2.1. Models

The model used in Zollman (Reference Zollman2007, Reference Zollman2010) was first introduced by Bala and Goyal (Reference Bala and Goyal1998). These authors consider the general conditions under which a network of learning agents will adopt the same beliefs and the conditions under which these learned beliefs will be successful. We will describe the exact version of the model employed by Zollman (Reference Zollman2007) and then the version employed by Zollman (Reference Zollman2010).

To motivate the model, consider the following situation. A group of clinicians all use drug X, with a known success rate, to treat a certain disease, but a pharmaceutical company has recently released drug Y, with an unknown success rate, for the same purpose. Each clinician has some belief, based on previous experience with drug X and new data on drug Y, as to which is more successful. Clinicians begin to prescribe the drug they prefer, all the while noting the success rates of the new drug. At the same time, clinicians communicate with colleagues about their practices and so gain information about the new drug’s success that way as well. If, as a result of this information, a clinician becomes convinced that she is doing the wrong thing, she will switch drugs. In a setup like this, the agents will learn from their neighbors, as well as from their own choices, which action to take in the future. As such, the structure of their communication network may influence the outcome of their inquiry.

This sort of scenario can be modeled using what is called a “two-armed bandit” model as follows. Imagine a network of agents who may take one of two actions, A or B. A yields a good outcome with probability $p_A=0.5$. Action B yields a good outcome with probability $p_B=0.5+ɸ$ where ɸ ∈ [0, 0.5]. However, agents do not have full information about the success of action B. They believe that it is successful either with probability $0.5+ɸ$ or with probability $0.5-ɸ$. In other words, they are unsure about whether this action is better or worse than A.

Agents in this model start with some randomly chosen belief about whether A or B is preferable (whether $p_B=0.5+ɸ$ or $p_B=0.5-ɸ$). This belief is modeled as a number between 0 and 1. For example, an agent might have a degree of belief of 0.8, meaning that she thinks there is a 0.8 probability that B is the better action (that $p_B=0.5+ɸ$). Agents then repeatedly choose which action to take. In each round, agents use their degree of belief to choose an act. If it is less than 0.5, they choose A; if it is greater than 0.5, they choose B.

Upon making a choice in each round, an agent is rewarded with a payoff selected from a binomial distribution of parameters p and n. What this means is that in a single round an agent will select her preferred action n times, and each time she will get a payoff of ‘1’ with probability p. For example, suppose that $p_B=0.65$, and an agent chooses action B in a round. If $n=100$, that agent will pick B 100 times, each time receiving ‘1’ with probability 0.65.

After acting, agents observe their outcomes and the outcomes of neighbors in their network. They use Bayesian updating to alter their degrees of belief about A and B on the basis of the success rates of these actions. This sort of updating involves the application of Bayes’s rule to an agent’s prior degree of belief, using all the evidence gathered by an agent and her neighbors.

Upon simulating this model, agents in a network will tend to slowly converge to the same action. In other words, all agents will learn to choose A or all agents will learn to choose B. Zollman (Reference Zollman2007) ran simulations of this sort, ending the simulations either when all agents chose A (thus ending the acquisition of new information that might lead them to choose B) or when all agents had a very high degree of belief that B was the better action (making their reversion to A vanishingly unlikely).

Zollman (Reference Zollman2010) employs a slightly more sophisticated version of this model. In this version, agent beliefs are modeled using what is called a beta distribution. This is a distribution on [0, 1] where the value of the distribution at a point represents the agent’s belief that that number is the real success rate of choice B. At the beginning of a simulation, agents are given a random beta distribution with parameters α and β chosen from the interval [0, 4]. These parameters determine the shape of the distribution, although it is not important for our purposes to understand the details of how they do so. In each round of simulation, these agents then use their distribution to decide which action they think will be better. After acting, as before, they use Bayesian updating to alter their distributions on the basis of their outcomes and the outcomes of their neighbors. Zollman (Reference Zollman2010) runs these simulations for 10,000 rounds before stopping them to see what agents have learned.

2.2. Zollman’s Results

The central result of Zollman (Reference Zollman2007), which is replicated in Zollman (Reference Zollman2010), is that the network structure in these models influences inquiry in significant ways. In particular, Zollman argues that a sparser network structure can benefit an epistemic community. More connected networks converge to uniform beliefs more quickly, but sparse networks are more likely to converge to the true belief.

