Computational learning mechanisms of information propagating in
social networks

Short title: Learning Rules in Social Networks

Julien Benistant<fnref n="1" />, <fnref n="2" />, Alireza Soltani<fnref n="3" />, Frédéric Moisan<fnref n="4" /> and Jean-Claude Dreher<fnref n="1" />, <fnref n="5" />

Affiliations

Corresponding authors: julien.benistant@univ-lille.fr; dreher@isc.cnrs.fr

Competing interests

The authors declare no competing interests.

Authors' contributions

J.B. created the models, performed model simulation, and analyzed the experimental and simulated data.

A.S. developed mathematical formulations and analyzes describing the relationship between the models.

J.B., A.S., F.M., and J.-C. D. wrote the paper.

Acknowledgement

This research has benefited from the financial support of IDEXLYON from Université de Lyon (project INDEPTH) within the Programme Investissements d'Avenir (ANR-16-IDEX-0005) and of the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-007) operated by the French National Research Agency. This work was also supported by grants from the Agence Nationale pour la Recherche to JCD (ANR-24-CE37-4261-01, ANR-24-NEUC-0001 and ANR-24-EXEN-0001), by MITI 2020 CNRS to JCD, and NSF CRCNS grant (BCS2423824) to A.S and JCD. We would like to thank Edmund Derrington and Kenan Huremovic for feedback on an early draft of the paper.

<footnote n="1">¹ Laboratory of Neuroeconomics, Institut des Sciences Cognitives Marc Jeannerod, CNRS, Lyon, France</footnote>

<footnote n="2">²Current affiliation: Univ. Lille, CNRS, IESEG School of Management, UMR 9221 - LEM - Lille
Économie Management, F-59000 Lille, France</footnote>

<footnote n="3">³ Department of Psychological and Brain Sciences, Dartmouth College, Hanover, NH, USA</footnote>

<footnote n="4">⁴ EmLyon Business School, GATE, CNRS, Lyon, France</footnote>

<footnote n="5">⁵ UFR Biosciences, Université Claude Bernard Lyon 1, Lyon, France</footnote>
**Abstract**

What mechanisms underlie how individuals integrate disparate and redundant information that spreads in social networks? The DeGroot (DG) model suggests that we adjust our beliefs according to the weighted average opinions of our social connections. In contrast, reinforcement learning (RL) models assume that learning occurs sequentially, depending on error-driven belief updating. Here, we compare these models directly using two datasets of information propagating in social networks, and determine which learning rule best accounts for the integration of information spreading in social networks. We found that variants of the DG model provide a better overall fit across participants, and are more effective in building consensus, and generating accurate estimates of the true state of the world. Furthermore, simulated models show how social network topology affects choice accuracy and consensus, depending on the learning rule. Together, our findings characterize the computational mechanisms used by individuals to propagate information in social networks.
Introduction

The rise of social media platforms has sparked great interest in understanding how social
networks contribute to the transmission of (mis)information across society. Research in network
science, computational social science and behavioral economics games played on networks has
started to explore the mechanisms underlying learning from information propagated in networks¹⁻
⁵. It is important to understand these mechanisms because it may help elucidate social phenomena
regarding opinion formation and dynamics⁶ such as the spread of fake news⁷, political consensus
and polarization⁸,⁹, vaccine hesitancy, voting behavior, financial markets¹⁰, and consumer
preferences¹¹. One important characteristic of information propagating through social networks is
that it is often entangled with other pieces of contradictory or redundant information. A
fundamental question is to determine the learning mechanisms used by individuals who interact
dynamically and repeatedly in real time in social networks. In such situations, our beliefs and
opinions are shaped by the information we receive from others, and in turn they also shape
collective dynamics over time. A further important question is to clarify how these social learning
mechanisms shape collective performance and consensus. Answers to these questions should shed
light on the computations that underlie how individuals integrate and spread information, including
disinformation and misinformation, in our interconnected world⁵,¹²⁻¹⁷.

The Degroot model has provided an influential model of social learning, and assumes that the transmission of information in social networks can be described by a simple heuristic based on weighting the opinions of direct connections¹⁸. More specifically, individuals update their beliefs by taking a weighted average of their own beliefs and those of their neighbors in the network (e.g., based on social relationships, social interactions and observations). A standard assumption in this case is that individuals assign equal weights to each of those beliefs. This computation is independent of the order that observations are made, and can occur all at once, similar to taking a snapshot. Importantly, recent behavioral economics studies have demonstrated that this simple heuristic effectively captures belief convergence within networks during dynamic communication¹⁹,²⁰, particularly in large networks²¹. The Degroot model that incorporates weighted averaging¹⁸,²², whereby the weights depend on the degree (i.e., number of connections of each neighbor), was supported by experimental evidence²³,²⁴. Although the Degroot averaging (weighted or unweighted version) has been one of the most prominent formalisms for social learning and opinion formation dynamics, it assumes that all agents update their opinions simultaneously, which may hold only in some controlled scenarios, in which the model is tractable. However, in many real-world situations, individuals do not update their opinions simultaneously²⁵. Instead, opinion updating often occurs asynchronously, with different agents updating their opinions at different times. This can be driven by the individuals’ environment itself or by some form of cognitive limitation to the aggregation of information from many sources.

An alternative, plausible social learning mechanism is based on reinforcement learning (RL), which has proven to be useful in decision neuroscience for its ability to model social learning from experience and from observation<sup>26-28</sup>. However, most social learning studies that use the RL framework have either studied strategic interactions using economic games in isolated dyads or small groups<sup>29-35</sup> or have studied contexts where individuals learn from observing the decisions of observees, who are performing the same task as the individual<sup>36-38</sup>. These studies have ignored the fact that in social networks, learning is distributed (i.e., we learn from and are being learned by others simultaneously), dynamic, entangled (i.e., sources may be more or less correlated) and heterogeneous (i.e., one's location in the network frequently constrains the quality or amount of information available)<sup>39,40</sup>. Taking these important aspects into consideration, a recent pioneering
study extended the RL framework to describe social learning in networks²⁴. It reported that learning
in social networks can be accounted for by sequential updating based on the difference between
one's own and others' beliefs, indicating that social learning is error-driven, similar to RL²⁴. Thus,
according to this study, the learning rule used in social networks to integrate private and propagated
information relies on a RL rule that takes into account relatively higher weight for more well-
connected neighbors. Note that the terminology used in the current paper differs from this previous
study, which did not test the classical DeGroot mechanism (see **Table S1** for a correspondence
between our nomenclature and the original names from the models in Jiang et al., 2023).

Given that error-driven learning in RL is computationally distinct from the belief or opinion
averaging in the DeGroot model, it remains unclear which of these two models can best explain
learning from information disseminated through social networks. To identify the computational
mechanisms that underlie learning in social networks, we focused on two datasets from similar
learning tasks in social networks (Fig. 1A). In both tasks, participants initially estimated the state
of the world (i.e., identifying the selected bag of marbles) based on their own private observation
and made an initial guess (Stage 0). Subsequently, over multiple stages, they were shown their
neighbors' guesses within the network and could revise their previous guess after each stage based
on this new information²¹,²⁴. Thus, these tasks involved gradual belief updating, based on private
observations and the choices of other individuals to whom a given participant was connected in the
network. Importantly, unlike many experimental studies that investigate social learning using
static, often simulated, sources of social information³²,⁴¹⁻⁴³ (e.g., social information that either
comes from earlier participants or is artificially manipulated), these tasks allowed us to capture the
complexity of multiple individuals who interact dynamically and repeatedly in real time, to allow
them to influence one another. Indeed, to determine how different social learning mechanisms
shape collective performance (e.g., consensus), we need to study the use of social information in
networked environments where social information arises endogenously from the behavior of
individuals who are interacting in the same network³⁸. Very few studies have taken such a dynamic
approach to study how cognitive processes shape, and are shaped by, collective dynamics over
time.

We first determined whether averaging or sequential error-driven updating better explained
learning in social networks. We compared variants of the DG and RL models to fit the data from
the two published behavioral datasets described above²¹,²⁴. The advantage of these two datasets,
with multiple participants interacting online in real time, is that they were acquired in well-
controlled experiments. This design allows for full knowledge of: (a) the full history of information
propagation, which is unavailable for real social networks; (b) the network properties (e.g.,
topology, size, relative degree, etc.). However, importantly, each of these experimental studies
tested only one of the two models. Jiang et al (2023) considered error-driven learning inspired by
the weighted DeGroot model, without testing the classical version of the DeGroot model, which
assumes simultaneous updating of opinions across agents. Choi et al (2023) tested the DeGroot
model but did not evaluate RL's ability to capture choice behavior. It is thus important to compare
directly how these learning rules (RL vs DG), and some variants of them, account for experimental
social network data.

