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Discovering Influencers in Opinion Formation over Social Graphs

The adaptive social learning paradigm helps model how networked agents are able to form opinions on a state of nature and track its drifts in a changing environment. In this framework, the agents repeatedly update their beliefs based on private observations and exchange the beliefs with their neighbors. In this work, it is shown how the sequence of publicly exchanged beliefs over time allows users to discover rich information about the underlying network topology and about the flow of information over graph. In particular, it is shown that it is possible (i) to identify the influence of each individual agent to the objective of truth learning, (ii) to discover how well informed each agent is, (iii) to quantify the pairwise influences between agents, and (iv) to learn the underlying network topology. The algorithm derived herein is also able to work under non-stationary environments where either the true state of nature or the network topology are allowed to drift over time. We apply the proposed algorithm to different subnetworks of Twitter users, and identify the most influential and central agents merely by using their public tweets (posts).


I. INTRODUCTION AND RELATED WORK
The social learning paradigm is a popular non-Bayesian formulation that enables a group of networked agents to learn and track the state of nature. It has motivated several studies in the literature with many useful variations under varied modeling assumptions (see, e.g., [1]- [12]). Under this framework, agents observe streaming data and share information with their immediate neighbors. Through a process of localized cooperation, the agents continually update their beliefs about the underlying state. These beliefs describe the agents' confidence on each possible hypothesis. The main question in social learning is whether agents are able to learn the truth eventually, i.e., whether the beliefs on the wrong hypotheses will vanish.
In principle, each agent in the network could consider pursuing a fully Bayesian solution to learn and track the state of nature. However, this solution is intractable and generally NP-hard [9], [10]. This is because it requires that each agent has access to the data from the entire network, in addition to their knowledge of the full graph topology. These pieces of information are rarely available in a decentralized setting. For V. Shumovskaia this reason, non-Bayesian approaches have been devised as effective alternatives [1]- [8]. In this formulation, the agents first perform a local Bayesian update using their newly received private observations, and then fuse their beliefs with those of their neighbors either linearly or geometrically [2], [5], [7], [8], [13]. This approach allows for diverse data models across the agents, and helps preserve the privacy of the individual observations. In this work, we adopt the Adaptive Social Learning (ASL) strategy from [5], which showed how to extend traditional non-Bayesian learning under fixed truth to dynamic scenarios where the state of nature is allowed to drift with time. Under ASL, the agents will be able to track these drifts rather effectively with performance guarantees. Now, given a collection of networked agents tracking the state of nature by means of the adaptive social learning (ASL) strategy, our main objective is to focus on two questions related to explainability and inverse modeling. In particular, by observing the sequence of publicly exchanged beliefs, we would like to discover the underlying graph topology (i.e., how the agents are connected to each other). We would also like to discover each agent's contribution (or influence) to the network's learning process.