Zollman (Reference Zollman2007) achieves these results by running the models described above for agents in different network structures. First he considers three networks with 3–10 agents—the cycle, the wheel, and the complete graph. Figure 1 shows these three types of networks (with eight agents in this case). He also considers every possible network formation with six agents. Figure 1 also displays two examples of six-agent network formations. For both sets of networks, and for both types of models, Zollman finds that more connections speed the learning of the community but decrease the chances that the community learns to take the better action.

Figure 1.

Various network formations. A, an eight-agent network with complete graph, wheel, and cycle configurations respectively; B, two six-agent networks, one more sparse, the other more connected.

Why these effects? As Zollman argues, in highly connected networks, runs of unlucky results can spread quickly and convince an entire community to start taking the wrong action. In other words, the group can too quickly learn the wrong thing. In sparser networks, such results are only seen by a few agents, so a single unrepresentative batch of data cannot simultaneously convince the entire community to perform the inferior, uninformative action. It is more likely that a few agents will maintain the better belief and thus enable their neighbors to learn from their continued data collection. In a real epistemic community, this translates to pockets of interacting researchers who may maintain different sets of beliefs but eventually share successful ones.

Zollman (Reference Zollman2010) provides extended results using the second class of models. He shows that, in particular, it benefits communities of epistemic agents to hold a diversity of opinions for some period of time. In sparse networks, this is achieved because agents are less likely to change each others’ minds quickly. It can also be achieved, Zollman argues, by ensuring that agents begin simulations with more extreme beliefs, which are harder to shift.

2.3. Impact

Zollman (Reference Zollman2007, Reference Zollman2010) has been widely cited both in philosophy and the social sciences. In this section, we motivate the claim made in the beginning of the article that the best way to understand the main epistemic role of his work is as ‘how-potentially’ modeling. His results have been taken, in a number of cases, to suggest that a sparse network structure in epistemic communities may indeed potentially help promote successful inquiry. Our intent here is to underpin the importance of discussing the robustness of said results and the application of these results to real world communities.

‘How-possibly’ models have received a great deal of attention from philosophers of science. These models typically work by showing that some phenomenon can possibly result from (often surprising) preconditions. For example, Quine (Reference Quine and Lee1936, Reference Lee2013) famously argued that without established linguistic convention, it is impossible to assign meaning to terms. Skyrms’s (Reference Skyrms2010) work on the evolution of signals showed that, contra Quine, conventional meaning can emerge through simple processes of learning and evolution. In other words, in answer to the question, is it possible to generate linguistic convention without existing linguistic convention? Skyrms’s models provide a definitive yes. When it comes to how-possibly modeling of this sort, there is no need to ensure that modeling results are robust. All that is required from the model is evidence that a phenomena can occur under the right conditions, not that this phenomena is likely or that it is, in fact, one we might see in the real world.

Zollman’s models do not fall under the heading of how-possibly modeling. Neither, though, are they so clearly representative of real world communities that they can be taken as directly informative of these communities. Rather, they present a phenomenon that has the potential to occur in a real epistemic community. In doing so, they warrant or direct us toward further research in this area. Their interest to philosophy of science has been largely in this vein.

To support this claim, we provide a few quotes illustrating the way Zollman and others have discussed these models. Zollman (Reference Zollman2007), for example, concludes by saying that “this model suggests that in some circumstances there is an unintended benefit from scientists being uninformed about experimental results in their field. … When we want accuracy above all else, we should prefer communities made up of more isolated individuals” (586). Zollman (Reference Zollman2010) concludes with, “Looking at these scientists from the perspective of individualistic epistemology, one might be inclined to criticize the scientists’ behavior. However, when viewed as a community, their behavior becomes optimal” (33). In these quotes he describes the results as applying to ‘scientists’ and ‘communities’, entities from real epistemic groups. And the work is not described as how-possibly modeling, that is, to disprove a claim like ‘communication always improves science’.Footnote ²

Others have echoed this move. With respect to the result that less connectivity may help a research community, Strevens (Reference Strevens2010) writes: “It might, for example, be preferable for scientists not to take into account too much information about their colleagues’ beliefs about a problem, if a few authoritative pronouncements would stifle much-needed diversity in the range of approaches to a problem. … The question how to tune attention to authority in the short term so as to find a level of diversity that maximizes correctness in the long term has been explored with considerable insight by Zollman (Reference Zollman2007)” (20). Wray (Reference Wray2013) reports that “Zollman found that full communication in a research community is sub-optimal, as lines of communication in a group not only aid in the spread of truth, but also facilitate the spread of errors” (78).