In addition, the topology of social networks may have a substantial impact on how we form
beliefs and make decisions. For example, different social network structures have been reported to
change the dynamics of norm formation, to determine the reciprocity that is achievable in
exchanges, and to affect how collective opinion and behavior are shaped ¹,⁴⁴⁻⁴⁷. Thus, we also
investigated whether distinct social network topologies affected the adoption of different learning
rules, individuals' accuracy and consensus building. To do this, we used experimental data to
compare the goodness-of-fit of the DeGroot and RL models across datasets featuring varying
network topologies. We also examined how factors such as network size influence information
integration in experimental social networks⁴⁸⁻⁵⁰. Finally, we examined similarities and differences
between RL and DeGroot models to identify conditions under which the two types of models
behave indistinguishably.

In brief, when fitting the DeGroot and RL models to the two datasets, we found that learning
behavior in both studies was best fitted by the DeGroot model (weighted or unweighted versions),
though there was heterogeneity in the adopted learning strategy. Moreover, simulation results
highlight the relative superiority of the weighted and/or unweighted DeGroot model in terms of
accuracy and consensus. Accuracy is defined as the proportion of nodes whose prediction about
the state of the world was correct while consensus referred to the proportion of nodes whose
prediction was the same as that of the majority of nodes in the network. We also found that the
effectiveness of different learning strategies depended on the underlying network topology, and the
way information was sampled in social networks. Finally, our theoretical analysis and simulations
of the social learning task in networks show that DeGroot and RL-based learning become nearly
indistinguishable at moderate learning rates, with the minimum-difference learning rate decreasing
in a power-law fashion as the degree increases. Together, our findings demonstrate that the
averaging approach of the DeGroot model outperforms the sequential error-driven learning models
in capturing learning within social networks.

Results

**DeGroot outperforms RL in capturing learning in social networks.** To determine whether the one-shot averaging mechanism described by DeGroot or a sequential error-driven updating RL model better captures social learning in networks, we fitted these models and their variants to the data obtained from two recent studies based on a distributed-learning task in social networks<sup>21,24</sup>. In the study by Jiang et al. (2023), seven participants were assigned to nodes in a network and tasked with predicting the source bag of colored marbles based on private signals and the predictions of others in the network (Fig. 1A). In the initial stage (Stage 0), each participant received a private signal in the form of a colored marble randomly drawn from one of two bags with different proportions of yellow and blue marbles. They made an initial guess about the bag's composition (whether it contained more yellow or blue marbles). Over the next three stages, participants observed the immediately previous predictions of their neighbors sequentially in a clockwise order, starting from a randomly selected neighbor. At the end of each stage, participants could revise their own prediction. Participants were then reassigned to a new network after each four-stage process. The study by Choi et al. (2023) used a similar social learning task in networks with a few key differences (Fig. 1B). Forty participants were assigned positions in a network and guessed which of two bags had been selected (one with more green marbles, the other with more red marbles). After receiving a private signal and making an initial guess, participants completed eleven stages in which they observed and adjusted their guesses based on neighbors' predictions. Unlike Jiang et al., participants could freely sampled neighbors' guesses, and each group was assigned to one of four network structures: Erdös-Reyni (ER), Royal Family (RF), Random Geometric Graph (RGG), and Stochastic Block (SB), which differed in homophily and centralization (Fig. S1). Participants played six rounds of this eleven-stage game, with participants'
network positions reshuffled after each round (see Methods for more details). Importantly, in both experiments, participants were shown the complete topology of the network.

We fitted data from the two experiments using five models: two based on the DeGroot framework, two on RL, and one control model (Fig. 2A, B). The first model, the naïve DeGroot (N-DG) model, assumes that individuals average the previous predictions of their neighbors with their own previous prediction (with equal weight) to update their beliefs about the true state of the world. The second model, the weighted DeGroot (W-DG) model, posits that individuals use a weighted average of their neighbors’ and their own previous predictions to update their beliefs. This weighting is based on the relative degree centrality (see Eq. 8 in Methods), which quantifies how well-connected each neighbor is within the individuals’ social network, such that the predictions of more connected neighbors have greater impact on belief updating. The third model, the naïve RL model (N-RL), uses the difference between the individual’s past estimate and each neighbor’s prediction (prediction error) multiplied by a learning rate to update the decision in a sequential fashion (one update per neighbor). The fourth model, the weighted RL model (W-RL), follows the same learning rule as the N-RL model, except that the learning rate was modulated by the relative degree centrality of the neighbor. One important difference between these two families of models is that the RL models are sensitive to the order in which the information is integrated, but the DG models are not (see Fig. 3). Finally, we included a control model that assumes that participants made predictions based solely on their private signal, and did not integrate any social information (Private Signal model). Each model was fitted to individual participant’s choice behavior to identify the best-fitting model for each participant and overall.

Our analysis revealed that the W-DG model provided the overall best fit for the Jiang et al. data (Fig. 4A). However, examination of individual participants showed that the behavior of approximately 40% of the participants was best explained by the W-RL model (Fig. 4B). This suggests heterogeneity in the learning strategies adopted by participants, or difficulty in distinguishing between the models. However, the latter hypothesis can be ruled out, as a model recovery analysis, which involved generating data with one model and fitting that data with all the models, showed that the models were could be clearly distinguished. Thus, the underlying models could be distinguished accurately (Fig. S2A).

Unlike the Jiang et al. study, in which neighbors’ predictions were presented in random order, Choi et al. allowed participants to view their neighbors’ predictions in any sequence they chose. As stated earlier, this difference significantly affects the outcomes of RL-based models, which are sensitive to the order of presentation (Fig. 3). To fit data from the Choi et al. study, we considered three possible patterns for the order in which participants visited their neighbors, as this order was not recorded during the experiment: (1) a random sequence, which we designated the N-RLr or W-RLr models; (2) a descending sequence based on degree centrality (most to least connected), which we designated the N-RLd or W-RLd models; and (3) an ascending sequence based on degree centrality (least to most connected), designated the N-RLa or W-RLa models respectively. We found that the best fitting model overall the Choi et al. data, among the 9 models (private signal model and 8 models that took into account the order in which neighbors’ decisions might have been sampled), was the N-DG model (Fig. 4C). However, the behavior of a significant fraction of participants was best explained by the W-DG or variants of the W-RL model (Fig. 4D), suggesting substantial heterogeneity in participants’ adopted strategies. Here also, model recovery analysis confirmed model identifiability (Fig. S2B).

By analyzing each network structure separately, we found that the N-DG model best explained participants’ behavior across all network structures, except for the RF structure, for
which the W-DG model performed best (Fig. 5A). This can be explained by the particularity of the RF network, in which there are a few highly connected central nodes and many poorly connected peripheral nodes. The optimal way to aggregate information in such networks would be to overweight information coming from the central nodes and underweight information coming from the peripheral nodes. This is exactly what the W-DG model does. Additionally, analysis of the best-fitting models for a minority of participants revealed distinct patterns across network structures. In the RF and SB structures, the descending variant of the RL model fit more participants' behavior than the ascending variant. Conversely, in the ER and RGG structures, the ascending variant of the RL model was a better fit for more participants than the descending variant (Fig 5B). These results suggest that network structure influenced both the learning algorithms individuals used and how they gathered information from their neighbors.

Together, the fitting of data from both studies with variants of the DeGroot and RL models demonstrates that the DeGroot model captures learning in social networks more effectively, but also highlights that there is heterogeneity in the learning strategies adopted by different participants.

**Impact of averaging vs. sequential error-driven updating on accuracy and consensus building.** Overall, our results indicate that despite their relative simplicity, the DG models more effectively capture how individuals update their beliefs within a social network. This begs the question of whether learning based on the DeGroot or RL leads to significantly different outcomes in terms of network-level accuracy and consensus. To answer this question, we simulated information propagation in the networks used in the two studies based on randomly sampled free parameters from a normal distribution that was derived from the estimated parameters of each dataset (see Methods for details). We followed this procedure to mimic to the inter-individual heterogeneity found with real participants, which will allow us to derive more generalizable conclusions. For these simulations, we assumed that every node in the network followed the same learning model and used the actual distribution of private signals observed by participants in each experiment. For both datasets, we conducted simulations using the four main models (N-DG, W-DG, N-RL, and W-RL) as well as the Private Signal model. Additionally, for the simulations based on the Choi et al. (2023) dataset, we included RL model variants that accounted for different potential patterns in the order in which participants might have visited their neighbors (random, descending, and ascending).

To evaluate model performance, we defined accuracy as the proportion of correct predictions (i.e., predictions consistent with the true state of the world) and consensus as the proportion of majority predictions. We performed random-effects linear regressions on accuracy and consensus over time, including a covariate controlling for network types for the simulations based on the Choi et al. dataset. Additionally, we controlled for standard errors clustered at the network level for simulations of Jiang et al. and at the round level for Choi et al. Comparisons across models were adjusted using Bonferroni corrections for multiple comparisons.