The question of explainability over graphs is actually a challenging task, and it has been receiving increasing attention (e.g., [14]- [20]). These works approach explainability from different perspectives, and with different aims. For instance, in [19], [20], the authors aim at building a framework for human understanding of why a black-box method arrives at a particular solution. The work [14] argues that a better understanding of multi-agent reinforcement learning can help to limit the search space. Other works suggest modifying the learning algorithm for better interpretability [17]. Overall, higher transparency and a better understanding of the solutions are generally crucial for critical applications, such as using artificial intelligence in healthcare or in the autonomous devices (such as vehicles).
We have performed some prior work on explainability for social learning [21], [22]. In these works, given the evolution of beliefs and assuming some partial prior knowledge about the distribution of private observations, it was shown how to identify pairwise influences inside the network (i.e., how strongly pairs of agents influence each other), as well as how to recover the underlying graph toplogy. In this work, we consider a more limited (and, therefore, more challenging) information scenario, where we do not require any information about the probability distribution of the observations (or their likelihoods). We will show that, under this more demanding scenario, we are still able to identify the contribution of each agent to truth learning, assess its level of informativeness, as well as learn the underlying graph and the pairwise influences between agents. The modeling conditions we consider are generally commonplace in real-world social networks, such as Twitter [23]- [30]. We consider a Twitter application in Section VII. In this application, users publicly exchange their opinions on Twitter, which are therefore observable. However, we do not have access to additional information about sources affecting people's opinion during these exchanges such as News or discussions occurring outside Twitter. Identifying the most influential users and their communication patterns can provide valuable insights in social network analysis [31]. Actually, the problem of identifying the most influential nodes in a network is increasingly relevant [32]- [37], especially following the rise of online social networks. Once identified, this information can be useful in many contexts. For example, it can be used to enhance recommendations for marketing purposes [32], [32]- [43], where the objective is to maximize the number of influenced nodes.
There have also been several important contributions to graph learning such as [21], [22], [44]- [56]. This is because the graph structure plays an important role in distributed learning [57]- [60], and its identification can provide valuable information about relationships within the network [21]. One approach is based on exploiting correlations and similarities between vertices [61]- [63]. Other works assume a defined model behind the graph signal, such as the heat diffusion process [46], [48], [49]. Structural constraints (such as sparsity, connectivity, symmetry) can also be introduced through regularization [45], [45], [50], [54]. The present work examines the problem of identifying a graph in the adaptive social learning setting. Learning the graph from social interactions requires a different approach, as already illustrated in [22], due to the special form of the non-stationary observation signals. Importantly, our study will relax certain assumptions introduced in that work.
The paper is organized as follows. We describe the adaptive social learning model in Section II. Then, we propose an algorithm for learning the combination matrix and the agents' informativeness in Section III. In Section IV, we justify the fact that each agent's contribution to truth learning is proportional to its relative centrality and its informativeness (the KL-divergence between the marginal likelihood of the truth and the other hypotheses). We provide theoretical performance guarantees of the algorithm in Section V. In Section VI, we illustrate performance in different settings by means of numerical simulations. Finally, in Section VII, we apply the algorithm to real-world data from Twitter.

