1 Introduction

Agent-based models (ABMs) have in recent years been increasingly utilized as a method for tackling socio-epistemic aspects of scientific inquiry [6, 18, 21, 23, 24]. The primary value of this approach is that it allows us to tackle questions that are difficult to answer with qualitative methods, such as historical case studies. One such question concerns the impact of different degrees and structures of information flow among scientists on their efficiency in acquiring knowledge. Zollman’s pioneering work in this domain [23, 24] suggested that a high degree of connectedness of a social network may be epistemically harmful, and that there is a trade-off between the success of agents and the time they need to reach a consensus. Even though it has been shown in [15] that this result, dubbed as “Zollman effect”, does not hold for a large portion of the relevant parameter space, structurally different ABMs have come to similar conclusions [9, 10].

The highly idealized nature of these ABMs makes it difficult to assess how relevant their findings are for actual scientific inquiry [15]. On the one hand, idealization and abstraction are necessary for simulations that aim at representing complex real world phenomena [14]. On the other hand, unless we include the most important ‘difference making’ factors figuring in the target phenomenon, the model might not represent any realistic scenario. Instead, it may represent only a logical possibility, uninformative about the real world.Footnote 1 Since due to their high degree of idealization such models operate at a significant representational distance from their intended target, we are faced with the methodological difficulty to judge their adequacy and epistemic merit. What we can do is to compare models with the same intended target that differ in various important respects, such as in the way they structure the phenomenon, in the way some of its elements are represented, in their degree of abstraction and idealization, etc. The possible gain may be that in virtue of this we are able to identify results that are robust under different modeling choices, and, based on that, to identify causes and explanatory relevant properties underlying these commonalities. Similarly, if specific types of models give rise to different outcomes, we may sharpen our understanding of their possible target phenomena and how they apply to them. For example, we may be able to identify distinct sub-classes of phenomena to which the different types of models apply.

The ABM presented in this paper has the same intended target phenomenon as the ones presented in [23, 24]. Nevertheless, the representation of social networks, the content of the exchanged information, and the behavior of agents is different, which raises the question whether the “Zollman effect” also occurs in our model. As will be demonstrated, it does not. Given the differences between our models, this result may not be surprising. Nevertheless, it suggests that further investigation is needed to determine to which target phenomena the results of each model apply.

Our model is based on a recently developed argumentative ABM of scientific inquiry (AABMSI) [3]. It represents scientific interaction as argumentative in nature. This means that instead of a simple information flow that lacks critical assessment, in AABMSI agents may refute previously accepted or new information in view of counterarguments. Second, information about a given scientific theory is represented as consisting of a set of arguments (rather than being fully aggregated in a single value, representing one’s credence in the given theory [23, 24] or the epistemic success of the theory [9, 10]). As a result, information sharing is represented as concerning particular parts of the given scientific theory, rather than an aggregated attitude about the whole theory. Third, receiving criticism triggers a search for defense of the given attacked argument. Fourth, the model takes into account that sharing information costs time and hence, that there is a trade-off between time spent on research and time spent on interaction.

In this paper we present an improved and more encompassing variant of AABMSI, aimed at examining the effects of different social networks on the efficiency of scientific inquiry. The model represents a situation in which scientists pursue different scientific theories, with the aim of determining which one is the best, and where they exchange arguments regarding their pursued theories. Compared to the model presented in [3], the current model introduces a number of improvements: (1) it employs the notion of social networks that is typically used in other ABMs of science, such as the complete graph, the wheel and the cycle [9, 10, 23, 24] – this allows for the representation of an increase in information sharing proportional to the population size, which is absent from AABMSI; (2) the heuristic behavior of agents is represented in a more adequate way; (3) an additional criterion of success has been introduced in order to examine the robustness of the results under different standards relevant in scientific practice; (4) the time cost of learning is now proportional to the amount of new information received by the agent; (5) the model has been computationally improved, allowing for statistical analysis of the data and for more reliable results.

Our findings suggest that a higher degree of connectedness leads to a more efficient inquiry, given a reasonable ratio between research time and time spent on communication. While our ABM is still too idealized to draw normative conclusions concerning scientific inquiry that would be useful, for instance, to policy makers,Footnote 2 it represents a step further in this direction.