For the networks in Jiang et al. (2023), the W-DG and N-RL models significantly outperformed the W-RL models in terms of choice accuracy (Fig. 6 and Table S2). In terms of consensus, the W-DG model significantly outperformed all other models except the W-RL model. For the networks in Choi et al. (2023), aside from the Private Signal model, which performed worst, the N-RL models were outperformed in both accuracy and consensus by the DG models and the W-RL models (Fig. 7 and Table S3). While we cannot definitively conclude that the DG models outperform the W-RL models overall in this dataset, the N-DG model achieved significantly better consensus than the W-RLa model. However, it did not outperform the W-RLr or the W-RLd
variants, which confirms that the sampling order has a significant impact on performance when an RL strategy is used. Similar patterns were observed when we analyzed the mean achieved accuracy and achieved consensus levels for both studies (Table S4). Overall, these simulations suggest that in both studies, the DeGroot models (N-DG or W-DG) foster more effective consensus-building than the sequential, error-driven N-RL models, although they are not always better than the W-RL models.

**Effects of network topology, size and edge density.** Given the potential influence of network characteristics, such as topology, size and edge density, on information propagation, we next investigated how these factors affect the simulated choice accuracy and consensus, as a function of the learning rule used by individuals. To that end, we examined the effects of network topology by conducting random-effects linear regressions on accuracy and consensus over task stages, separately for each network type in the Choi et al. (2023) study. For ER, RGG, and SB networks, the results were similar and consistent with the aggregate analysis (Fig. S3 and S4). However, in the RF networks, characterized by prominent hubs, the N-DG model exhibited the lowest accuracy but achieved the highest consensus. These findings indicate that the effectiveness of different learning strategies depends on the underlying network structure, and the way information is sampled in social networks.

In addition, to directly examine the effects of network size, we generated new ER networks with varying numbers of nodes (10, 20, 30, 40, and 50) and ran simulations of the Choi et al task using each model (see Methods for details). We found that overall, larger networks led to higher mean accuracy but lower mean consensus (Fig. S5). Notably, there were no significant differences between the 8 models tested in these simulations (not including the private signal model that always underperformed). This suggests that the impact of network size on accuracy and consensus building is not influenced by the learning rule.

Next, we examined the effects of edges density, defined as the mean degree centrality (i.e., the average number of connections per node). To do so, we generated new Erdős–Rényi networks with mean degree centralities ranging from 3 to 8, and conducted simulations similar to those used to assess the impact of network size (see Methods). We observed that both accuracy and consensus improved with higher average connectivity (Fig. S6). As with increasing network size, there were no significant differences between the 8 models tested in these simulations (other than for the Private signal model).

Together, these results show that while the network size and edge density influence choice accuracy and consensus in the studied social networks, these effects do not differ significantly between the DeGroot and RL learning models. In contrast, accuracy and consensus-building are more likely to vary between the two families of models depending on the network topology.

**Similarities and differences between learning in the DeGroot and RL models.** Although the results from model fitting showed that DeGroot models were superior in capturing social learning behavior, they also revealed considerable variation in the learning strategies adopted by individuals (Fig. 4). This suggests that the DeGroot learning rule and sequential error-driven updates based on RL may be more closely related than expected. To explore the relationship between these learning strategies and how their underlying computations can be distinguished, we compared the theoretical behavior of the naïve DeGroot model and the naïve RL model during the social network learning task (Fig. 8). To simplify, we examined the naïve (non-weighted) DeGroot and RL models. These results extend to weighted versions of these models, as weighting has a similar impact on the influence of each observation.
According to the naïve DeGroot model, the estimate of agent *i* about the state of the world, $E_s^i$, is updated immediately according to the agent *i*'s own private observation at stage 0 ($O^i$). At later stages, the estimate is equal to the mean of its own and its neighbors' predictions (see Learning models in Methods). Thus, at the end of stage 1, the DeGroot estimate is equal to:

$$E_1^i = (O^i + \sum_{j=1}^N O^j)/(N + 1) \quad (\text{Eq. 1})$$

where *N* is the number of neighbors of agent *i*.

In contrast, after the initial stage, updates in the naïve RL model occur after each observation *j* of a neighbor's prediction/choice, $E_s^i(j)$, which results in the following estimate for the RL agent at the end of stage 1:

$$E_1^i(N) = O^i(1 - \alpha)^N + \sum_{j=1}^N O^j \alpha(1 - \alpha)^{(N-j)} \quad (\text{Eq. 2})$$

where $\alpha$ is the learning rate, and neighbors are sorted according to the sequence in which their predictions/choices are presented to agent *i*. Eq. 2 indicates that the belief update in the RL model is strongly affected by the order in which neighbors' estimates (in this case, the initial observations) are revealed to the agent (see Fig. 3 for an illustration of the order effect). Specifically, depending on the value of $\alpha$, the agent's own observation vs. each of their neighbor's observations have different weights on the final estimate of the RL agent.

To illustrate this, we computed the weight of private observation and each neighbor on agent *i*'s value estimate (at the end of stage 1), as a function of the learning rate and the number of neighbors (Fig. 8). Overall, we found that more recent observations (most recently visited neighbors) had larger weights than less recent observations. Moreover, as the learning rate increased, the weight of more recent observations became gradually larger and at some point, exceeded the weight of the agent's own observation. For example, in the simplest case of $N = 1$ (Fig. 8A), at $\alpha = 0.5$ own vs. neighbor's observations have equal weight, corresponding to averaging observations, as occurs in the DeGroot model. If $\alpha < 0.5$, an agent's own observation has greater weight on the final estimate and vice versa when $\alpha > 0.5$, making the estimate of the RL model different from that of the DeGroot model. With a larger number of edges ($N>1$), no learning rate results in equal weighting for all individuals in the agent's estimate (Fig. 8B-F). However, at intermediate learning rates, the weight of the agent's own observation and those of the neighbors become sufficiently similar to make the estimates of the RL and DeGroot models potentially indistinguishable.

To determine this condition, we computed the learning rate at which the variance of agents' weights is minimal (minimum-difference learning rate). The variance (dashed curve in Fig. 8) is a good measure for comparing weights in the RL model with averaging in the DeGroot model for two reasons: (1) the mean weight across all agents is always equal to 1/(N+1) because these weights sum up to 1; and (2) variance naturally measures the spread around this equal weighting. We found that the minimum-difference learning rate (vertical dashed line in Fig. 8) occurs around the value at which the weight of the agent's own observation and that of the most-recently visited neighbor intersect. Importantly, as the number of connections becomes larger, the learning rates at which the estimates from the two models become close to each other (minimum-difference learning rate) decreases, and this decrease follows a power law function (Fig. S7A). Overall, these results demonstrate that for very low and high learning rates, the order in which the neighbors are visited
can result in the largest deviations between the predictions of RL and DeGroot models. Very low learning rates ($\alpha < 0.1$) favor the agent's own observations on the estimated value, with similarly low weights for the observations of others (primacy effect). High learning rates ($\alpha > 0.4$) increase the weight of the most recently visited neighbor (recency effect) (Fig. S7A). Moreover, our results suggest that for a significant range of learning rates, around the minimum-difference learning rate, and when there are larger numbers of connections, the difference between the RL and DeGroot models may not be distinguishable using a binary readout (e.g., blue vs. yellow) as in the current experimental paradigms.

Similar logic can be extended to later stages for which the starting point is the revised predictions about the state of the world according to each agent. The only difference for later stages is that, due to observations of neighbors' estimates and the learning process, the predictions of agents in the network may begin to converge (consensus building). If convergence occurs, the difference in the weights assigned by an agent to the value estimates of neighboring agents becomes irrelevant, because all agents begin to act according to the same set of predictions, which leads to similar estimates by DeGroot and RL agents. In other words, the distinction between these two types of learning emerges only when individuals in the network have differing observations or estimates. Once opinions converge during consensus building, the updated values according to both models become identical, making the two learning rules similar.

While it remains unclear how individuals adopt a specific learning rate, our theoretical considerations predict distinct patterns of differences between the behavior of the DG and RL models. Specifically, they suggest that for very high or very low learning rates, the RL model's behavior is significantly influenced by the order of observations, which makes it difficult for the DG model to capture the behavior of participants who adopt RL adequately. Consequently, for individuals with extremely low or high learning rates, the DG model is expected to provide a lower quality of fit. To test this, we fitted choice data from the Jiang et al. (2023) and Choi et al. (2023) studies using the DG and RL models, and calculated the frequency at which each model provided the best fit for individual participants. Plotting the difference in model fit frequency as a function of the estimated learning rate in the RL model revealed an inverted U-shape. This confirms our prediction, because it shows that the DG models can account for the behavior of participants with intermediate learning rates better, because of the similarity between the DG and RL agents with intermediate learning rates (Fig. S7B, C).

