Consequently, agents with a positive (negative) status tend to obtain increased (decreased) status, which means that we observe the Matthew effect in status attribution among agents. This was also the explicit intention of Manzo and Baldassarri [27] when modeling the imitation heuristic in skill perception.
As already noted, we allow for heterogeneous behavior among agents when investigating the parameterized status models below (see Sect. "The Matthew effect in the status models: a numerical analysis"). That is, some agents place greater weight on their own assessments of agents’ skills than do other agents in the network.
The sour grapes heuristic in deference attribution
The third step in the dynamics of the status models is that each agent decides the amount of deference she will assign to the agents she will interact with in the network. Specifically, if the favor was repaid by a peer agent in the previous period, the amount of deference equals the agent’s perception of this agent’s skill. However, if the favor was not repaid by the agent, we follow Gould [21] and Manzo and Baldassarri [27] and account for that “it is painful to pay attention to another person if the favor is not repaid” (p. 1149 in [21]) when modeling the amount of deference that the agent assigns to the other agent.
In accordance with Manzo and Baldassarri [27], the amount of deference that agent \(i\) assigns agent \(j\)—if agent \(i\) will interact with agent \(j\) in the current period and if agent \(i\) and agent \(j\) interacted with each other in the previous period—depends on whether the favor in the previous period was repaid. If the favor was repaid, the amount of deference that agent \(i\) assigns agent \(j\) equals agent \(i\)’s perception of agent \(j\)’s skill. However, if the favor was not repaid, the amount of deference that agent \(i\) assigns agent \(j\) equals agent \(i\)’s perception of agent \(j\)’s skill minus agent \(i\)’s sensitivity to attachment differences, \(0\le {s}_{i}\le 1\), times the attachment difference between agent \(i\) and agent \(j\) in the previous period:
There are two extremes in the sour grapes heuristic in (10). The first extreme is that the agent is not sensitive at all to other agents’ nonreciprocal behavior (\({s}_{i}=0\)), and the other extreme is that the agent is maximally sensitive to other agents’ behavior (\({s}_{i}=1\)).
Furthermore, if agent \(i\) will interact with agent \(j\) in the current period but agent \(i\) and agent \(j\) did not interact with each other in the previous period, the amount of deference that agent \(i\) assigns agent \(j\) equals agent \(i\)’s perception of agent \(j\)’s skillFootnote 2:
In the parameterized status models examined below (see Sect. "The Matthew effect in the status models: a numerical analysis"), some agents are more sensitive to unrepaid favors than other agents in the network are.
The averaging heuristic in status update
The fourth step in the dynamics of the status models is to calculate the status of each agent based on all deferential gestures that have been exchanged in the network. In contrast to Gould [21] and Lynn et al. [26], Manzo and Baldassarri [27] do not use an equally weighted averaging heuristic when updating an agent’s status. Instead, they rely on a temporally weighted averaging heuristic in the sense that a deferential gesture that was received further back in time is given less weight than a deferential gesture that was received more recently. We also use a temporally weighted averaging heuristic in our models, although we formulate the heuristic differently from Manzo and Baldassarri [27].
Specifically, agent \(j\)’s status equals a weighted average of all deference attributions received:
where \({v}_{j,i,{t}_{i}^{*}}\) is the weight given to the deference attribution received from agent \(i\) in period \({t}_{i}^{*}\), and \({t}_{i}^{*}\) is the period in which the latest deference attribution was received from agent \(i\):
\({H}_{1}\left[\bullet \right]\) in (13) is determined by the sirens heuristic in (6), and the weight in (12) is
which means that a more distant deference attribution received from agent \(i\) is given a lower weight in the heuristic in (12), where the rate of downweighting is determined by \(d\ge 0\). Recall that all agents in the network interacted with each other in period \(t=0\). Hence, if the sirens heuristic in (6) equals zero in all periods \(t>0\), then the latest deference attribution was received in period \(t=0\) (see (13)).
There are two extremes in the downweighting function in (14). The first extreme is that there is no downweighting (\(d=0\)), which means that the weight coincides with the weights in Gould [21] and Lynn et al. [26]. The second extreme is that no weight is given to deference attributions received in previous periods (\(d\to \infty\)).