As to the related result that diversity of beliefs among agents can promote inquiry, Douven and Kelp (Reference Douven and Kelp2011) say that “Zollman’s intriguing work shows that, from a socio-epistemic perspective, it may be important to maintain epistemic diversity in a community of agents, at least for a while, and that, for that reason, it is not always best if all agents have access to all information available in their community; for the same reason, a certain dogmatism on the part of the agents … may be beneficial” (277). Muldoon (Reference Muldoon2013) writes that “Zollman showed how transient diversity in beliefs, whether fostered by limited communication or stubborn scientists, can help ensure that scientific consensus tracks the truth” (123).

In the quotes above, authors describe these models as potentially informing ‘scientists’ and ‘communities’. Our goal here is not to claim that these authors are making any sort of mistake in describing Zollman’s results as they do. We are simply pointing out that the models are taken as potentially telling us something about the real world, rather than showing that some phenomenon is possible in principle. The question arises then: how seriously should we take these results? Should they point us to further research in this area? Should they increase our confidence that communication might lead to poorer inquiry? In the next sections we analyze the robustness of Zollman’s findings with an eye to these questions.

3.1. Success Rates

In Zollman (Reference Zollman2007, Reference Zollman2010), the probability that action A is successful is always $p_A=0.5$. The probability that action B is successful is always chosen to be $p_B=0.501$. This means that in models from Zollman (Reference Zollman2007), agents believe that B has a success rate of either 0.501 or 0.499.

Obviously, 0.5 and 0.501 are very close values. In this section, we show that in many cases as p _B increases by even a relatively small amount, the Zollman effect vanishes. The intuitive explanation for this is that when p _B is higher, trials of action A and action B are less likely to have similar results, so agents have an easier time distinguishing them. As a result, a sparse network structure does not help the community learn correct beliefs but still slows learning significantly.

Figure 2 shows outcomes of our simulations when we vary p _B but keep population size and n fixed at values considered in Zollman’s original models, 10 and 1,000 respectively. The x-axis tracks the value of $p_B-0.5$, the differential success rate for the better action. The y-axis shows, for the complete and cycle networks, how frequently they converge to the successful action.

Figure 2.

Simulation results showing the rates of convergence (i.e., portion of our simulations for which the agents converged to the better action) as a function of p _B for the complete and cycle networks of size 10 with n = 1,000.

As discussed, as p _B increases, the Zollman effect decreases. In other words, the difference in likelihood of successful convergence between the two network configurations decreases. This difference is 0 for $p_B\ge 0.525$. For $p_B=0.51$ there is a difference of less than 2% in the successful convergence rates. This occurs because, as p _B increases, all networks are able to successfully converge to correct beliefs with very high probability, so sparse networks are not particularly helpful.

Zollman (Reference Zollman2010) says, with respect to the Zollman effect and different values of p _B, “The degree of difference here should not be taken too seriously; it can be altered by modifying the difference in objective probabilities of the different methodologies. However, the ordering of the graphs remains the same—the cycle is superior to the wheel which is superior to the complete graph” (17). Zollman (Reference Zollman2007) includes the following footnote, “The results for both reliability and speed are robust for [the cycle, wheel, and complete networks] across … difference in payoff between the good and uninformative actions. Although these different modifications do effect the ultimate speed and reliability of the models, for any setting of the parameters the relationship between the three networks remains the same” (n. 7). His claim is that the effect occurs, to varying degrees, for all parameter values in these models. Given our findings, we think it more accurate to say that the effect does not arise for significant portions of the relevant parameter space. Note, though, that we never see the effect reverse so it may be that it holds but to such a slight degree that it is undetectable given the limits of our computational setup.

3.2. Number of Trials per Round

Figure 2 uses Zollman’s choices of n = 1,000 for the number of times per round that each agent tries her preferred action. In expanding the parameter space investigated for these models, we found that decreasing n actually increases the Zollman effect across models. Figure 3 shows the simulation results for a population of size 10 with p _B held fixed at 0.51, where n ranges from 10 to 10,000. As this figure shows, the difference in rates of successful convergence for the cycle and complete networks is more significant for lower n.

Figure 3.

Simulation results showing the influence of n, or the number of trials per round, on convergence for complete graphs and cycles for a population of size 10 with $p_B=0.51$.

In these simulations, for larger n, probabilistically it is more likely that the data obtained by each actor will reflect the true underlying probabilities of success for each action, and ditto for the combined information between actors. For smaller n, actors are more likely to receive strings of data where the less preferable action is successful, or the preferable one is unsuccessful, as a result of the law of small numbers. In a situation like this, network sparsity helps the entire community avoid these misleading strings.Footnote ⁴

3.3. Number of Agents

We also looked at larger populations than Zollman considered in his original papers. In these larger populations, we found that the significance of the Zollman effect, again, decreased. Once more, we ran simulations of complete and cycle networks for various parameter values and measured what proportion of simulations (again, out of at least 10,000) converged to the better action.