These findings demonstrate that, under certain conditions, learning behaviors based on one-shot averaging according to the DeGroot model and sequential error-driven updating of the RL model become similar. This takes place when learning rates are moderate and decreases as the number of neighbors increases, or when consensus is reached after several rounds of information exchange and updating. This explains when and why both models can be equally effective in capturing learning in social networks.
Discussion

The DeGroot model has been used extensively to account for learning in networked environments, while the RL models have been used in observational learning studies in non-networked environments³⁶⁻³⁸. However, these two frameworks have not been directly compared to study how information is integrated in social networks. Here, our models’ estimations show that the family of DeGroot models captures learning in social networks better than error-driven learning. Indeed, despite the diversity of learning strategies adopted by individual participants, the averaging rule of the DeGroot models provided an overall better fit than the sequential updating mechanisms of RL models. Using simulations, we show that the DeGroot models were more effective in building consensus than RL models. Moreover, the network topology influenced the difference in consensus and accuracy between the two families of models. Below, we discuss these different aspects of our study.

Role of the order of information integration for the RL model

We observed that when information from neighbors is presented non-sequentially, as is
common in many experimental settings¹⁹⁻²¹,⁵¹, the RL models' predictions vary, and depend on the
order in which beliefs are updated (Fig. 3). That is, when updating the neighbors' prediction in a
clockwise or counterclockwise order, the probability given by the RL models varies dramatically
whereas it remains constant for the DG models. Although the DeGroot models provided an overall
better fit for the two datasets²¹,²⁴, differences between the two studies could have influenced
participants' adoption of learning strategies. In particular, Jiang et al. controlled the presentation
order of neighbors' beliefs, to ensure it was random. In contrast, Choi et al. allowed participants to
choose freely the order in which they visited neighbors and learned their opinions. Based on this
latter dataset, our simulations demonstrate that the order of information integration has an effect
on RL models (Fig. 7) because the simulations of the ascending/descending and random variants
of N-RL and W-RL models differed in their level of accuracy and consensus. This ordering and
the resulting differential weighting could be influenced by attentional processes, as suggested by
studies focused on value-based decision making⁵²,⁵³ and learning⁵⁴. Thus, to fully understand how
individuals aggregate information in real life social network settings, it would be necessary to
determine the order in which they sample each piece of information⁵⁵.

Effect of network topology on accuracy and consensus

A number of studies have shown that the social network structure is an important driver of collective adaptation and cooperation⁵⁶⁻⁵⁹. Yet, it remained unclear whether social network topologies affect the learning rules of individuals when information is disseminated in social networks. Here, we propose that the learning rule used by the majority of individuals in different social networks can, at least in part, explain the influence of network topologies on choice accuracy and consensus. Our theoretical account and simulations of behavior in the observational-learning task demonstrated that inconsistencies between the DeGroot and RL models lead to significantly different outcomes in terms of global consensus and accuracy, depending on the network structure. Although the DeGroot model and its variants consistently provided the best fit for the data in both studies, we found that network topology influenced the sampling strategy of those participants adopting RL in the Choi et al. (2023) study²¹. This latter study involved larger networks with a variety of topologies, including Erdös-Reyni, Royal Family, Random Geometric Graph and Stochastic Block structures. Specifically, in networks with Royal Family and Stochastic Block topology, the descending variant of the RL model explained participants' behavior better than the
ascending variant (Fig. 5). In contrast, in the Erdos-Renyi and Random Geometric Graph structures, the ascending variant of the weighted RL model provided a better fit for a larger number of participants than the descending variant. These results suggest that network topology influences how participants sample information from their neighbors. In networks with distinct communities or prominent hubs, participants tend to begin by sampling information from the more connected neighbors, whereas in networks without such structures, they appear to be more likely to start with less-connected neighbors.

## Distinguishing RL and DG computations

We identified specific scenarios in which the two families of models (DeGroot vs RL) produced similar behavioral outcomes (Fig. 8). This occurred at moderate learning rates that decreased with an increasing number of neighbors in a power-law manner (Fig. S6). Moreover, the two learning strategies would behave similarly once consensus is reached after multiple rounds of information exchange and updating. This explains why in some cases, both models can be equally effective in capturing learning in social networks. It also calls for caution in future experiments aiming to distinguish the computations of these models, and suggests that the paradigms used should encourage either low or high learning rates.

Importantly, averaging models may be particularly valuable when sequential error-driven integration of information is difficult, but the task does not tax working memory heavily$^{60,61}$. Such difficulty could happen when determining the appropriate learning rates is not feasible, while small or large learning rates enhance the impact of specific observations on estimation: high learning rates create a recency effect, whereas low learning rates lead to a primacy effect$^{62}$.

From a broader perspective, our study highlights the importance of understanding the mechanisms through which we balance the costs and benefits of information integration. In social learning, this represents a trade-off between the more cognitively demanding DeGroot model, which optimally integrates information, and the less demanding, error-driven updating typical of RL. In social networks with a large number of edges between nodes, the averaging of opinions, as in the DeGroot model, is likely to be cognitively costly because it requires maintaining recent observations in working memory. Conversely, the sequential updating approach of RL may reduce working memory load of aggregating information. Accordingly, in social networks with a large number of social connections (such as online social platforms), RL models could be more efficient than DG models in terms of cognitive load, while in real world physical social networks, with less connections, DG models could be more efficient than RL ones. This latter prediction may be tested with a real-world network dataset, by comparing learning with the RL and DG models.

## Conclusion

The current study considered the computational properties that underlie individual learning behavior in networked environments. Our findings point to a novel perspective that grounds important social phenomena, frequently investigated only at the collective level, such as information propagation (e.g., fake news), belief polarization, echo chamber effects, etc. Our approach explicitly models the learning processes of individuals in networks. It also provides mechanistic insights into collective adaptation and decision-making, thereby linking individual-level learning to group-level adaptation$^{63,64}$. Moreover, our study determines how cognitive processes shape, and are shaped by collective dynamics over time. Instead, most studies on social influence have only considered static and simulated sources of social information, and have been unable to accommodate the complexity of many individuals interacting dynamically and repeatedly
in real time³²,⁴¹⁻⁴³. Our models of the individual learning processes engaged in dynamic interactions between networked individuals that interact repeatedly in real time and respond to information propagated by others make it possible to describe, understand and predict collective outcomes. Together, our results suggest that encouraging a more DeGroot-like integration of information (e.g., by providing explicit and automatic aggregate opinions of one's network neighbors) could foster greater consensus within networks.

Our theoretical and simulation results have implications for understanding misinformation propagation, and the computational challenges faced by individuals when trying to estimate the state of their world, based on entangled information in social networks. Misinformation encompasses false, inaccurate or misleading content that is not necessarily shared with the intent to deceive. Users may endorse and share such information because of their uncertainty about its veracity⁶⁵⁻⁶⁷ while only a very small percentage of people purposely endorse sharing misinformation online⁶⁸. Recent studies indicate that people rate true news as more accurate than false news and are better at rating false news as false than at rating true news as true⁶⁹,⁷⁰. Thus, people may not lack the skills to detect false news. Yet, they may still fall for them because of a number of reasons. In the settings we considered here, in which information does not concern veracity of news, we suggest that people may still be misled about the true state of the world because of the particular set of private information instantiated in the network at the initial stage of learning, and because of the difficult nature of the integration to be done on uncertain information. Understanding the computations and motives underlying decisions to propagate inaccurate information should be useful to design more efficient intervention programs to mitigate misinformation⁶⁸,⁷¹⁻⁷³.
## Methods

To compare the predictions of DeGroot and RL models, we fitted choice data from two recent social learning tasks with interacting individuals in networks, conducted by Jiang et al. (2023) and Choi et al. (2023). We also performed simulations on choice accuracy and consensus, including the effects of network size and number of edges. Finally, we report theoretical analyzes analyzing how learning rates and number of neighbors influence estimated value in RL.

### Experimental paradigm.

In the study by Jiang et al. (2023), seven participants were randomly assigned to different nodes in a network, with one participant inside an fMRI scanner and the remaining six outside. The participants' task was to guess which of two bags of colored marbles (with a majority of blue or yellow marbles), had been randomly selected after receiving a private signal at the initial stage (or stage 0), followed by updating their decision over three stages when they observed their neighbors' guesses on the previous stage. Specifically, for the first/initial stage, participants received a private signal, which was a colored marble randomly drawn from one of two possible bags containing yellow and blue marbles in different proportions. They then had to make an initial guess regarding whether the selected bag contained more blue or yellow marbles. During each of the following three stages, participants were sequentially exposed to the previous predictions of all their neighbors in the network in a clockwise order, starting from a randomly selected neighbor. At the end of each stage, participants were able to revise their previous guesses regarding which bag had been selected. At the end of each round which consisted of four stages, participants were randomly assigned to a new network (Fig. 1A). A total of 209 participants participated in the experiment and played a total of 40 games (networks) each.