II. SOCIAL LEARNING MODEL
We refer to Fig. 1 and consider a collection of agents N performing peer-to-peer exchanges of beliefs according to some combination matrix A with non-negative entries, [A ] ,k = a k ≥ 0. Agent is able to communicate with agent k when a k is positive; this scalar refers to the weight that agent k assigns to the information received from agent . We assume that the matrix A is left-stochastic and corresponds to a strongly connected graph [2], [5], [7]. The first assumption means that the entries on any matrix column k ∈ N add up to one, ∈N a k = 1. The second assumption means that there exists a path with positive weights between any two agents, and there is at least one agent in network that does not ignore its own observation, i.e., a kk > 0 for at least one k ∈ N . This implies that the combination matrix is primitive, i.e., for any , k ∈ N , there exists t > 0 such that [A t ] ,k > 0. It follows from the Perron-Frobenius theorem [64,Chapter 8] that the power matrix A t converges to u1 T as t → ∞ at an exponential rate, where u is the Perron eigenvector that satisfies: where the u denote the individual entries of u. Each of these entries describes the centrality of the corresponding agent in the graph (i.e., its level of contribution to the inference task). We assume that there exists one true state of nature θ belonging to a finite set of hypotheses, denoted by Θ. Initially, each agent k starts with a private belief vector µ k,0 ∈ [0, 1] |Θ| , where each entry µ k,0 (θ) describes how confident agent k is that θ corresponds to the true hypothesis θ . As befitting of a true probability mass function, the total confidence sums up to one, θ µ k,0 (θ) = 1. To ensure that no hypothesis is excluded beforehand by any of the agents, we assume that µ k,0 (θ) > 0, ∀θ ∈ Θ.
At each time instant i, each agent k observes a measurement ζ k,i . We assume initially that each agent k ∈ N has access to private likelihood functions, L k (ζ|θ), which describe the distribution of the observation ζ conditioned on each potential model θ. The observations ζ k,i are assumed to be independent and identically distributed (i.i.d.) over time. In order to be able to distinguish the true hypothesis θ from any other hypothesis θ = θ , we need to assume that there exists at least one clearsighted agent k ∈ N that has strictly positive KL-divergences relative to the true likelihood, i.e., D KL (L k (θ) ||L k (θ )) > 0, for all θ = θ ∈ Θ. The following boundedness assumption on the likelihood is common in the literature [6], [21]; it essentially amounts to assuming that the likelihoods share support regions.
Next, we describe the ASL strategy from [5]. At each time step i, each agent k performs a local update based on the newly received observation and forms the intermediate (public) belief: Here, δ ∈ (0, 1) is a step-size parameter that controls the adaptation capacity, i.e., δ controls the importance of newly received data relative to the past history. This intermediate belief is shared with the follower agents of k, i.e., with all agents for which a k > 0. Subsequently, agent k fuses the beliefs received from its neighbors, i.e., from all agents for which a k > 0. We denote this set by N k . One fusion rule is to fuse the beliefs geometrically as follows in order to obtain the private beliefs [2], [5], [7], [13]: The term in the denominator in (4) is used to normalize µ k,i (θ) to a probability mass function. Once (4) is performed, the true state θ can be estimated by agent k at time i using the maximum a-posteriori construction over either the private or public beliefs, for example: It was shown in [5], [21] that this social learning scheme has a powerful performance guarantee. Specifically, the probability of error goes to zero for both the private and public beliefs as the step-size δ approaches zero, namely, and Thus, the agents' confidence on hypothesis θ being the true hypothesis converges to one. In other words, the agents are able to learn the truth eventually.
Following [21], we introduce two matrices Λ i and L i in order to represent the recursions (3)-(4) in a more compact matrix form as follows: The matrices are of size |N | × (|Θ| − 1), and their entries are log-belief and log-likelihood ratios and given by: where the reference state θ 0 ∈ Θ can be chosen at will by the designer. The matrices L i are i.i.d. over time due to the assumed i.i.d. assumption on the observations. Again, following similar previous approaches [5], [21], we introduce the following condition on the higher-order moments of L i .
Assumption 2 (Higher-order moments of L i ). The second, third, and fourth-order moments of L i are finite. Moreover, the second-order moment of L i is uniformly positive-definite for all i ≥ 0, i.e., there exists τ > 0 such that: Iterating (8), we find that For large i, it can be shown that Λ i converges in distribution to a limit value given by [21,Lemma 1]: For further analysis, we introduce the following useful result for log-beliefs ratios. The standard laws of large numbers cannot be applied directly to the expression for Λ i in (12) since there are dependencies between the random variables.
Lemma 1 (Law of large numbers for log-belief ratios). After a sufficient number of iterations i (i.e., for i > M 1), the average of the log-belief matrix converges in probability as follows: Proof. See Appendix A.
III. INVERSE LEARNING FROM PUBLIC BELIEFS In this section, we introduce an algorithm for learning the graph combination matrix A from observations of the public beliefs, as well as for assessing the level of informativeness of the various agents.