The learning-by-doing heuristic in skill updating
The fifth step in the dynamics of the status models is updating the agents’ skills in the two models with time-varying skills. In one of the models, it is assumed that an agent’s skill is a function of her status in the sense that higher status implies a higher probability of receiving research funding, which improves the agent’s skill; in the other model, higher skill in the previous period implies a higher probability of receiving research funding in the current period, which improves the agent’s skill. In the absence of skill-improving funding, the agent’s skill deteriorates in both models. The idea behind the mechanism is that a research grant gives the agent the opportunity to spend more time on her own research and thereby improve her skill. However, a lack of research funding makes it more difficult for the agent to conduct research that is close to the research frontier, and her skill declines.
Specifically, agent \(j\)’s skill equals her skill in the previous period plus the change in her skill, \({\Delta Q}_{j,t}\):
where
if skill depends on status, or
if skill depends on previous skill, where \(\Delta Q>0\), \(P\left(\bullet \right)\) is probability, and \(N\left(\text{0,1}\right)\) is the standard normal distribution. Hence, it is less likely that a random number generated from the standard normal distribution is greater than the agent’s status (see (16)) or skill (see (17)) if the agent has a higher status in the scientific community (see (16)) or is more skillful as a researcher (see (17)). Consequently, it is more likely that the agent receives skill-improving funding if the agent has a higher status in the scientific community (see (16)) or is more skillful as a researcher (see (17)).
Recall that the agents’ skills are constant in one of the status models:
This step completes the description of the dynamics of the status models. In Sect. "The Matthew effect in the status models: a numerical analysis", we return to the models when we examine under which conditions the Matthew effect in status attribution is a generic property of the models in the sense that the Matthew effect is the expected outcome of a model. However, before doing so, we first review cumulative advantage (disadvantage) and the Matthew effect.
Cumulative advantage (disadvantage) and the Matthew effect
Cumulative advantage (disadvantage) and the Matthew effect are defined in Sects. "Defining cumulative advantage (disadvantage) and the Matthew effect: the case of two agents" (when only two agents in the network are considered) and "Defining cumulative advantage (disadvantage) and the Matthew effect: the case of many agents" (when all agents in the network are considered), and an intuitive explanation of how the Matthew effect is measured is provided in Sect. "Measuring the Matthew effect".
Defining cumulative advantage (disadvantage) and the Matthew effect: the case of two agents
Assume that there is a network consisting of \(N\) agents, two of which are referred to as agent \(A\) and agent \(B\), and assume that their respective status in period \(t\) are \({S}_{A,t}\) and \({S}_{B,t}\). In the context of the three status models presented in Sect. "Three status models", the agents in the network are researchers. Moreover, a natural measure of the inequality between these two agents is the distance between their respective status in period \(t\):
where we calculate the distance between agent \(A\)’s and agent \(B\)’s status in periods \(t=0\), \(t=1\), and so on up to and including period \(t={t}_{max}\), resulting in the time series \({\left\{{\mathcal{D}}_{\left\{A,B\right\},t}\right\}}_{t=0}^{{t}_{max}}\).
If the trend in the time series \({\left\{{\mathcal{D}}_{\left\{A,B\right\},t}\right\}}_{t=0}^{{t}_{max}}\) slopes upward, then the trajectories of agent \(A\)’s and agent \(B\)’s status diverge over time. This means that there is interindividual divergence of the status trajectories, also known as the Matthew effect. Conversely, if the trend in the time series \({\left\{{\mathcal{D}}_{\left\{A,B\right\},t}\right\}}_{t=0}^{{t}_{max}}\) slopes downward, then the trajectories of agent \(A\)’s and agent \(B\)’s status converge over time. This means that we have an interindividual convergence of the status trajectories.
Beginning with the Matthew effect, the interindividual divergence of the trajectories of agent \(A\)’s and agent \(B\)’s status has one of three possible causes. First, agent \(A\) experiences a cumulative advantage because the time series \({\left\{{S}_{A,t}\right\}}_{t=0}^{{t}_{max}}\) with status slopes upward, whereas agent \(B\) experiences a cumulative disadvantage because the time series \({\left\{{S}_{B,t}\right\}}_{t=0}^{{t}_{max}}\) with status slopes downward (or agent \(A\) experiences a cumulative disadvantage and agent \(B\) experiences a cumulative advantage).