Figure 4 shows results of these simulations. For this figure we look at the Zollman effect for the parameters n = 1,000 and $p_B=0.501$, matching Zollman’s original models. The Zollman effect is strongest for smaller networks, but as network size increases it drops off. We find similar results for other parameter values as well. Note that while in this figure the effect occurs at all values of n, this is not so for many other parameter values.

Figure 4.

Rate of convergence by population size for complete graph and cycle, $p_B=0.501$, n = 1,000.

What is going on in this case? For these larger networks, the effects of connectivity are relatively unimportant because actors are able to gain so much information about their target inquiry from their larger cohort. Thus in almost every case, these larger networks converge to correct beliefs about the world. This ends up meaning that sparser networks do not give agents a notable advantage in terms of accuracy.

Again, sparse networks tend to learn slowly, and this is especially the case for large networks. Figure 5 shows the difference in speed of convergence on average for cycle versus complete networks. The cycle is always a slower learner, and this speed difference increases as population size increases, although the cycle’s advantage in rate of convergence diminishes. This makes sense, as in larger networks the tendency to perform action A takes longer to propagate along a cycle, and the difference in connectivity between cycle and complete networks is much larger. Thus as network size increases, the potential benefit of decreased communication goes down, and the cost (i.e., speed of convergence) associated with reducing network connectivity climbs dramatically. For a population of 100 agents, the convergence rate for the complete network is 99.12%, versus 100% for a cycle, while the average round at which the community achieves convergence is 20 for the complete network and 1,977 for the cycle.

Figure 5.

Speed of convergence in cycle and complete networks by population size, $p_B=0.501$. Vertical bars show 1 standard deviation.

One argument Zollman (Reference Zollman2007) gives is that there is a trade-off, in such networks, between learning quickly and learning well. More connected networks learn quickly but may fail to converge to correct beliefs, while sparser networks are more accurate but slower. When the Zollman effect is nonexistent, this trade-off, of course, also disappears. Sparse networks still learn more slowly than densely connected ones, but there is no particular benefit from sparse network structure.

To drive the point home, we include one more figure. Figure 6 shows the Zollman effect as a function of both the difference between the success rates of the two theories and population size for n = 1,000. As is evident from this figure, which only shows p _B up to 0.7, for the majority of the parameter space there is no effect.

Figure 6.

Zollman effect as a function of p _B and population size, n = 1,000.

5.1. When Does Network Structure Matter in the Models?

In Bala and Goyal style models of epistemic networks, less connectivity can improve the accuracy of learning in the long term, but this only happens for certain areas of parameter space. In particular, the effect occurs under circumstances in which learning is more ‘difficult’ for the agents in that the community has more trouble distinguishing between two alternative theories.

The learning situation can be more difficult in this way when the parameters have the following features:

1. 1. The two actions that agents may take are more similar in terms of success rates (low p _B);

2. 2. The population size is smaller; and

3. 3. The amount of data collected each round is smaller (low n).

In the first case, more data are necessary to accurately determine which action is the preferred one. In the second and third cases, agents receive less data each round. In all three cases, (relative) sparsity of data allows for misleading strings of information to sway epistemic communities to incorrect beliefs, meaning that there is an opportunity for network structure to influence inquiry.

To illustrate this claim consider figure 8. For each set of parameters—n, network size, and p _B—we average convergence times for the cycle and wheel network using our data from section 3. This average gives us a proxy for how difficult it is for networks of agents to converge under each set of parameter values. We then compare this measure of speed to the strength of the Zollman effect. The data are divided into smaller networks (6, 10, and 20) and larger ones (50, 100) since there is a significant difference in the strength of the effect in these two sets of networks. Note that the y-axis is on a logarithmic scale to make the trend more visible. As is evident, there is a clear correlation between networks that take more time to converge and those where connectivity matters. This supports the claim above that network structure matters more when inquiry is trickier.

Figure 8.

Average speed of convergence over cycle and complete networks as a function of Zollman effect size.

There is something unintuitive about this observation. When good information is harder to come by, this is exactly the situation in which, for these models, it is potentially useful to decrease the amount of information flowing between agents at each time step. The way to think about this is to observe that sparsity in epistemic networks can provide the very benefit that Zollman outlined (helping groups of agents to avoid preemptively converging to incorrect beliefs about the world), but it will only provide it when agents are already in a more difficult situation for inquiry. When agents have enough good data, decreasing connectivity only slows learning without providing any benefit.