The paper is structured as follows. In Sect. 2 we explicate the main features of our ABM. In Sect. 3 we present the main results of the simulations. In Sect. 4 we compare our model with an argumentation-based ABM introduced in [8], as well as with other ABMs of scientific interaction. We conclude the paper in Sect. 5 by suggesting further enhancements of our model and future research avenues.

2 The Model

Agents in our model represent scientists who inquire into a number of rivaling theories in a given scientific domain. Theories are represented by sets of arguments which scientists can discover and investigate. This is modeled in terms of an argumentative landscape where scientists can investigate arguments by spending time on them, which allows them to discover other arguments or argumentative attacks between arguments of different theories. In certain intervals scientists evaluate theories on the basis of their knowledge about the argumentative landscape and decide which theory to pursue further. Additionally, scientists are situated in communication networks. By communicating with other scientists they enhance their knowledge about the argumentative landscape which may help them to make more informed evaluations.

In the remainder of this section we describe the main components of our model in more detail: the underlying argumentative landscape, the behavior of agents and the notion of social networks.Footnote 3

2.1 The Argumentative Landscape

An abstract argumentation framework, as introduced by Dung [7], is a directed graph \(\mathcal {AF} = \left\langle \mathcal {A}, {\leadsto }\right\rangle \) where \(\mathcal {A}\) is a set of abstract entities called arguments and \({\leadsto }\) is an attack relation between arguments. Where \(a,b\in \mathcal {A}\), a attacks b if \(a \leadsto b\). A set of arguments \(\mathcal {S}\) is conflict-free if there are no \(a,b \in \mathcal {S}\) such that \(a \leadsto b\).

For our purposes it is useful to add another relation between arguments, the discovery relation \(\hookrightarrow \). It represents possible paths scientists can take to discover arguments:Footnote 4 if \(a\hookrightarrow b\), then b can be discovered by the scientists if a has been previously discovered.

A theory is represented by a conflict-free set of arguments connected in a tree-like graph by discovery relations. Argumentative attacks exist only between arguments of different theories. Formally, an argumentative landscape is a triple \(\mathcal {L} = \left\langle \mathcal {A},\leadsto ,\hookrightarrow \right\rangle \) where the set of arguments \(\mathcal {A}\) is partitioned into m theories \(\left\langle \mathcal {A}_1,\ldots ,\mathcal {A}_m\right\rangle \) such that for each \(i\in \{1,\ldots , m\}\) the theory \(T_i = \left\langle \mathcal {A}_i, a_i,\hookrightarrow \right\rangle \) is a tree with root \(a_i\in \mathcal {A}_i\) and

$${\leadsto } \subseteq \bigcup _{\begin{array}{c} 1 \le i, j \le m \\ i \ne j \end{array}} (\mathcal {A}_i \times \mathcal {A}_j) \quad \text{ and } \quad {\hookrightarrow } \subseteq \bigcup _{1 \le i \le m} (\mathcal {A}_i \times \mathcal {A}_i).$$

This definition of the attack relation \(\leadsto \) ensures that each theory is conflict-free and that attacks only occur between members of different theories.

At the beginning of the simulation only the roots of each theory (representing a basic underlying hypothesis) are visible to the agents. During the simulation the agents gradually discover arguments and attacks between them (see also Sect. 2.2). Each argument a has a degree of exploration: \(\mathfrak {expl}(a)\in \{0,\ldots , 6\}\) where 0 means that the argument has not been discovered and 6 means that the argument has been fully researched and it cannot be explored further. By spending time on an argument a, a scientist increases its degree of exploration. The higher the degree of exploration of a, the higher is the likelihood that the connected arguments will become visible. This concerns both arguments connected to a via the discovery relation in the same theory and arguments that attack a or that are being attacked by a. If \(\mathfrak {expl}(a) = 0\) [resp. \(\mathfrak {expl}(a) = 6\)] no [resp. all] relation[s] from a to other arguments have been discovered.

Agents only have subjective knowledge of the landscape, which consists of arguments, their respective degrees of exploration, and attacks. During the simulation agents gain knowledge, on the one hand, by exploring the landscape, and on the other hand, by communicating with other agents. This way two agents may have different knowledge about the degree of exploration of an argument or about discovered relations. See also Sect. 2.3.