Second, both agents experience a cumulative advantage, but agent \(A\)’s (or agent \(B\)’s) time series \({\left\{{S}_{A,t}\right\}}_{t=0}^{{t}_{max}}\) with status is more strongly upward-slo** than agent \(B\)’s (or agent \(A\)’s) time series \({\left\{{S}_{B,t}\right\}}_{t=0}^{{t}_{max}}\) with status. Third, both agents experience a cumulative disadvantage, but agent \(A\)’s (or agent \(B\)’s) time series \({\left\{{S}_{A,t}\right\}}_{t=0}^{{t}_{max}}\) with status is less strongly downward-slo** than agent \(B\)’s (or agent \(A\)’s) time series \({\left\{{S}_{B,t}\right\}}_{t=0}^{{t}_{max}}\) with status.
The interindividual convergence of the trajectories of agent \(A\)’s and agent \(B\)’s status also has one of three possible causes: (i) agent \(A\) (or agent \(B\)) experiences a cumulative advantage, and agent \(B\) (or agent \(A\)) experiences a cumulative disadvantage; (ii) both agents experience a cumulative advantage, but agent \(A\)’s (or agent \(B\)’s) time series of status is more strongly upward-slo** than agent \(B\)’s (or agent \(A\)’s) time series of status; or (iii) both agents experience a cumulative disadvantage, but agent \(A\)’s (or agent \(B\)’s) time series of status is less strongly downward-slo** than agent \(B\)’s (or agent \(A\)’s) time series of status. Hence, we have the same three causes as above when we observed the Matthew effect. The difference, of course, is whether agent \(A\) or agent \(B\) had the higher status in period \(t=0\).
Stated differently, there is no one-to-one correspondence between intraindividual change in status—that is, cumulative advantage or cumulative disadvantage—and interindividual convergence or divergence of the trajectories of the agents’ status—that is, in the case of divergence, the Matthew effect. The reason is that the intraindividual change in status is a microlevel phenomenon, whereas the interindividual change in status is a meso- or macrolevel phenomenon.
Defining cumulative advantage (disadvantage) and the Matthew effect: the case of many agents
How should the Matthew effect be defined if all \(N\) agents in the network are considered rather than only two of them? Bask and Bask [6] argue that after taking the average of the slopes of all of the time series showing how the distance between two agents’ status trajectories evolves over time, restricting attention to the pairs of status trajectories that were initially close to each other, the average slope should be positive when the Matthew effect is in play. Bask and Bask [6] also demonstrated that the positivity of the Lyapunov characteristic exponent, \(\lambda >0\), is the quantitative operationalization of the Matthew effect.Footnote 3
Hence, in the context of the status models presented in Sect. "Three status models", if the Lyapunov characteristic exponent is positive, \(\lambda >0\), then the Matthew effect occurs in the dynamic process described by a particular parameterization of the model. This means that the expected outcome of the model is an interindividual divergence of status trajectories, although individual agents may experience a mix of cumulative advantages and disadvantages in status attribution. It should be emphasized that a nonpositive Lyapunov characteristic exponent, \(\lambda \le 0\), does not mean that the Matthew effect is never observed among agents; it only means that the Matthew effect is not the expected observation.
Finally, a positive Lyapunov characteristic exponent, \(\lambda >0\), encapsulates what DiPrete and Eirich [14] argue is a well-defined inequality-generating process because it is “capable of magnifying small differences over time [because] it [is] difficult for an individual or group that is behind at a point in time […] to catch up” (p. 272). This means that a particular parameterization of a status model that is characterized by the Matthew effect is able to magnify small differences in status attribution over time. In fact, the Lyapunov characteristic exponent is a measure of the rate of separation of infinitesimally close (status) trajectories.
Measuring the Matthew effect
How is the Lyapunov characteristic exponent, \(\lambda\), measured? Essentially, there are two groups of methods in the literature for measuring the Lyapunov characteristic exponent: direct and Jacobian methods [28]. The feature in common in the first group of methods is that the definition of the Matthew effect presented by Bask and Bask [6] in a direct way is implemented in the estimation procedure (cf., [33]).