A. Problem Statement
The data available from the social network might be limited for various reasons, including privacy. Therefore, in this work, we assume that we can only observe the evolution of the public beliefs over time: This assumption is motivated by the fact that agents share their intermediate beliefs computed by (3) during the collaborative process, in contrast to the private beliefs in (4). Here, by i 0 we underline that we are observing Λ i after a sufficient amount of iterations, i.e., after Λ i has reached its steady-state distribution (13). A good illustration for this setting is the social network of Twitter users. From each post (or tweet) that a user posts to their followers, we can extract an intermediate belief ψ k,i based on sentiment analysis, a.k.a., opinion mining. After that, each user k reads the posts of its followers and constructs the private belief µ k,i according to (4).
Next, we will show that by observing public beliefs, we can recover many useful network properties such as (i) the combination matrix, which determines the confidence levels that agents have about each other, (ii) the KL-divergences D KL (L k (θ )||L k (θ)), which assess the capacity of each agent to distinguish the true hypothesis from the other possibilities, (iii) the pairwise influences of agents on each other, and the (iv) global influencers across the network. We allow both the network and the true state θ to drift over time, and the algorithm will be able to track these changes too.

B. Algorithm Development
The previous work on learning the combination matrix A in [21] assumes that the expected log-likelihood matrixL EL i is known. It can be verified that Using this knowledge along with (8), the following objective function was then minimized to learn A : In this work, we do not assume thatL is known beforehand, and will instead estimateL at each iteration i by using where A will be the estimate that is available for A at that point in time. Accordingly, the cost function is replaced by: where we introduced The corresponding risk function is then given by: We will show in Lemma 2 that the unique minimizer of this risk function gets closer to the true combination matrix as M grows. Now, assume that we observe a sequence of public beliefs (15). We define the objective function as a sum over N observed time indices i 0: To solve (22), we apply stochastic gradient descent (SGD) with constant step-size µ > 0. At each iteration i, the estimate A i for the combination matrix is updated via: We sample the observations in the direct order i, i+1, i+2 . . . . This online nature of the algorithm, as well as the use of a constant step-size instead of a vanishing step-size, will allow the algorithm to track changes in A . We list procedure in Algorithm 1.

IV. GLOBAL INFLUENCE IDENTIFICATION
In this section, we establish a strong connection between the probability of error for truth learning and the network divergence. The network divergence is defined in terms of the Perron eigenvector of A , and the KL-divergences between the likelihoods: Note that this quantity is a function of the hypothesis θ. As remarked before (5), the true state estimator can be deduced from the public beliefs. The probability of error is defined as the probability of selecting a wrong hypothesis θ = θ : Note that, if θ does not maximize a public belief, then there exists at least one θ = θ such that Thus, we can equally define the probability of error p k,i as: For this section alone, we will additionally assume that the observations {ζ k,i } are independent over space (and not only time). Now, we know from [5, Theorem 3] that each random Algorithm 1: Graph social learning (GSL) Data: At each time i: Combination matrix update: Log-likelihoods matrix update: variable (for any k) of the form (26) can be approximated by a Gaussian random variable in the steady state with the following moments: for some finite and constant covariance matrix, C. Thus, the probability of error (27) becomes the probability of the Gaussian random variable (28) assuming negative values for at least one θ ∈ Θ. This Gaussian random variable concentrates around its mean (i.e., the network divergence in (24)), which is positive. The larger network divergence is, the smaller the probability of error for each individual agent will be. If we examine expression (24) for the network divergence, we observe that each individual agent k contributes