2.2 Basic Behavior of Agents

Our model is round based. Each round every agent performs actions that are among the following:

  1. 1.

    the agent investigates the argument a she is currently situated at (i.e. she increases \(\mathfrak {expl}(a)\)) and while doing so she gradually reveals outgoing discovery relations as well as attacks from and to a;

  2. 2.

    the agent explores her current branch of the theory further by moving along a discovery relation to a neighboring argument (that she can see);

  3. 3.

    the agent leaves her current theory and moves to an argument of a rivaling theory (that she can see).

Every round each agent decides (based on a certain probability) whether to stay on her current argument (option 1) or to move to a new argument in her direct neighborhood (relative to the discovery relation, option 2). If she has reached a leaf of her branch, which is fully explored (i.e., \(\mathfrak {expl}(a)=6\)), she backtracks on this branch to find an argument that is not fully explored. In case this fails, she moves to another not fully explored argument in the same theory.

Additionally, every 5 rounds agents consider whether they are still working on the theory they consider the best (with respect to their current subjective knowledge of the landscape). Depending on this decision, they continue with 1 and 2, or move to an alternative theory and start to explore that one (3). Their decision is based on an evaluation of the degree of defensibility of a theory.

The degree of defensibility of a theory is the number of defended arguments in this theory, where –informally speaking– an argument a is defended in the theory if it is not attacked or if each attacker b from another theory is itself attacked by some defended argument c in the current theory.

Let us give a more precise formal definition. First, we call a subset of arguments A of a given theory T admissible iff for each attacker b of some a in A there is an \(a'\) in A that attacks b (we say that \(a'\) defends a from the attack by b). Since every theory is conflict-free, it can easily be shown that for each theory T there is a unique maximally admissible subset of T (with respect to set inclusion). An argument a in T is said to be defended in T iff it is a member of this maximally admissible subset of T.Footnote 5 The degree of defensibility of T is equal to the number of defended arguments in T.

Fig. 1.
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Argumentation Framework 1

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Fig. 2.
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Argumentation Framework 2

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Example 1

Figure 1 depicts a situation with three theories as it might occur from the perspective of a given agent: \(T_1\) consisting of arguments e and f (white nodes), \(T_2\) consisting of arguments a, b and g (gray nodes), and \(T_3\) consisting of arguments c and d (dark gray nodes). The arrows represent attacks, we omit discovery relations. We are now interested in the degrees of defensibility our agent would ascribe to the given theories. The table shows which arguments are defended in each theory and their corresponding degree of defensibility. The only defended argument in this situation is f in theory 1. Note for instance that in \(T_3\) the argument d is not defended since no argument in \(T_3\) is able to defend it from the attack by b. Although the argument f in \(T_1\) attacks b, it doesn’t count as a defender of d for theory \(T_3\) when determining the defended arguments in \(T_3\) since in our account a theory is supposed to defend itself.

Figure 2 depicts the situation after an attack from a to f has been discovered. Consider theory \(T_2\). In this situation a defends b from the attack by f, b defends a from the attack by d, a defends g from the attack by e and g defends a from the attack by c. Hence, all arguments are defended resulting in a degree of defensibility of 3.

A scientist decides to work on another theory \(T'\) instead of her current theory T if the degree of defensibility of \(T'\) surpasses the one of T by a certain margin. We represent the objectively best theory as a fully defended one.

Moreover, agents are equipped with heuristic abilities. An agent that encounters an attack from an argument b on the argument a, she is currently working on, will try to find a defense for this argument. For this she will consider all the arguments in her subjective knowledge that belong to the same theory as a. If there is an argument \(a'\) among these that can potentially defend a from the attack, she will begin to investigate it. That \(a'\) potentially defends a from b means that there is an attack from \(a'\) to b but this attack has not yet been added to the subjective representation of the landscape of our agent (e.g., since \(\mathfrak {expl}(a')\) is too low or since it has not been communicated to her). This means that agents are equipped with ‘professional hunches’ which help them to tackle problems in their theories.Footnote 6

2.3 Social Networks

Besides discovering the argumentative landscape by exploring it on her own (see Sect. 2.2), an agent can share information about the landscape with other agents.