Unfortunately, direct methods might provide a biased estimate of the Lyapunov characteristic exponent, \(\widehat{\lambda }\), because these methods assume that the differences in status attribution among agents grow at a constant rate [17]. Hence, if the initial rate of separation of the status trajectories is slower than the rate of separation a few periods later, then the estimate of the rate of separation, \(\widehat{\lambda }\), of the status trajectories would be upwardly biased if too many separation periods are utilized in the estimation procedure of the Lyapunov characteristic exponent. The focus in the literature has therefore shifted from direct to Jacobian methods (cf., [20]).Footnote 4
Furthermore, even though the mathematical description of a particular status model has been specified, as in the present paper, the calculation of the Lyapunov characteristic exponent, \(\lambda\), by utilizing the Jacobians for this model might not be achievable from a practical point of view because the model is too complicated. For example, the models presented herein are potentially of high dimensionality. Moreover, the Heaviside step functions in the models might also complicate the calculation of the Lyapunov characteristic exponent. Therefore, we rely on a Jacobian method that assumes that the mathematical description of a specific model is not known when calculating the Lyapunov characteristic exponent.
At first glance, it might look like a limitation to not theoretically derive the value of the Lyapunov characteristic exponent, \(\lambda\), when, in fact, the mathematical description of the status model is known. However, this is not a limitation because we will employ an estimator of the Lyapunov characteristic exponent, \(\widehat{\lambda }\), that converges to the true value of the Lyapunov characteristic exponent in probability. That is, a consistent estimator of the Lyapunov characteristic exponent is used. Moreover, a statistical framework for inference will be used when examining the models presented herein for the Matthew effect.
For both groups of estimation methods—direct and Jacobian methods—one must first accurately reconstruct the dynamics in phase space to be able to estimate the Lyapunov characteristic exponent, \(\widehat{\lambda }\). In technical terms, this means that it is necessary to have a proper embedding in the sense that the dynamics are completely unfolded in the reconstructed phase space to be able to estimate the Lyapunov characteristic exponent. Two important contributions in embedology—Whitney [39] and Takens [38]—form the foundation for how one can properly reconstruct the dynamics in phase space and thereby be able to consistently estimate the Lyapunov characteristic exponent [34].
Specifically, Whitney [39] showed that a map** from an \(n\)-dimensional manifold to a \(2n+1\)-dimensional phase space is an embedding. That is, the image of the \(n\)-dimensional manifold is completely unfolded in a \(2n+1\)-dimensional phase space, with the implication that the dynamics in phase space can be properly reconstructed using \(2n+1\) independent measurements generated by the dynamic process. Almost half a century after Whitney [39] presented his embedding theorem, Takens [38] showed that instead of using \(2n+1\) independent measurements generated by the dynamic process, it is sufficient to use the time-delays of one measurement to have an embedding.Footnote 5 That is, a univariate time series is sufficient to accurately reconstruct the dynamics in phase space (see also [15]). This means that we need only a univariate time series generated by the status model to be able to determine whether a particular parameterization of the model is characterized by the Matthew effect.
To summarize, Bask and Bask [6] argue that a dynamic inequality-generating process—such as the dynamic processes described by the status models presented in this paper—is characterized by the Matthew effect when the Lyapunov characteristic exponent is positive, \(\lambda >0\). In other words, \(\lambda >0\) is the quantitative operationalization of the Matthew effect. Furthermore, although it would be possible in theory to derive the value of the Lyapunov characteristic exponent from the mathematical descriptions of the models presented herein, this exercise is not feasible in practice owing to the potentially high dimensionality of the models. For this reason, we rely on a numerical method when estimating the Lyapunov characteristic exponent, \(\widehat{\lambda }\).
The Matthew effect in the status models: a numerical analysis
The status models are implemented and simulated in MATLAB for a large number of model parameterizations and thereafter studied numerically using NETLE 4.1 for the Matthew effect in status attribution.Footnote 6, Footnote 7 The chosen parameterizations are presented in Sect. "Parameterizations of the status models", and the models are studied numerically in Sect. "Findings from the numerical analysis of the status models".Footnote 8
Parameterizations of the status models
Behavioral parameters in the status models: \({\varvec{h}}\), \({\varvec{w}}\) and \({\varvec{s}}\)
We vary the three behavioral parameters in the status models. The first is the parameter for an agent’s propensity to interact with status-dissimilar others, \(h\). This parameter determines the degree of homophily among agents, which is the tendency of agents to interact more with those who are similar to them. Specifically, when there are small differences among agents, the parameter values are \(h\in \left[\text{0.1,0.2}\right]\) or \(h\in \left[\text{0.8,0.9}\right]\), representing low and high propensities, respectively, to interact with status-dissimilar others. When there are large differences among agents, the parameter values are \(h\in \left[\text{0.1,0.5}\right]\) or \(h\in \left[\text{0.5,0.9}\right]\), once again representing low and high propensities, respectively, to interact with status-dissimilar others.