The second study by Choi et al. utilized a similar observational-learning task but with a few differences (Fig. 1B). Similar to the first study, 40 participants were randomly assigned positions in a network and had to guess which of two bags of colored marbles (one containing more green marbles and the other more red marbles). In the first stage, participants received a private signal and made an initial guess, similar to the procedure in Jiang et al. In the subsequent eleven stages, participants could observe their neighbors' previous predictions and adjust their guesses accordingly. Unlike the first study, the presentation of neighbors' guesses was not randomly determined by the experimenter; instead, participants could freely sample their neighbor's predictions. In addition, each group of 40 participants was randomly assigned to one of four distinct network structures for the entirety of the experiment: Erdös-Reyni (ER), Royal Family (RF), Random Geometric Graph (RGG), and Stochastic Block (SB). These networks vary significantly in their degree of homophily and centralization (refer to Fig. S2 for a graphic representation of different networks). Participants played a total of 6 rounds of this twelve-stage game with their position in the network being reshuffled at the end of each round.

**Learning models.** We considered multiple versions of RL and DeGroot models. The naïve RL and naïve DeGroot and their weighted versions, the weighted RL and weighted DeGroot models. All models were fitted to choice data from Jiang et al. and Choi et al. studies to compare the ability of these models to capture learning in social networks. We also considered a baseline model without any social learning, the Private Signal model.
### Naïve DeGroot model (N-DG)

According to the naïve DeGroot model, the estimate of agent *i* about the state of the world at the end of stage *s* of the task, $E_s^i$, is computed at once, using the agent's own private observation in the initial stage, $O^i$, and the predictions of the agent's and its neighbors' predictions in the subsequent stages, $C_s^i$. More specifically, in the initial stage (s=0), where agents privately observe the outcome of drawing a ball from one of the two bags, their beliefs are equal to their private observation:

$$E_0^i = O^i \quad (\text{Eq. 3})$$

where $O^i$ is equal to 1 or 0 if the observed ball was blue (red) or yellow (green), respectively. Based on the initial estimates, each agent makes a prediction/guess at the end of the initial state which we assume is the same as the private observation: $C_0^i = O^i$.

After the initial stage, each agent observes the predictions of their $N_i$ neighbors from the previous stage. Agents update their estimates according to these predictions and their own previous prediction as follows:

$$E_s^i = \frac{c_{s-1}^i + \sum_{j \in \text{neighbors of } i} c_{s-1}^j}{N_i + 1} \quad (\text{Eq. 4})$$

where *N* defines the set of neighbors of agent *i* in the network, $C_{s-1}^j$ are the predictions of neighbors *j* ($j \in$ neigbors of *i*) at the end of preceding stage (*s*-1). According to this update rule, the order of observations does not affect the updated estimates. These estimates are then used to make predictions about the state of the world using the following probability:

$$P_s^i(C_s^i = 1) = \frac{1}{1 + \exp(-\beta E_s^i)} \quad (\text{Eq. 5})$$

where $\beta$ is the inverse temperature (with $\beta>0$) determining the sensitivity of prediction to the estimates about the state of the world.

### Naïve RL model (N-RL)

The naïve RL model also uses the same framework for making predictions but unlike the naïve DeGroot model, updates in the RL model happen after each observation of a neighbor's prediction:

$$E_s^i(1) = E_{s-1}^i + \alpha (C_{s-1}^j - E_{s-1}^i) \quad \text{for } j = 1$$

$$E_s^i(j) = E_s^i(j-1) + \alpha (C_{s-1}^j (\text{end}) - E_s^i(j-1)) \quad \text{for } j > 1 \quad (\text{Eq. 6})$$

where $E_s^i(j)$ denotes estimate of agent *i* about the state of the world after observation *j* and during stage *s* of the task using the RL model, $j \in$ neigbors of *i*, and $\alpha$ is the learning rate that determines the strength of update based on the difference between the previous estimate and observed prediction (prediction error). In the initial stage, where agents privately observe the outcome of drawing a ball from one of the two bags, their initial belief, $E_0^i$, depends on their private signal, $O^i$, and a free parameter $\gamma$ ($0 \le \gamma \le 1$):

$$E_0^i = \gamma \quad \text{if } O^i = 1$$

$$E_0^i = -\gamma \quad \text{if } O^i = 0 \quad (\text{Eq. 7})$$

### Weighted DeGroot model (W-DG)

The W-DG model operates on the same principles as the N-DG model but incorporates weights based on the relative degree centrality of each neighbor or individual. While various metrics, such as eigenvalue or betweenness centrality, could be used for weighting, recent evidence supports using relative degree centrality because it is both a prominent characteristic of each node and easy to compute²⁴. This weighting is applied starting from stage 2 of the learning game, as neighbors
in stage 1 were not yet aware of their own neighbors' previous guesses, resulting in the following update rule:

$$E_s^i = C_{s-1}^i \cdot RD_i + \sum_{j \in \text{neighbors of } i} RD_j \cdot C_{s-1}^j \quad (\text{Eq. 8})$$

in which $RD_j = \frac{\text{Degree of } j}{\text{Sum of Degrees of all } j \text{ and } i}$. For the initial guess, the W-DG model follows the same rule as the N-DG model.

### Weighted RL model (W-RL)

The W-RL model is similar to the N-RL model but the prediction error is weighted by the relative degree of centrality of the observed neighbor:

$$E(j)_{i,t} = E(A)_{i,t} + \theta \cdot RD_j \cdot (R(j)_{j,t} - E(j)_{i,t-1}) \quad (\text{Eq. 10})$$

where $\theta$ is the learning rate, distinct from $\alpha$ (used in stage 1), and $RD_j$ the relative degree of centrality weight, as defined above.

### Private Signal model (Priv. Sig.)

The Priv. Sig. model assumes that agents' prediction is only based on their private observation in the initial stage. Like in Equation 7, we assume that the agent estimation of the state of the world, at any stage $s$, is based on her private signal and a free parameter $\gamma$ ($0 \le \gamma \le 1$):

$$E_s^i = \gamma \quad \text{if } O^i = 1 \\ E_s^i = -\gamma \quad \text{if } O^i = 0 \quad (\text{Eq. 11})$$

**Model estimation and selection.** Our estimation procedure follows the approach outlined in Jiang et al. Specifically, we utilized the VBA toolbox, and employed the same parameters as those used in their study, to estimate models for both the Jiang et al. and Choi et al. datasets. For model selection, we performed Bayesian Model Selection (BMS) using the VBA toolbox (Variational Bayesian Analysis) in a random-effects analysis, relying on the free energy as a lower bound for model evidence. Exceedance Probability (EP) measurements were used to identify the model most frequently favored in each of the two samples$^{74}$.

For Choi et al., the participants' order of integration of their neighbors' prediction was not established in the design, therefore we had to consider several variations of the RL models. We implemented three variations. The first assumed that participants integrated each neighbor's prediction in a random order (N(W)-RLr). The second assumed they began with the prediction of the most well-connected neighbor (based on the number of edges of the node) and ended with the least well-connected (N(W)-RLd, descending). The third assumed the opposite, starting with the least well-connected neighbor, and finishing with the most well-connected (N(W)-RLa, ascending). In the last two variants, when two or more neighbors had the same number of connections, their order of integration was randomized. Unfortunately, these variations are post-hoc considerations, as the Choi et al. study does not provide information on the order in which participants observed their neighbors' predictions at each stage.

**Network simulations.** To evaluate the performance of the DeGroot and RL strategies under different conditions, we simulated network behavior based on the experimental setups of the Jiang et al. and Choi et al. studies. For these simulations, we used the distributions of private signals as well as the network structures from the two studies. In each simulation we considered that all nodes of a given network performed according to the same learning model. In total, we ran 100
simulations for each of the four models. For the Jiang et al. study, we ran 100 simulations for each network present in their dataset and each model (43 networks in total), leading to a total of 17200 simulations. For the Choi et al. study, we ran 100 simulations for each network (of 40 nodes), round and model, over 11 stages and 4 sessions per network, leading to a total of 1036800 observations. The parameters of each model were randomly drawn from a normal distribution for each simulation. The parameters of the distribution we used are provided in **Table 1** below and are derived from the mean and standard deviation of the estimated values of both datasets.