At the start of a simulation agents are divided into local collaborative networks, each consisting of exactly five individuals working on the same theory. During the simulation each agent gathers information (i.e., the degree of exploration of arguments, discovery and attack relations) on her own. Agents of the same collaborative network have the same subjective knowledge of the landscape since whenever an agent learns something new, this is communicated with the other agents in the same collaborative group.

Additionally, the collaborative groups form a community network. These have one of the following three structures: a cycle, in which each collaborative group is connected to exactly two other groups, a wheel which is similar to the cycle, except that a unique group is connected to every other group, and a complete graph where each group is connected to all other groups (Fig. 3). Every five rounds, randomly chosen representative agents of the collaborative groups communicate along the communication channels of the community network. The different network structures allow us to represent varying degrees of information flow in the scientific community, with the cycle representing the lowest and the complete graph the highest degree of information sharing.Footnote 7

Fig. 3.
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A cycle, a wheel and a complete graph. Each node is a collaborative group, while the edges represent communication channels

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Representative agents do not share their whole knowledge of the landscape with agents from other collaborative networks. Instead, they share the knowledge they have obtained recently by their own exploration of the landscape, which consists of the argument they are currently at, its neighboring arguments connected via the discovery relation, their respective degrees of exploration, and attacks to and from their current argument. One way of interpreting this limited knowledge sharing is by considering this to be a situation where the agent writes a paper or gives a presentation on her current research results.

An alternative interpretation of collaborative groups is that they represent one individual working on 5 different arguments (papers, hypotheses, etc.) in a given theory. Whenever this individual communicates with other individuals from the community network it exchanges the information from the neighborhood of one of the 5 arguments she is currently engaged with.

Our model takes into account that information sharing and especially receiving information is time costly: agents who receive information (the representative agents of each collaborative network) are blocked from further exploration for a certain number of rounds. The costs of information sharing are proportional to the amount of new information obtained.Footnote 8

Finally, we distinguish two types of information sharing, characterizing two types of scientists. A reliable agent shares all her recent discoveries, whereas a deceptive agent withholds the information about attacks on her current theory. This way, agents receiving information from a deceptive agent from theory T might come to a more favorable interpretation of T than if they would have communicated with a reliable agent [4].

3 The Main Findings

We will now specify the parameters used in the simulations and then present our most significant results.

3.1 Parameters Used in Simulations

We have run simulations with a landscape consisting of 3 theories. The landscape is created in three steps: First, each theory is represented by a tree (as explained in Sect. 2.1) of depth 3 such that each node has 4 children (except for the leaves) resulting in 85 arguments per theory. Second, with a chance of 0.3, each argument gets randomly attacked by an argument from another theory. Third, for every argument in theories \(T_2\) and \(T_3\) that attacks an argument in theory \(T_1\) and that is not attacked by an argument from \(T_1\) we add an attack from some random argument in \(T_1\). Thus, it is made sure that \(T_1\) is the objectively best theory and as such is fully defended from all attacks. In this way we wish to represent a scenario in which theories that are rivals to the best one are worse, though not completely problematic (as it would be the case with pseudo-scientific theories).

Simulations were run 10.000 times for each of the scenarios with 10, 20, 30, 40, 70 and 100 agents. The scenarios are created by varying:

  1. 1.

    the community network: in the form of a cycle, a wheel and a complete graph;

  2. 2.

    two types of information sharing: reliable and deceptive.Footnote 9

A simulation stops when one of the theories is completely explored. At this point all the agents have one more chance to make their final evaluation and choose their preferred theory. We then evaluate whether the agents have been successful according to the following criteria (where \(T_1\) is the objectively best theory):

  1. 1.

    monist criterion, according to which a run is considered successful if, at the end of the run, all agents have converged onto \(T_1\);

  2. 2.

    pluralist criterion, according to which a run is considered successful if, at the end of the run, the number of agents working on \(T_1\) is not smaller than the number of agents working on any of the other theories.