Regarding the parameters for an agent’s propensity to imitate other agents’ deferential gestures (i.e., a shortcut for assessing the skills of other agents), \(w\), and an agent’s sensitivity to attachment differences (i.e., it is painful to pay attention to another agent if the favor is not repaid), \(s\), we examine the same sets of parameter values as for an agent’s propensity to interact with status-dissimilar others. That is, when there are small differences among agents, the parameter values are \(w,s\in \left[\text{0.1,0.2}\right]\) or \(w,s\in \left[\text{0.8,0.9}\right]\); when there are large differences among agents, the parameter values are \(w,s\in \left[\text{0.1,0.5}\right]\) or \(w,s\in \left[\text{0.5,0.9}\right]\). For small (large) parameter values, agents have low (high) propensities to imitate other agents’ deferential gestures (\(w\)) and low (high) sensitivities to attachment differences (\(s\)).
Other parameters in the status models
In addition to the behavioral parameters, we vary three nonbehavioral parameters in the status models. The first parameter is the size of the network with agents, where the examined values are \(N=25\) and \(N=100\). The second parameter is the rate of downweighting of deference attributions received in previous periods, where the investigated values are \(d=-\text{ln}0.5\) and \(d=-\text{ln}0.9\), meaning that the weight given to deference attributions received in the previous period equals \(0.5\) and \(0.9\), respectively. The third parameter is the degree of perfection of agents’ perceptions of other agents’ skills, where the examined values are \({\sigma }_{e}=0.1\) and \({\sigma }_{e}=0.5\). Finally, the change in skill in the models with time-varying skills is \(\Delta Q=0.025\).
Findings from the numerical analysis of the status models
Main findings
What are the main findings after having studied the three status models numerically? See Tables 1, 2, 3, 4, 5, 6 for the presence of the Matthew effect in status attribution for different parameterizations of the models.
First, the Matthew effect in status attribution is not a generic property of the status model with constant skills for 85.9% of all examined parameterizations. Hence, for those parameterizations, the expected outcome of the model is not an interindividual divergence of status trajectories. However, this does not mean that the Matthew effect never occurs for those parameterizations; it only means that the Matthew effect is not the expected outcome of the model. Nevertheless, the Matthew effect is a generic property of the model and, for that reason, is the expected outcome for 14.1% of the scrutinized parameterizations.
The story differs regarding the status models with time-varying skills. When agents’ skills depend on their status, the Matthew effect in status attribution is a generic property of the model for 28.1% of all investigated parameterizations. Thus, the Matthew effect is twice as common in this model as in the model with constant skills. The story is even more striking when agents’ skills depend on their previous skills. In this model, the Matthew effect in status attribution is a generic property of the model for 43.0% of the scrutinized parameterizations, which means that the Matthew effect is trice as common in this model as in the model with constant skills.
Hence, by adding a learning-by-doing mechanism to the status model with constant skills, an interindividual divergence of status trajectories is much more often observed among agents. This finding should not come as a surprise because of the reinforcement effect that is present in the models with time-varying skills. That is, depending on the model, a more skilled agent or an agent with higher status has a higher probability of receiving research funding that improves her skill and increases her status. In contrast, once again, depending on the model, a less skilled agent or an agent with lower status has a lower probability of receiving research funding and therefore a higher probability of a deteriorated skill that decreases her status. Thus, we have an interindividual divergence of status trajectories.
For the two status models with time-varying skills, the reinforcement effect is stronger when agents’ skills depend on their previous skills. The reason is that when agents’ skills depend directly on their status, the skills are directly affected by deference attributions received from other agents, which, in turn, are affected by unrepaid favors. The latter mechanism weakens the reinforcement effect.
Behavioral parameters
To determine whether a specific behavioral parameter has a statistically significant effect on the probability of the Matthew effect in status attribution being present in a status model, a probit regression model is estimated for each of the three status models, where the behavioral and nonbehavioral parameters are the independent variables in the regressions.Footnote 9 See Tables 7, 8, 9 for the estimation results.