In Kummerfeld and Zollman (Reference Kummerfeld and Zollman2016) type models, again, network effects only show up when it is more difficult for agents to distinguish between alternative theories. When the mean of the payoff distribution for the more successful action is similar to the payoff for the unsuccessful action, both a Zollman effect and a reverse effect are seen for different levels of exploration. As this mean becomes larger (i.e., the better action is more obviously better), the different networks become equally successful. When the standard deviation of the payoff distribution is large—when it is harder to determine that the better action is, in fact, better—again both sorts of network effects are seen. And as standard deviation decreases, networks matter less.

5.2. Real World Implications

As argued previously in the article, these sorts of epistemic network models can, and have, been employed as tools that outline effects that might potentially occur in real world communities. Now we ask: should we take the potential for decreased connectivity to improve community outcomes seriously as a result of these modeling results? Do these results mean that further effort should be expended on this potential effect?

First, obviously, our results imply that we should only be looking for this effect in certain types of epistemic communities—those where inquiry is more difficult for various reasons. Previous modeling work has, similarly, shown that different epistemic situations correspond to different optimal network structures (Zollman Reference Zollman2013). We see this with the Kummerfeld and Zollman (Reference Kummerfeld and Zollman2016) results, where in exploratory communities, connection improves inquiry. Likewise, Holman and Bruner (Reference Holman and Bruner2015) look at Bala and Goyal style networks where ‘biased’ agents simply attempt to sway the scientific community to their views. Again, in these models connected, rather than less connected, networks are more successful.

We also think that our results should decrease confidence, at least to some degree, that epistemic network effects of the sort observed in these models regularly happen in real world communities. Why? We do not have a good sense of which real world communities are well represented by which epistemic network models. This is because these models are highly simplified. They abstract away from many relevant details of such communities. This allows them to provide an explanatory clarity that might be absent from more detailed models, but it also means that the model-world match is loose at best. As a result, we cannot say with confidence that we expect real world epistemic communities to generally fall under the area of parameter space where the Zollman effect occurs. We are unsure whether they correspond to this area of parameter space, or some other area, or some other models with different assumptions. Our results, and previous results showing the sensitivity of epistemic network effects to structural changes, should then attenuate the degree to which we worry about network effects in real epistemic communities. We do not mean to make an overly strong statement here. This is especially the case because there is at least one other model that finds a Zollman-like effect under some structural changes (Grim Reference Grim2009) and two experiments with some evidence for a similar effect (Mason, Jones, and Goldstone Reference Mason, Jones and Goldstone2008; Jönsson, Hahn, and Olsson Reference Jönsson, Hahn and Olsson2015). But because we have identified many conditions under which the Zollman effect does not occur, and because we think many epistemic communities might fall under these conditions, our confidence in the importance of the effect decreases to some degree.

On the flip side, highly robust phenomena are more likely to have applicability to the real world because the conditions under which they occur are more likely to hold.Footnote ⁶ When epistemic network effects are highly robust, it makes sense to take them more seriously as important findings for real world communities. There are a few results from investigations of epistemic network models that do seem robust in this way. For example, the observation by Zollman (Reference Zollman2010) that transient diversity of beliefs is necessary to good scientific practice holds for all the models discussed. Our work suggests another possibility—that network structure matters only when inquiry is more difficult. Suggestively, Mason et al. (Reference Mason, Jones and Goldstone2008) find that when experimental subjects are tasked with searching a space for areas that yield high payoff, less connected networks seem to do better for more difficult searches.

There is one last thing to mention about the suggestion that, in some cases, it might improve inquiry to decrease communication among scientists. As noted in the last section, there is something unintuitive about the idea that in just those cases in which good data are hard to come by—small communities, small samples, theories that are hard to differentiate in terms of quality—agents should decrease the amount of data flowing between them. In fact, there are better solutions, in these cases, to the problem. More stubborn or exploratory agents, who keep investigating both theories, even if one seems less promising, will not preemptively settle on a poor theory. Standards for the amount of data necessary to confirm or deny a theory, such as those being implemented in the social sciences that face replicability crises, will likewise do the trick. These measures ensure that enough data go into each scientist’s judgment that a theory is or is not the best available one, while simultaneously allowing each scientist to take advantage of all the data available. In other words, there are ways to prevent preemptively settling on a poor theory apart from ignoring good data.