Jiang et al, 2023

<table><thead><tr><th rowspan="2"></th><th colspan="2">Priv. Sig.</th><th colspan="2">N-DG</th><th colspan="2">W-DG</th><th colspan="2">N-RL</th><th colspan="2">W-RL</th></tr><tr><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th></tr></thead><tbody><tr><th>β</th><td>1.467</td><td>0.300</td><td>4.627</td><td>0.378</td><td>5.482</td><td>0.501</td><td>6.904</td><td>0.539</td><td>9.282</td><td>0.669</td></tr><tr><th>γ</th><td></td><td></td><td></td><td></td><td></td><td></td><td>0.665</td><td>1.327</td><td>0.528</td><td>1.305</td></tr><tr><th>α</th><td></td><td></td><td></td><td></td><td></td><td></td><td>0.198</td><td>0.541</td><td>0.151</td><td>0.620</td></tr><tr><th>θ</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.736</td><td>1.241</td></tr></tbody></table>

Choi et al, 2023

<table><thead><tr><th rowspan="2"></th><th colspan="2">Priv. Sig.</th><th colspan="2">N-DG</th><th colspan="2">W-DG</th><th colspan="2">N-RLr</th><th colspan="2">N-RLd</th></tr><tr><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th></tr></thead><tbody><tr><th>β</th><td>1.475</td><td>0.890</td><td>4.259</td><td>0.868</td><td>3.892</td><td>0.940</td><td>6.232</td><td>1.048</td><td>5.862</td><td>1.152</td></tr><tr><th>γ</th><td></td><td></td><td></td><td></td><td></td><td></td><td>0.480</td><td>1.410</td><td>0.507</td><td>1.488</td></tr><tr><th>α</th><td></td><td></td><td></td><td></td><td></td><td></td><td>0.112</td><td>1.554</td><td>0.133</td><td>1.556</td></tr></tbody></table>

<table><thead><tr><th rowspan="2"></th><th colspan="2">N-RLa</th><th colspan="2">W-RLr</th><th colspan="2">W-RLd</th><th colspan="2">W-RLa</th></tr><tr><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th><th>Mean</th><th>Std. Dev.</th></tr></thead><tbody><tr><th>β</th><td>6.040</td><td>1.120</td><td>7.305</td><td>1.227</td><td>6.939</td><td>1.146</td><td>7.773</td><td>1.288</td></tr><tr><th>γ</th><td>0.551</td><td>1.378</td><td>0.558</td><td>1.358</td><td>0.534</td><td>1.434</td><td>0.556</td><td>1.363</td></tr><tr><th>α</th><td>0.170</td><td>1.588</td><td>0.195</td><td>1.645</td><td>0.192</td><td>1.399</td><td>0.190</td><td>1.676</td></tr><tr><th>θ</th><td></td><td></td><td>0.360</td><td>2.039</td><td>0.412</td><td>2.087</td><td>0.380</td><td>2.049</td></tr></tbody></table>

**Table 1.** The parameters of each model (Priv. Signal, N-DG, W-DG, N-RL, W-RL) for Jiang's et al., 2023 dataset and (Priv. Signal, N-DG, W-DG, N-RLr, N-RLd, N-RLa, W-RLr, W-RLd, W-RLa) for Choi's et al. 2023 dataset were randomly drawn from a normal distribution for each simulation. The table reports the parameters of the distribution derived from the mean and standard deviation of the estimated values of both datasets.

To generate Erdős-Reyni networks with 10, 20, 30, 40, and 50 nodes, we used a MATLAB algorithm to create 100 networks for each size. From these, we selected one network per size with an average degree of 4, following the procedure used in the Choi et al. study. We then applied the same simulation process as in previous analyzes, with the exception that the correct bag and private signal were randomly drawn at the start of each of the 100 simulations. To generate Erdős-Rényi networks with nodes exhibiting an average degree centrality of 3, 4, 5, 6, 7, or 8, we used a MATLAB algorithm to create 100 networks with varying connectivity. Specifically, we constructed 40×40 adjacency matrices and introduced random links between node pairs with a connection probability *p*, which was incrementally varied from 0.05 to 0.25 in steps of 0.01. For each value of *p*, a network was simulated, and we selected the first network whose average degree
centrality matched one of the target values. The same simulation procedure used for varying
network size was then applied to these networks.
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<figure><img src="image_1.png" /><figcaption>**Figure 1.** (A) Experimental paradigm from Jiang et al., 2023. In each round participants had to predict which bag of 7 colored marbles has been selected. One contained more blue marbles and the other more yellow marbles. Participants first received a private information which was the color of a randomly selected marble from the chosen bag. This information stayed the same during the round. After making a first prediction solely based on this information, they were sequentially informed about the previous prediction of each of their direct neighbors in their social network. Then, they could revise their previous predictions. This last step was repeated three times. Accordingly, a round comprised of three stages plus an initial guess. (B) Experimental paradigm from Choi et al., 2023. Participants were first associated with one of four networks for the whole experiment. Then, in each round participants had to predict which bag of 10, colored marbles had been selected. One was mostly filled with green marbles and the other mostly red marbles. Participants first received a private information which was the color of a randomly selected ball from the chosen bag. This information remained the same during the round. After making a first prediction solely based on this information, they were informed about the previous predictions of all of their direct neighbors from their social network and they could then revise their previous prediction. This last step was repeated eleven times. Accordingly, a round comprised 11 stages plus the initial guess.</figcaption></figure>
A

$$ \textbf{N-RL} \quad E_s^i(j) = E_s^i(j-1) + \alpha(C_{j,s-1} - E_s^i(j-1)) $$

$$ \textbf{W-RL} \quad E_s^i(j) = E_s^i(j-1) + \alpha \cdot RD_j(C_{j,s-1} - E_s^i(j-1)) $$

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

**Sequential update (N-RL and W-RL)**

$$ \textbf{B} \quad \textbf{N-DG} \quad E_s^i(end) = \frac{\sum C_{j,s-1} + C_{i,s-1}}{N+1} $$

$$ \textbf{W-DG} \quad E_s^i(end) = \sum RD_j \cdot C_{j,s-1} + RD_i \cdot C_{i,s-1} $$

<figure><img src="image_6.png" /></figure>

**Averaging (N-DG and W-DG)**

**Figure 2.** (A, B) Illustration of error-driven sequential updating in the reinforcement learning models (A) and averaging in DeGroot models (B) in a social network. A participant in the dashed node aims to use previous predictions of her neighbors to update her belief about the state of the world. A. In the N-RL and W-RL models, individual *i* updates her beliefs (*E*<sub>*i*,*s*</sub>) about the likelihood of a bag being selected by incorporating the previous prediction of each neighbor *t* (from a total of *N* neighbors). This update is based on a prediction error comparing her previous belief (*E*<sub>(1)<sub>*i*,*s*</sub>) with the neighbor's past prediction (*C*<sub>*j*,*s*−1</sub>), which is weighted by a learning rate *α* (and weighted by the relative degree of centrality of the neighbor (*RD*<sub>*j*</sub>) for W-RL). This update is sequential as it is performed for each neighbor *j* (in the example we show the equation for neighbor 2). B. In the N-DG and W-DG models, the integration is not gradual/sequential because the participant uses the average of her neighbors' predictions (including her own past prediction, *C*<sub>*i*,*s*−1</sub>) to update beliefs about the likelihood of a bag being selected. Only in the W-DG, the average is weighted by the neighbor's relative degree of centrality (*RD*<sub>*j*</sub>) and the participants' own relative degree centrality (*RD*<sub>*i*</sub>).
<figure><img src="image_7.png" /></figure>

Clockwise update

<figure><img src="image_8.png" /></figure>

Counterclockwise update

<figure><img src="image_9.png" /><figcaption>**Figure 3.** Simulations illustrating the effect of the order of information aggregation on the N-RL and W-RL models. An individual at a given node (top panel, dashed circle) has to integrate her neighbor's previous prediction sequentially in the N-RL and W-RL models. The order of the sequence matters as it can lead to different outcomes. In the top left panel, we considered a situation in which the participant integrates the information in a clockwise fashion and in the top right panel in a counterclockwise direction. Based on experimental data from Jiang et al, we randomly selected one unique participant of one unique network. We then simulated 1000 participant predictions with either one of the four models (parameters value: $\beta = 4.5$, $\gamma = 0.7$, $\alpha = 0.25$). In the four bottom panels, we plotted the mean predictions (of the blue color) simulated with the clockwise update and the counter-clockwise update for all four models. The effect of the order of integration dramatically affects the predictions of the two RL models. The mean prediction that blue is the true state of world significantly differs across stages (except for stage 0 ($p = 0.178$ for N-RL and $p = 0.549$ for W-RL), $p$-values <0.001 one-sided t-tests). The simulations based on the two DG models show that they are unaffected by the order ($p$-values goes from 0.225 to 1, two-sided t-tests).</figcaption></figure>
A