In the status model where agents’ skills are constant, two of the behavioral parameters have a statistically significant effect on the probability of observing the Matthew effect in status attribution. First, the probability of the Matthew effect is larger when agents’ propensities to imitate other agents’ deferential gestures (\(w\)) are low than when they are high (see Table 7). Hence, if agents imitate other agents’ evaluations of colleagues in the network to a lesser extent, then it is more likely that an interindividual divergence of status trajectories is observed than when agents imitate other agents’ evaluations to a greater extent. In other words, when agents give more consideration to other agents’ evaluations, the differences in the evaluations decrease, and the Matthew effect in status attribution is less likely to occur.
Second, the probability of the Matthew effect is also larger when agents’ sensitivities to attachment differences (\(s\)) are \(low\) than when they are high (see Table 7). Thus, if agents are less sensitive to unrepaid favors from colleagues in the network, then it is more likely that an interindividual divergence of status trajectories is observed than when agents are more sensitive to unrepaid favors. Stated differently, when agents are more sensitive to other agents’ nonreciprocal behavior and, for this specific reason, penalize such behavior to a greater extent, the differences in agents’ evaluations of colleagues in the network decrease, and the Matthew effect in status attribution is less likely to occur.
The parameter for the imitation heuristic in skill perception also has a statistically significant effect on the probability of observing the Matthew effect in the status model where agents’ skills depend on their status. Specifically, the probability of the Matthew effect in status attribution is larger when agents’ propensities to imitate other agents’ deferential gestures (\(w\)) are low than when they are high (see Table 8). However, in contrast with the status model where agents’ skills are constant, the parameter for the sour grapes heuristic in deference attribution (\(s\)) does not have a statistically significant effect on the probability of observing the Matthew effect in the status model where agents’ skills depend on their status.
In the status model where agents’ skills depend on their previous skills, none of the behavioral parameters have a statistically significant effect on the probability of observing the Matthew effect in status attribution. In fact, the overall fit of the probit regression model is not statistically significant (see Table 9). Finally, it is worth noting that the parameter for the sirens heuristic in partner selection (\(h\)) is not statistically significant in any of the three probit regression models (see Tables 7–9). That is, the degree of status-based homophily in a network of agents does not have a statistically significant effect on the probability of observing the Matthew effect in status attribution in any of the status models.
Discussion
Most people argue in favor of a meritocratic system in which research grants are awarded based on researchers’ skills rather than their status in the scientific community. Although it might often be the case that a highly skilled researcher is also a researcher of high status, and vice versa, this is certainly not always the case. One lesson we learned from the analysis in Manzo and Baldassarri [27] is the importance of social influence for the magnitude of the status-skill gaps among agents (but see [32] for an empirical study on how social influence can correct the discordance between popularity and quality, which translates into the discordance between status and skill in this paper). Nevertheless, we found that if grant-providing bodies focus on researchers’ skills rather than their status when financing research proposals, then this comes at the cost of a higher probability of observing the Matthew effect in status attribution. This was the general lesson learned from analyzing the status models that were developed herein in the tradition of Gould [21], Lynn et al. [26], and Manzo and Baldassarri [27].
Specifically, we found that the Matthew effect in status attribution is a generic property of the status model where skills depend on status (28.1% of all examined parameterizations) twice as often as in the model with constant skills (14.1%) and that the Matthew effect is a generic property of the model where skills depend on previous skills (43.0%) trice as often as in the model with constant skills. These findings are due to the reinforcement effect of the learning-by-doing mechanism. That is, some researchers experience a cycle of more research funding, improved skills, higher status, and even more research funding, whereas other researchers experience a cycle of less research funding, deteriorated skills, lower status, and even less research funding. The former researchers experience a series of cumulative advantages, and the latter researchers experience a series of cumulative disadvantages, resulting in the Matthew effect in academic recognition. The reinforcement effect is weaker when researchers’ skills depend directly on their status because status is affected by unrepaid favors from other researchers.
Regarding the impact of a specific behavioral parameter on the likelihood of the Matthew effect in status attribution, the following findings merit attention. First, the degree of status-based homophily does not have a statistically significant effect on the probability of observing the Matthew effect in any of the status models. Second, in the status model where researchers’ skills depend on their previous skills, none of the behavioral parameters have a statistically significant effect on the probability of observing the Matthew effect. In other words, the Matthew effect in status attribution is likely to be observed, irrespective of researchers’ behavior. Third, in the status model where researchers’ skills depend on their status, if researchers give more consideration to other researchers’ evaluations of colleagues in the network, then the differences in the evaluations decrease, and the Matthew effect is less likely to occur.