with a term of the form which is scaled by the Perron entry u k . This entry reflects the centrality of agent k, and serves as a measure of how well it is connected to other nodes in the network. The larger the value of (29) is, the stronger the contribution of this agent will be towards moving the network away from an erroneous decision. We therefore say that (29) helps convey the amount of information that agent k has about θ agreeing with θ . Since (29) is a function of θ, we can define the level of informativeness of agent k to the learning process by considering the aggregate of its contributions for all θ, which we denote by This quantity serves as a measure of influence, since agents with large I k contribute the most to learning the truth by the network.
In what follows, we describe how to estimate the quantities I k by the learning algorithm. First, to obtain the Perron eigenvector for A i , we need to normalize any of its eigenvectors corresponding to the eigenvalue at 1. Subsequently, we identify θ k,i that maximizes ψ k,i (θ). From (7) we know that θ k,i tends to θ almost surely as i → ∞ and δ → 0. After a sufficient number of iterations i N of the Algorithm 1, we let j denote the index within the hypothesis set Θ that corresponds to: Returning to (16), we can then approximate the KLdivergences by where L i N is an estimate forL.
To conclude, the sequence of public beliefs contains rich information about the network. Using the GSL algorithm, we are not only able to identify the graph topology, but can also find answers to the explainability question: which agents were the most responsible (or the main drivers) for the overall network learning process?
V. THEORETICAL RESULTS First, we establish some useful properties of the risk function (21).
Lemma 2 (Risk function properties). For any δ > 0, there exists M such that with probability at least 1 − δ and after sufficient number of iterations i, the risk function J i (A) is strongly convex and has Lipschitz gradient with constants ν i and κ i given by: where λ min (·) and λ max (·) are the minimum and maximum eigenvalues. In other words, it holds that: Moreover, the difference between the true combination matrix A and the unique minimizer A min,i of J i (A) is on order of: Proof. See Appendix B.
Result (37) states that the gap between A and the unique minimizer of J i (A) is negligible as M grows.
To investigate the steady-state performance of recursion (23), we adopt the following independence assumption, which is typical in the study of adaptive systems [65]- [67].
Assumption 3 (Separation principle). We denote the estimation error by A i A − A i , and assume the step-size µ is small enough to allow A i 2 F to attain a steady-state distribution. The separation principle states that the error A i is independent of the observations Λ i , . . . , Λ i−M , conditioned on the history of previous observations.
The following theorem shows that in steady-state, the meansquared error is O(µ) + O (δ/M ) in expectation.
Theorem 1 (Steady-state performance). For any δ > 0, there exists M such that with probability at least 1 − δ, the mean-square deviation (MSD) converges exponentially fast with asymptotic convergence rate: where In the limit, the MSD satisfies: Proof. See Appendix C.
Naturally, the convergence rate for learning the combination matrix is dependent on the strong convexity constant, ν. Usually, the limiting MSD expression is of the form O (µ), so that we can reduce the deviation by using arbitrary small stepsize µ. In our case, we also have the additional term O (δ/M ), which is due to the difference (37) between the unique minimizer and the true combination matrix. We can control the number of samples M . Also, in accordance with (7), the ASL strategy [5] has theoretical guarantees of truth learning in the small δ regime, so that the term O (δ/M ) remains relatively small. Finally, we study how well the algorithm approximatesL.
Theorem 2 (Steady-state log-likelihood learning). For any δ > 0, there exists M such that with probability at least 1 − δ, the MSD converges exponentially fast with where is independent of i due to i.i.d. observations.
Proof. See Appendix D.  Since we simultaneously learn the combination matrix A andL, the limiting MSD forL has similar expression to Theorem 1. The final MSD is controlled by the step-size µ and it decreases as we use higher number of samples M , and as δ approaches zero.