<figure><img src="image_10.png" /></figure>

B

<figure><img src="image_11.png" /></figure>

C

<figure><img src="image_12.png" /></figure>

D

<figure><img src="image_13.png" /></figure>

**Figure 4.** Comparison of the goodness-of-fit of different models in capturing experimental data from Jiang et al. (A, B) and from Choi et al. (C, D). (A) Exceedance Probability (EP) of each of five models (Priv. Sig: Private Signal, N-DG: naive DeGroot, W-DG: weighted DeGroot, N-RL: naive RL, W-RL: weighted RL). The EP quantifies the likelihood that a model best explains the participants’ behavior. The weighted DeGroot model provides the best fit. (B) Plot shows the frequency that each model provides the best fit for individual participants for Jiang et al data. (C) Exceedance Probability (EP) of each of eight models with N-DG, W-DG being the same than in (A). The N(W)-RL<sub>t/d/a</sub> models correspond to the naïve (weighted) RL models with different sequence of integration for the neighbors’ prediction. R corresponds to random as the neighbors’ predictions are integrated in a random order. D corresponds to descending as the neighbors’ prediction are integrated in a descending order, from the most connected to the least connected (i.e., number of edges of the neighbors’ node). A stand for ascending where the neighbors’ predictions are integrated in an ascending order, from the least connected to the most connected. The naïve DeGroot model provides the best fit. (D) Plot shows the frequency that each model provides the best fit for individual participants for Choi et al data.
<figure><img src="image_14.png" /><figcaption>**Figure 5.** Comparison of the goodness-of-fit of different models in capturing experimental data from Choi et al. separately for each network structure indicated on the top. **A.** Exceedance probability (EP) of each of nine models (Priv. Sig: Private Signal, N-DG: naive DeGroot, W-DG: weighted DeGroot, N(W)-RL<sub>r</sub>: naive (weighted) RL random order, N(W)-RL<sub>d</sub>: naive (weighted) RL descending order, N(W)-RL<sub>a</sub>: naive (weighted) RL ascending order) in the Erdös-Renyi, Royal Family, Random Geographic Graph and Stochastic Block networks. **B.** Plots show the frequency that each model provides the best fit for individual participants for Choi et al. data, and for each network.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 6. Choice accuracy and consensus contrasts between each simulated model in Jiang et al.</figcaption></figure>

A. Graphic representation of choice accuracy. It is defined as the mean proportion of nodes on a network whose predictions are consistent with the true state of the given network. B. Graphic representation of choice consensus. It is defined as the mean proportion of nodes on a network who predicted the same dominant option in a given stage of the game. (A, B) We simulated predictions for each node on each network from Jiang et al using the private signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the naïve Reinforcement Learning (N-RL) and weighted Reinforcement Learning (W-RL) models. The matrices represent the contrast of the marginal effect of accuracy (A) and consensus (B) between the models in the x-axis and the models in the y-axis. Blue (red) colored cell corresponds to contrast with negative (positive) values, the more intense it is the more negative (positive) the value is. Stars represent significant p-values with Bonferroni corrections for multiple comparisons (10 comparisons in total). With, * p < 0,05, ** p < 0,01 and *** p < 0,001.
<figure><img src="image_16.png" /><figcaption>**Figure 7.** Choice accuracy and consensus between each simulated model in Choi et al. (A, B) We simulated predictions for each node on each network from Choi et al using the private signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the random, ascending and descending naïve Reinforcement Learning (N-RLr,d,a) and weighted Reinforcement Learning (W-RLr,d,a) models. The matrices represent the contrast of the marginal effect of accuracy (A) and consensus (B) between the models in the x-axis and the models in the y-axis. Blue (red) colored cell corresponds to contrast with negative (positive) values, the more intense it is the more negative (positive) the value is. Stars represent significant p-values with Bonferroni corrections for multiple comparisons (36 comparisons in total). With, \* p < 0.05, \*\* p < 0.01 and \*\*\* p < 0.001.</figcaption></figure>
A

<figure><img src="image_17.png" /></figure>

B

<figure><img src="image_18.png" /></figure>

C

<figure><img src="image_19.png" /></figure>

D

<figure><img src="image_20.png" /></figure>

E

<figure><img src="image_21.png" /></figure>

F

<figure><img src="image_22.png" /><figcaption>**Figure 8.** Averaging of observations in the naive RL model. Plots show the weight of initial observation by different agents on the estimated value in RL as a function of the learning rate, separately for different number of neighbors ($N$ equal to 1 to 6 in panels A to F; $N_i$ denotes neighbor $i$ in order their guesses are revealed). The gray dot-dashed curve shows the variance and the vertical dashed line indicates the minimum-difference learning rate. As the learning rate increases, the weights of more recently visited neighbors become gradually larger than the weight of agent's own observation. Moreover, as the degree of centrality increases, the overall variance decreases, making the predictions of the RL and DeGroot models less distinguishable across a broader range of learning rates.</figcaption></figure>
Supplementary materials

<figure><img src="image_23.png" /><figcaption>**Figure S1.** Four networks tested in Choi et al., 2023. Participants were randomly attributed to one of four networks for the whole experiment. The four networks vary significantly in their degree of homophily and centralization.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>**Figure S2. Confusion matrixes.** The cells of the matrices show the mean exceedance probability obtained by Bayesian model selection on data generated under the corresponding model. The higher the exceedance probability is, the more likely the model explained the simulated data. Here, our analysis shows a perfect recovery of each true generative model. The simulations were performed for every participant with the models' parameters being those estimated for that participant. **A.** Confusion matrix for Jiang et al. data. **B.** Confusion matrix for Choi et al. data. The r, d and a versions of the N(W)-RL models correspond to different orders of integration of the neighbors' prediction, thus they do not correspond to a formal model that can be estimated. Each version should be estimated as the corresponding N(W)-RL model.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>**Figure S3.** Contrast of the marginal effect of choice accuracy between each simulated model across networks in Choi et al. We simulated predictions for each node on each network from Choi et al. using the private signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the random, ascending and descending naïve Reinforcement Learning (N-RLr,d,a) and weighted Reinforcement Learning (W-RLr,d,a) models. The matrices represent the contrast of the marginal effect of accuracy between the models in the x-axis and the models in the y-axis for each of the four networks in Choi et al. experiment. Blue (red) colored cell corresponds to contrast with negative (positive) values, the more intense it is the more negative (positive) the value is. Stars represent significant p-values with Bonferroni corrections for</figcaption></figure>
multiple comparisons (36 comparisons in total). With, * p < 0,05, ** p < 0,01 and *** p < 0,001.

<figure><img src="image_26.png" /><figcaption>**Figure S4. Contrast of the marginal effect of choice consensus between each simulated model across networks in Choi et al. We simulated predictions for each node on each network from Choi et al. using the private signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the random, ascending and descending naïve Reinforcement Learning (N-RLr,d,a) and weighted Reinforcement Learning (W-RLr,d,a) models. The matrices represent the contrast of the marginal effect of consensus between the models in the x-axis and the models in the y-axis for each of the four networks in Choi et al. experiment. Blue (red) colored cell**</figcaption></figure>
corresponds to contrast with negative (positive) values, the more intense it is the more negative (positive) the value is. Stars represent significant p-values with Bonferroni corrections for multiple comparisons (36 comparisons in total). With, \* p < 0,05, \*\* p < 0,01 and \*\*\* p < 0,001.

<figure><img src="image_27.png" /><figcaption>**Figure S5.** Mean achieved choice consensus and accuracy over simulations with varying network size. We simulated predictions for each node on randomly generated Erdös-Reyni (ER) networks. We considered ER networks with 10, 20, 30, 40 and 50 nodes (Nd). For the simulations we used either the Private Signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the random, descending, ascending naïve Reinforcement Learning (N-RLr/d/a) and the random, descending, ascending weighted Reinforcement Learning (W-RLr/d/a) models. The parameters used for the simulation for each model on each network were the same (N = 100 simulations per network and per models. See Methods for details). Choice accuracy (A) is defined as the proportion of nodes on a network whose predictions are consistent with the true state of the given round. Here, we displayed the mean achieved accuracy for each network (Stage 11). Choice consensus (B) is defined as the proportion of nodes on a network who predicted the same dominant option in a given stage of the game. Here, we displayed the mean achieved consensus for each network size (Stage 11).</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Figure S6.** Mean achieved consensus and accuracy over simulations with varying mean number of edges. We simulated predictions for each node on randomly generated Erdös-Reyni (ER) networks. We considered ER networks with 40 nodes and with an average number of edges (degree of centrality) for each node equals to 3, 4, 5, 6, 7 and 8. For the simulations we used either the Private Signal (Priv. Sig.), naïve DeGroot (N-DG), the weighted DeGroot (W-DG), the random, descending, ascending naïve Reinforcement Learning (N-RLr/d/a) and the random, descending, ascending weighted Reinforcement Learning (W-RLr/d/a) models. The parameters used for the simulation for each model on each network were the same (N = 100 simulations per network and per models. See Methods for details). Choice consensus is defined as the proportion of nodes on a network who predicted the same dominant option in a given stage of the game. Here, we displayed the mean achieved consensus for each mean number of edges (Stage 11). Choice accuracy is defined as the proportion of nodes on a network whose predictions are consistent with the true state of the given round. Here, we displayed the mean achieved accuracy for number of edges (Stage 11).</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>**Figure S7.** Predictions about the difference between the DeGroot and RL models as a function of the learning rate in the RL. (A) Plot shows the learning rates at which the variance of all agents' weights on RL agent's estimated value is minimal for a given number of neighbors. The minimum-difference learning rate decreases with the number of neighbors ($N$) as a power law. (B, C) Plots show the difference between the frequency of the best DG and RL models for each individual plotted against the learning rate of the best RL model in the Jiang et al. (B) and Choi et al. studies (C). Each dot represents a simulated individual, and the red line represents the best fit of a second-degree polynomial.</figcaption></figure>
<table><caption>**Table S1.** Original names of some of the models in Jiang et al., 2023 and the name used in our paper.</caption><thead><tr><th>Original name</th><th>Updated name</th></tr></thead><tbody><tr><td>DeGroot learning model</td><td>Weighted Reinforcement Learning model (W-RL)</td></tr><tr><td>Baseline learning model</td><td>Naive Reinforcement Learning model (N-RL)</td></tr></tbody></table>