In other words, what we have provided herein is an epistemically possible how-possibly explanation for the Matthew effect in real-world scientific communities rather than just a possible explanation (cf., [22]). The explanation is epistemically possible because it is grounded in social psychology and able to produce the Matthew effect. However, a how-possibly explanation goes a step further than a mere possible explanation by constructing a model that illustrates how the Matthew effect occurs. This is also what we have offered herein—status models that under certain parametrizations generate the Matthew effect in academic recognition. Based on these models, we have concluded that advocating for a meritocratic system increases the likelihood of the Matthew effect in status attribution. Moreover, which is key to have a how-possibly explanation, we also discuss to what extent the behavioral parameters in the models affect the likelihood of the Matthew effect in academic recognition.
In the numerical analysis of the status models, we used the quantitative measure of the Matthew effect proposed by Bask and Bask [6]. While the social sciences have extensively discussed the Matthew effect across various domains, a quantitative measure had not been established in the literature prior to Bask and Bask [6]. This gap partly resulted from the absence of a shared and precise definition of the Matthew effect among scholars [14]. For example, some scholars have associated the Matthew effect with inequality. However, even though the Matthew effect is closely related to inequality, the measure in Bask and Bask [6], \(\lambda\), is not exactly an inequality measure. Instead, an inequality measure such as the Gini index measures the degree of inequality in status between agents at a specific point in time, whereas \(\lambda\) measures how the degree of inequality changes over time between agents with similar status. Hence, \(\lambda >0\) can be associated with both a low and a high Gini index. Our use of \(\lambda\) to explore the presence of the Matthew effect in a theoretical model is novel in the literature.
Although we have performed a methodical numerical analysis of the status models, a more thorough analysis of the models should be pursued in future research. First, we analyzed the models using only two spans of values of the behavioral parameters \(h\), \(w\) and \(s\); a small span of values (e.g., \(\left[\text{0.1,0.2}\right]\)) and a large span of values (e.g., \(\left[\text{0.1,0.5}\right]\)), representing small and large differences among agents, respectively. In a more in-depth analysis of the models, a greater number of value spans should be studied (e.g., \(\left[\text{0.05,0.1}\right]\), \(\left[\text{0.05,0.15}\right]\), …, \(\left[\text{0.05,0.95}\right]\)). Second, unlike the approach taken in this paper, one should explore combinations of value spans of varying lengths (e.g., spans of sizes \(0.05\), \(0.1\), etc., for \(h\) should be studied in combination with spans of sizes \(0.05\), \(0.1\), etc., for \(w\) and \(s\)). Third, even though we consider it to be of lesser importance, varying the nonbehavioral parameters in the models could also be beneficial. Notably, conducting a numerical analysis as described here requires substantial computational resources, although an empirical validation of the models might help in narrowing the interesting spans of values of the behavioral parameters and, thereby, the need for such resources (see the review of the literature on the empirical validation of agent-based models by [25]).
Finally, the status models need to be externally validated to obtain a better grasp of the generalizability of our findings. For example, we believe that the status models are also applicable in labor market contexts other than those of scientific communities. The agents in the models could be workers in the labor market with skills that are determined by their human capital, where either their status in the labor market or their task-specific skills determine whether they are given the opportunity to invest in their human capital via task-specific training. Moreover, the status models could be incorporated into a suitable labor market model for further analysis, including an empirical validation of the models.
The external validation of the status models and a more thorough numerical analysis of the same models are saved for future research.
Data availability
The data generated through simulations, as well as the software used to produce these data, are available upon request from the author.
Notes
That an agent’s status can take negative values is a result of how the status models are initialized (see also [27]). Specifically, according to (4), an agent’s status is a function of her skill that is generated by a normal distribution centered at zero (i.e., \(Q\sim N\left(0,{\sigma }_{Q}\right)\)). However, what is important here is not the absolute values of agents’ status but instead the relative values. That is, what is important is whether an agent has a higher or lower status compared with a peer agent.
In an alternative formulation of the sour grapes heuristic, the symmetry concern (cf., (10)) is applicable regardless of when agent \(i\) last interacted with agent \(j\), potentially with some downweighing based on the recency of the last interaction. The non-symmetry concern (cf., (11)) is then relevant only if the two agents have never interacted with each other.