VI. COMPUTER SIMULATIONS
In this section, we illustrate how well the proposed algorithm is able to identify the true combination matrix A and the expected log-likelihood matrixL. We also experiment with different M in (18) to see how the convergence rate changes. Additionally, we will see how well the learned KL-divergences identify the influences (30).
We generate a graph with |N | = 20 agents according to the Erdos-Renyi model with an edge probability of p = 0.2. We set the adaptation hyperparameter to δ = 0.05. Then, we generate the combination weights (see Fig. 2a) with uniform weights in the column, such that the resulting matrix is leftstochastic. We consider |Θ| = 5 states, where the likelihood models L k (θ) for each agent k ∈ N are assumed to follow a binomial distribution with randomly generated parameters. We generate likelihood models such that we observe only 3 agents with high informativeness. Later, we illustrate that the algorithm allows for identifying these agents.
First, we consider how well the combination matrix is learned for different M ∈ {1, 10, 50}. We additionally compare with [21], where the expectationL was assumed to be known beforehand. For M = 50, we use µ = 0.1, for M = 10, we use µ = 0.01, and for M = 1, we use µ = 0.001 for better convergence. In Fig. 3, we plot the reconstruction error with respect to the iteration number: We notice that the higher M improves the limiting MSD as reflected in Theorem 1.
Next, we illustrate how well the learned L i approximates L for different M ∈ {1, 10, 50}. In Figure 4, we show how the reconstruction error evolves with i: We notice that with M growing, we can approximateL more precisely, which aligns with the result of Theorem 2, and explains Figure 3.  To reduce the variance and therefore for better interpretability, we plot the rolling mean with window size equal to 50.
We illustrate the recovered combination matrix in Fig. 2b . We use M = 50 since it has better convergence. Comparing Fig. 2a and 2b, we observe an almost perfect recovery of the true combination weights. Figure 5 illustrates how well the learned KL-divergences and combination matrix can recover the global influences (30). For better interpretability, we normalize the values so that they add up to one. We see that for some agents, the algorithm does not perfectly recover these components, but yet allows us to identify that the first agents are driving the learning the most. This property allows us to search for agents that are the most contributing to learning the true state.
Finally, Fig. 6 illustrates the connection between the probability of error (25) and the presence of agents with a significant contribution I k (30) to the network divergence. We plot the rate of correctly classified states at each point i: VII. APPLICATION TO TWITTER DATA In this section, we will apply the graph social learning algorithm introduced in this manuscript to actual datasets. We   choose Twitter as a suitable social media platform where we can analyze the posts of users and observe their interactions. In particular, we aim to detect the centrality (i.e., Perron entry) and influence of agents over a network, and identify the most influential agents. We choose this objective as a more realistic goal for Twitter data, rather than trying to recover the entire combination matrix. This is primarily due to the fact that we can only have access to the true adjacency matrix of the users on Twitter, that is to say, we only know who follows whom, but we do not know the weights (confidence levels) agents assign to each other. In the literature, there have been multiple studies trying to analyze the influence of Twitter accounts on various topics. The work [23] analyzed the propagation of information through Twitter networks, and how different users take part in the effective dissemination of ideas. Moreover, the work [24] analyzes the influential users on Twitter by investigating the linguistic aspects of the tweets of users, such as the grammatical structure and vocabulary of the tweets. However, these studies lack a mathematical foundation to analyze the "influence" of agents, and rely largely on heuristics. In this context, the work [25] aims to leverage ten different attributes of Twitter accounts and their tweets to develop a Twitter-based influence measure. Similarly, the work [26] uses descriptive statistics of the Twitter accounts such as their follower counts, frequency of their posts and the total number of replies, likes and retweets of users. However, what we propose in this study is significantly different. In the proposed algorithm, we do not need to have access to any of these features and the various statistics. In fact, the only input the proposed algorithm needs is the publicly shared tweets of the users. This information is sufficient to learn the centrality of users over a subnetwork of Twitter users, and to identify the most influential user for the formed opinions, using the described mathematical model of social learning.
To identify the influence of agents in a network, we first need to i) create a subnetwork of Twitter users that is strongly connected, ii) obtain the tweets (posts) of the users in the created network, and iii) process the text in the tweets to obtain the log-belief ratios Λ i over time. After these preprocessing operations, we run the proposed algorithm, and estimate the underlying combination matrix of the network of users and the likelihood models of these users. Then, we compute the Perron eigenvector of the estimated combination matrix. We refer to this vector as the "learned" Perron vector, not to be confused with the "original" Perron vector of the true combination matrix of the network of users. Note that, there is no ground truth regarding the confidence agents place on each other, hence one cannot possibly know the true combination matrix. Therefore, we heuristically form a combination matrix to obtain its Perron vector, through the procedure of Section VII-A. However, as an indisputable ground truth data for a given subnetwork, we know the agents that have the highest number of followers within that subnetwork. In our experiments, these agents happen to coincide with the highest original Perron vector entries.
We run experiments on three different user subnetworks of Twitter users by using Twitter API, where we utilize the Tweepy library to build up our queries in Python. Quantitatively, we show that we can identify the most central user in all three networks, i.e., the largest entries of the learned Perron vector and the original Perron vector. Note that in these experiments with real data, we do not have access to the likelihood models of the users, hence we do not have "ground truths" for the influence of agents, i.e., u k × D KL (L k (θ )||L k (θ)). However, to obtain a qualitative measurement, we construct the aforementioned three networks (see Fig. 7) around famous public figures (the first network is built around Elon Musk (CEO of Twitter, 118.5M followers), the second network is built around Maggie Haberman (journalist, 1.6M followers), and the third network is built around Ben Shapiro (columnist, 5M followers). In all of these networks, we recover these public figures as the most influential users.