The monist criterion is the standard notion of success, often employed in other ABMs of science, e.g. [23, 24]. The pluralist criterion, on the other hand, is motivated by the philosophical conception of scientific pluralism, according to which a parallel existence of multiple theories in a given scientific domain is considered epistemically and heuristically beneficial e.g. [5]. This means that the convergence of all scientists onto the objectively best theory isn’t a primary epistemic concern for pluralists. Rather, what matters is that the best theory is one of the most actively researched theories.Footnote 10

3.2 Results

In this section we describe the most important findings of the simulations.

Fig. 4.
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Pluralist criterion of success, reliable and deceptive agents

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Fig. 5.
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Monist criterion of success, reliable and deceptive agents

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Fig. 6.
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Time needed

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Reliable vs. deceptive agents. With respect to both criteria of success, reliable agents are clearly more successful than the deceptive ones (Figs. 4 and 5), while being only slightly slower (Fig. 6).Footnote 11

The degree of connectedness. In case of reliable agents, the complete graph tends to outperform the wheel and cycle, with respect to both criteria of success (Figs. 4 and 5), as well as with respect to the speed of exploration (Fig. 6). In other words, a higher degree of connectedness tends to lead to a more efficient inquiry.

In case of deceptive agents, the situation is a bit trickier. On the one hand, higher degrees of connectedness are also beneficial for the success according to the monist criterion (Fig. 5). However, the effect of connectedness is inverse for the pluralist criterion given populations with up to 70 agents (Fig. 4). A possible explanation for this asymmetry between the two success criteria is that deceptive agents in more connected networks share more false positives. As a result, if one theory is explored by a larger number of agents than either of the other theories, for the agents on this theory it will be easier to attract the whole population to it, leading to a fast, possibly wrong, convergence. In contrast, deceptive agents in the less connected networks will spread less information among each other, resulting in fewer cases of wrong convergence. For populations larger than 70 agents these premature convergences of the complete graph are prevented by the fact that each theory has on average enough researchers praising it and pointing out problems with the other theories so that the false positives are debunked as such more often.

Size of the community. Larger populations tend to be more efficient if agents are connected in the complete graph. In contrast, larger populations of agents in less connected networks perform similarly to smaller ones with respect to the pluralist criterion, and drastically worse with respect to the monist criterion. The latter results from the lack of sufficient information flow, which is why larger populations fail to converge on any (including the best) theory.

4 Discussion

In this section we will first comment on some of our results and then we will compare our model with other ABMs designed to tackle similar questions.

We focus on two particularly interesting results: the impact of deceptive information sharing in contrast to the reliable one, and the impact of the degree of connectedness on reliable agents. Concerning the former, deceptiveness has not been studied in ABMs of science to a large extent.Footnote 12 This is, however, an important issue for the efficiency of scientific inquiry. For instance, withholding results that would undermine one’s theory, typical for scientific fraud, is an example of deceptive information sharing. Even though deception is clearly problematic from the perspective of ethics of scientific conduct, its effects on the efficiency of the given scientific community aren’t immediately clear. For instance, one could assume that presenting one’s theory in positive light, in spite of the early problems can attract new researchers and help in developing it further. Our findings suggest that this is in general not the case, i.e. that deception tends to be epistemically harmful. More precisely, assuming that the whole community consists of deceptive scientists, and that scientists prefer theories that have the highest number of defensible results, deception leads to significantly less successful (though slightly faster) inquiry. Whether these results hold also under other assumptions remains to be examined in future research.

Our findings suggest that increasing the degree of connectedness of the communication network tends to be epistemically beneficial. This contrasts with findings obtained in other ABMs [9, 10, 23, 24], according to which agents connected in a cycle perform better than agents connected in a complete graph. In order to see whether and in which sense our results challenge the latter results we first need to highlight some differences between our approach and these ABMs.