When there are only two agents in the network, referred to as agent \(A\) and agent \(B\), the Lyapunov characteristic exponent, \(\lambda\), measures how the distance between the status trajectories change over time: \({\mathcal{D}}_{\left\{A,B\right\},t\ge 0}\cong \epsilon \bullet \text{exp}\left(\lambda \bullet t\right)\), where \(\epsilon >0\) is small because the initial distance between the two agents’ status, \({S}_{A,0}\) and \({S}_{B,0}\), is short. Hence, when \(\lambda >0\), we have an interindividual divergence of the status trajectories—that is, the Matthew effect—whereas when \(\lambda <0\), we have an interindividual convergence of the status trajectories.
To understand why Jacobian methods have been in focus regarding estimating the Lyapunov characteristic exponent, imagine the dynamic process \(f\left(\bullet \right)\) that amplifies the distance between two states of the process, \({S}_{t}\) and \({S}_{t}^{\prime}\), where the initial distance \(\Vert {S}_{0}-{S}_{0}^{\prime}\Vert\) between them is short: \({S}_{t}-{S}_{t}^{\prime}={f}^{t}\left({S}_{0}\right)-{f}^{t}\left({S}_{0}^{\prime}\right)\cong D{f}^{t}\left({S}_{0}\right)\left({S}_{0}-{S}_{0}^{\prime}\right)\), where \(D{f}^{t}\left({S}_{0}\right)=Df\left({S}_{t-1}\right)Df\left({S}_{t-2}\right)\cdots Df\left({S}_{0}\right)\) and \(Df\left(\bullet \right)\) is the Jacobian of \(f\left(\bullet \right)\). The question is then how the distance \(\Vert {S}_{t}-{S}_{t}^{\prime}\Vert\) between the states is amplified when time approaches infinity, \(t\to \infty\). To answer this question, the Lyapunov characteristic exponent is measured: \(\lambda \equiv \underset{t\to \infty }{\text{lim}}\frac{{\text{log}}_{e}\left|D{f}^{t}\left({S}_{0}\right)\right|}{t}=\underset{t\to \infty }{\text{lim}}\frac{1}{t}\sum_{\tau =0}^{t-1}{\text{log}}_{e}\left|Df\left({S}_{\tau }\right)\right|\). Under suitable conditions, the limits in the definition of \(\lambda\) exist and are independent of state \({S}_{0}\) [7, 15].
Takens’ embedding theorem (with more than 16,000 Google Scholar citations in April 2024) is a well-known theorem in mathematics and the natural sciences but is still a less-known theorem in the social sciences.
NETLE 4.1 was developed by R. Gençay, C.-M. Kuan and T. Liu, and is available at http://www.cs.bsu.edu/homepages/tliu/download/index.html, where Bask and Bask [5, 6] contain detailed descriptions of the estimation methodology in NETLE 4.1. Shintani and Linton [35] derived the asymptotic distribution of a neural network estimator of the Lyapunov characteristic exponent and thereby provided the statistical framework for inference that is used in NETLE 4.1.
An R package that employs an estimator of the Lyapunov characteristic exponent that converges to the true value of the Lyapunov characteristic exponent in probability is DChaos, available at https://cran.r-project.org/web/packages/DChaos/index.html.
Time series with 500 periods of the agents’ status are generated for each parameterization of each status model, where the first 100 periods are discarded from the time series. The Lyapunov characteristic exponent is then estimated using the time series for the agent with the status closest to zero in the initialization period and is estimated with 5, 6, 7 and 8 inputs to the neural network, where the number of hidden units, in each case, runs from 5 to 8 units. Hence, 16 neural networks are estimated for each parameterization of each model, where the estimate of the Lyapunov characteristic exponent associated with the neural network that minimizes the Schwarz Information Criterion is selected.
The results are qualitatively the same irrespectively of whether the nonbehavioral parameters are included or not among the independent variables in the regressions. Moreover, the results are also qualitatively the same if logistic regression models or linear regression models instead are estimated.
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Bask, M. Skill, status and the Matthew effect: a theoretical framework. J Comput Soc Sc (2024). https://doi.org/10.1007/s42001-024-00298-z
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DOI: https://doi.org/10.1007/s42001-024-00298-z