<table><caption>**Table S2.** Contrasts of simulations' accuracy and consensus between models. The results are based on mixed-effect linear regressions with standard error being clustered at the network level. All statistical tests are Bonferroni corrected. * p<0.05; ** p<0.01; *** p<0.001.</caption><caption>Jiang et al</caption><thead><tr><th></th><th>Accuracy</th><th>Consensus</th></tr></thead><tbody><tr><td>N-DG vs PS</td><td>0.049<br>(0.027)</td><td>0.085 ***<br>(0.009)</td></tr><tr><td>W-DG vs PS</td><td>0.049<br>(0.026)</td><td>0.096 ***<br>(0.010)</td></tr><tr><td>N-RL vs PS</td><td>0.028<br>(0.018)</td><td>0.068 ***<br>(0.008)</td></tr><tr><td>W-RL vs PS</td><td>0.001<br>(0.020)</td><td>0.092 ***<br>(0.006)</td></tr><tr><td>W-DG vs N-DG</td><td>0.001<br>(0.005)</td><td>0.011 *<br>(0.004)</td></tr><tr><td>N-RL vs N-DG</td><td>-0.020<br>(0.018)</td><td>-0.017 *<br>(0.005)</td></tr><tr><td>W-RL vs N-DG</td><td>-0.048<br>(0.020)</td><td>0.007<br>(0.005)</td></tr><tr><td>N-RL vs W-DG</td><td>-0.021<br>(0.015)</td><td>-0.028 ***<br>(0.006)</td></tr><tr><td>W-RL vs W-DG</td><td>-0.048 *<br>(0.016)</td><td>-0.004<br>(0.006)</td></tr><tr><td>W-RL vs N-RL</td><td>-0.027 ***<br>(0.005)</td><td>0.023 ***<br>(0.003)</td></tr><tr><td>N</td><td>86000</td><td>86000</td></tr><tr><td>N clusters</td><td>21500</td><td>21500</td></tr></tbody></table>
<table><thead><tr><th></th><th>(1)</th><th>(2)</th><th></th><th>(1)</th><th>(2)</th></tr><tr><th></th><th>Accuracy</th><th>Consensus</th><th></th><th>Accuracy</th><th>Consensus</th></tr></thead><tbody><tr><td>W-DG vs N-DG</td><td>-0.021<br>(0.003)</td><td>-0.021 ***<br>(0.003)</td><td>N-Rla vs N-Rlr</td><td>-0.009<br>(0.005)</td><td>-0.001<br>(0.003)</td></tr><tr><td>N-Rlr vs N-DG</td><td>-0.023<br>(0.005)</td><td>-0.043 ***<br>(0.004)</td><td>W-RLr vs N-Rlr</td><td>0.012<br>(0.003)</td><td>0.01 ***<br>(0.002)</td></tr><tr><td>N-RLd vs N-DG</td><td>-0.041<br>(0.005)</td><td>-0.055 ***<br>(0.003)</td><td>W-RLd vs N-Rlr</td><td>0.018<br>(0.004)</td><td>0.021 ***<br>(0.002)</td></tr><tr><td>N-Rla vs N-DG</td><td>-0.031<br>(0.008)</td><td>-0.044 ***<br>(0.005)</td><td>W-Rla vs N-Rlr</td><td>0<br>(0.006)</td><td>0.001<br>(0.004)</td></tr><tr><td>W-RLr vs N-DG</td><td>-0.011<br>(0.006)</td><td>-0.032 ***<br>(0.005)</td><td>N-Rla vs N-RLd</td><td>0.009<br>(0.007)</td><td>0.011<br>(0.004)</td></tr><tr><td>W-RLd vs N-DG</td><td>-0.005<br>(0.004)</td><td>-0.022 ***<br>(0.003)</td><td>W-RLr vs N-RLd</td><td>0.03<br>(0.005)</td><td>0.023 ***<br>(0.004)</td></tr><tr><td>W-Rla vs N-DG</td><td>-0.023<br>(0.009)</td><td>-0.042 ***<br>(0.006)</td><td>W-RLd vs N-RLd</td><td>0.036<br>(0.003)</td><td>0.033 ***<br>(0.003)</td></tr><tr><td>N-Rlr vs W-DG</td><td>-0.002<br>(0.004)</td><td>-0.022 ***<br>(0.002)</td><td>W-Rla vs N-RLd</td><td>0.018<br>(0.008)</td><td>0.014<br>(0.006)</td></tr><tr><td>N-RLd vs W-DG</td><td>-0.02<br>(0.005)</td><td>-0.034 ***<br>(0.003)</td><td>W-RLr vs N-Rla</td><td>0.021<br>(0.004)</td><td>0.012 ***<br>(0.002)</td></tr><tr><td>N-Rla vs W-DG</td><td>-0.011<br>(0.006)</td><td>-0.023 ***<br>(0.003)</td><td>W-RLd vs N-Rla</td><td>0.027<br>(0.007)</td><td>0.022 ***<br>(0.003)</td></tr><tr><td>W-RLr vs W-DG</td><td>0.01<br>(0.004)</td><td>-0.011 **<br>(0.003)</td><td>W-Rla vs N-Rla</td><td>0.009<br>(0.002)</td><td>0.002<br>(0.002)</td></tr><tr><td>W-RLd vs W-DG</td><td>0.016<br>(0.004)</td><td>-0.001<br>(0.002)</td><td>W-RLd vs W-RLr</td><td>0.006<br>(0.005)</td><td>0.01 **<br>(0.003)</td></tr><tr><td>W-Rla vs W-DG</td><td>-0.002<br>(0.007)</td><td>-0.021 ***<br>(0.004)</td><td>W-Rla vs W-RLr</td><td>-0.012<br>(0.005)</td><td>-0.01<br>(0.003)</td></tr><tr><td>N-RLd vs N-Rlr</td><td>-0.018<br>(0.004)</td><td>-0.013 ***<br>(0.003)</td><td>W-Rla vs W-RLd</td><td>-0.018<br>(0.008)</td><td>-0.02 ***<br>(0.004)</td></tr><tr><td>N</td><td></td><td></td><td></td><td>1036800</td><td></td></tr><tr><td>N clusters</td><td></td><td></td><td></td><td>86400</td><td></td></tr></tbody></table>

**Table S3.** Contrasts of simulations' accuracy and consensus between models. The results are based on mixed-effect linear regressions with standard error being clustered at the round level. All statistical tests are Bonferroni corrected. * p<0.05; ** p<0.01; *** p<0.001.
Jiang et al (2023)

<table><thead><tr><th></th><th>Priv. Sig. N-DG</th><th>W-DG</th><th>N-RL</th><th>-</th><th>-</th><th>W-RL</th><th>-</th><th>-</th></tr></thead><tbody><tr><th>Achieved mean accuracy</th><td>52.54%</td><td>57.29%</td><td>56.80%</td><td>53.85%</td><td>-</td><td>47.64%</td><td>-</td><td>-</td></tr><tr><th>Achieved mean consensus</th><td>64.99%</td><td>84.08%</td><td>86.39%</td><td>81.66%</td><td>-</td><td>86.98%</td><td>-</td><td>-</td></tr></tbody></table>

Choi et al (2023)

<table><thead><tr><th></th><th>Priv. Sig. N-DG</th><th>W-DG</th><th>N-RLr</th><th>N-RLd</th><th>N-RLa</th><th>W-RLr</th><th>W-RLd</th><th>W-RLa</th></tr></thead><tbody><tr><th>Achieved mean accuracy</th><td>60.32%</td><td>78.58%</td><td>77.43%</td><td>76.54%</td><td>76.15%</td><td>77.32%</td><td>79.42%</td><td>80.34%</td><td>78.37%</td></tr><tr><th>Achieved mean consensus</th><td>59.58%</td><td>70.67%</td><td>69.26%</td><td>71.62%</td><td>71.08%</td><td>71.93%</td><td>75.19%</td><td>76.00%</td><td>73.62%</td></tr></tbody></table>

**Table S4.** Achieved mean accuracy and consensus across simulations. The achieved mean accuracy corresponds to the mean proportion of nodes in all of the simulated networks that predicted, in the last stage, the correct state of world (i.e., color of the bag). The achieved mean consensus corresponds to the mean proportion of nodes in all of the simulated networks that made the same predictions as the dominant prediction in the last stage (accurate or not).