First, while in the latter agents are directly connected in the given networks, we employ a more structured approach by distinguishing between collaborative (local research) groups and communal networks between these groups. Note that in the real world it is impossible for each member of a larger scientific community to invest time in communicating with all the other members for the simple fact that communication (such as, e.g. reading papers) costs time, which could otherwise be spent on doing research. Thus, we have started from the assumption that scientists only have limited time for interacting with members of other collaborative groups, while they fully share information within their own collaborative projects. Altogether this means, however, that the highest degree of connectedness examined in our model gives rise to less information flow than in models in which agents are directly linked into a complete graph. Second, the content of the information shared among our agents is more localized and patchy since only representative agents exchange information about local aspects of their theory, namely their current argument and its neighborhood. As a result, agents from different collaborative groups end up having more diverse subjective representations of the landscape than e.g. agents in [23, 24]. Third, our representation of interaction includes a critical component. This is important for a model designed to examine the efficiency of scientific knowledge acquisition due to the fact that criticism has been shown to be truth conducive since it allows for false beliefs to be exposed as such [2]. Finally, similarly to ABMs that employ epistemic landscapes [9, 10, 21], our argumentative landscape allows for the representation of the process of discovery and its timeline. However, unlike the models employing epistemic landscapes, the information that agents encounter via an argumentative landscape is defeasible. This feature allows not only for the representation of critical interaction, but also for specific heuristic behaviors,Footnote 13 such as the search for a defense of an argument in case it has been attacked.Footnote 14 In this sense, our respective results might concern a different kind of scenario and thus refer to different target phenomena.

Having explicated the differences between our and other ABMs, it is important to notice though that there is no reason to assume that our model introduces more problematic idealizing assumptions than the previous ABMs of scientific interaction when it comes to the representation of a typical scientific inquiry. To the contrary, it includes a number of assumptions directly relevant for its adequate representation. Thus, our findings suggest that the results obtained by others models might not hold for usual cases of scientific inquiry. Instead, they may hold only for some very specific contexts. Which subclass of the phenomenon of scientific inquiry each of these models reliably represents, remains to be tackled in future research.

Finally, let us compare our model with Gabbriellini and Torroni’s (G&T) ABM [8]. Their aim is to study polarization effects, e.g., in online debates. Similarly to our approach, their model is based on an abstract argumentation framework. Agents start with an individual partial knowledge of the given framework and enhance their knowledge by means of communication. Since G&T do not model inquiry, their agents cannot discover new parts of the graph by means of ‘investigating’ arguments. Rather, they exchange information by engaging in a dialogue. This way, agents may learn about new arguments and attacks but also remove attacks. Whether new information is incorporated in the knowledge of an agent depends on the trust relation between the discussants. The beliefs of agents are represented by applying Dung-style admissibility-based semantics to the known part of the argumentation framework of an agent. This is quite different from our model where the underlying graph topology is given by several discovery trees of arguments representing scientific theories and attacks between them. This additional structure of the argumentation graph is essential since we do not model the agents’ beliefs in individual arguments but rather evaluative stances of agents that inform their practical decision of which theory to work on. While an admissibility-based semantics would lead to extensions that feature unproblematic sets of arguments from different theories (ones that form conflict-free and fully defensible sets), in our approach agents pick theories to work on. For this, they compare the merits of the given theories, pick the one that is most defended, and employ heuristic behavior to tackle open problems of theories. It will be the topic of future research to include dialogue protocols that are relevant for scientific communication, such as information-seeking, inquiry and deliberation dialogues [20].

5 Conclusion

In this paper we have presented an argumentative ABM aimed at modeling the argumentative nature of scientific inquiry. The model is designed to examine how different kinds of social networks affect the efficiency of scientists in acquiring knowledge. Under the assumption that, in order to conduct their inquiry, scientists only have limited time to spend on communicating with others, our results suggest that a high information flow tends to be epistemically beneficial.

A variety of enhancements can be added to our model in order to make it apt for tackling similar or related questions. First, our current notion of the degree of defensibility represents scientists who prefer theories that exhibit a greater number of defensible results than their rivals. An alternative notion of defensibility would punish theories for having more anomalies (indefensible arguments) than their rivals, thus representing scientists who stick to their theories as long as they are not too anomalous (irrespective of how many positive results they have). Second, adding an explanatory relation and a set of explananda [16] would allow for a more refined representation of the desiderata of scientific theories and evaluative procedures which agents perform when selecting their preferred theory (e.g. in addition to the degree of defensibility, agents can take into account how much their current theory explains, or how well it is supported by evidence). Furthermore, a number of enhancements available from the literature on argumentation frameworks, such as probabilistic semantics [17], values [1], etc. can be introduced in future variants of our ABM. In addition to examining the impact of social networks, the model can be used to examine different heuristic behaviors and evaluations that guide scientific inquiry.