A. Network formation
To construct a network, we first select a popular Twitter account A, such as an influential CEO or a journalist whom we expect to be influential among other users. Next, in order to select a strongly connected network, we build a network starting from a less centered Twitter account B that is followed by that popular user A. Starting from account B, we construct a subnetwork of depth 2, i.e. we identify 1k followers of account B, and 1k followers of each of those followers. While satisfying these conditions, we filter users who post frequently, so that they provide sufficient data, and who are less centered (they have less than 10k followers). Among all of these identified follower-following relationships, we construct a network, and verify that the network is strongly connected. Following this procedure, the networks constructed for Elon Musk, Maggie Haberman and Ben Shapiro are of sizes 20, 26 and 28, respectively.
After constructing the network, we obtain its adjacency matrix by finding out who follows whom. Since there is no way in real world experiments to determine the confidence agents assign to each other, there is no ground truth for the combination matrix. Therefore, we assume that agents assign uniform weights to their neighbors (i.e., to the people they follow). This corresponds to applying the averaging rule [66,Chapter 14] to the adjacency matrix.

B. Obtaining user posts
In order to obtain the tweets of the users, we first build up our query for the Twitter API. This query includes a specific keyword for each network so that we do not fetch unrelated tweets. For the network containing Elon Musk, we choose the keyword to be "coin OR bitcoin OR crypto-currency", for the tweets between 01.01.2017 and 01.05.2022. In this case, we would like to see the influential agents in that network in shaping the opinion of Twitter users on cryptocurrency related matters. For the other networks, we choose the keyword to be "Trump" for the tweets between 01.01.2017 and 31.01.2021, and "Biden" for the tweets between 01.01.2021 and 01.05.2022. Hence, we aim to determine the influence of agents in determining the attitude of their respective networks regarding the issue of "the current president of the United States".

C. Obtaining the log-ratio beliefs of users
After obtaining the tweets of the users, we need to process the text in those tweets to obtain the intermediate belief vectors. For this purpose, we use a language model based on Roberta, which is trained with around 124M tweets, and fine tuned with the TweetEval benchmark for the sentiment analysis task [68]. We can feed a text to this model and obtain three different probabilities for three different sentiments of the input text: p 0 for "Negative", p 1 for "Neutral", and p 2 for "Positive". We eliminate the neutral labels and recalculate the probability of the positive sentiment of a text as p p 2 p 0 + p 2 (46) and accordingly, the probability of the negative sentiment as   In Table I, we show sample tweets from the network related to Bitcoins. When p is close to 1, this means that the text asserts a positive attitude towards Bitcoin, and when p is closer to 0, this means that the text asserts a negative attitude.
To construct the log-belief ratio of an agent at a day i, we incorporate the sentiment of all tweets within this day. We denote the number of tweets user k shares at iteration i by N k,i . Among those tweets, we denote the positive sentiment probability of each tweet t by p k,i,t , so that p k,i,t approximates belief ψ k,i,t (θ 0 ), and 1 − p k,i,t as approximates belief ψ k,i,t (θ 1 ). Here, θ 0 is the hypothesis that the underlying topic (e.g., Bitcoin, the current president of the United States, etc.) is "good", and θ 1 is the counter-hypothesis. Then, we construct log-belief of each agent k as: Note that if N k,i = 0 for some agent k at some iteration i, we set Λ k,i to previous value of it, i.e. Λ k,i = Λ k,i−1 .

D. Adjustments to the algorithm
We have made some adjustments to the graph social learning algorithm in Algorithm 1 to cope with real life data. In particular, instead of performing gradient updates at each iteration, as in the online stochastic learning algorithm, we use stochastic mini-batches. Namely, we select a window size W , and average the gradients calculated in that window, and then perform the gradient step. In our experiments, we observe that this practice reduces the noise in the gradients due to real life data. Furthermore, we use 1 regularization to promote sparsity. The motivation for sparsity is to get rid of unnecessary links between different users in the graph. These hyperparameters in the experiments are determined with a grid search on one of the networks, and then the parameters are used in the other two networks. These hyperparameters are: for the stochastic mini-batch size, W = 30, for the stepsize δ = 0.0001, for the learning rate µ = 0.0003 and for the

E. Experimental results
We compare the Perron vectors of the learned combination matrix and the original combination matrix (which is constructed by assigning uniform weights to users each user follows). In these comparisons of Perron vectors, we show that we can correctly estimate the most central agents in all three networks. These comparisons can be seen in Fig. 8. Secondly, we show the influence plots of the agents in these three networks in Fig. 9. These figures, "qualitatively" show that we can indeed identify the popular accounts (Elon Musk, Maggie Haberman and Ben Shapiro) as the most influential agents in their respective networks. For instance, in Fig. 9a, we can identify Elon Musk as the most influential Twitter user of his network of users.

VIII. CONCLUSIONS
In this study, we show that a sequence of publicly exchanged beliefs in the adaptive social learning protocol contains rich information about the underlying model. We present an algorithm for learning the agents' informativeness in terms of KLdivergences between likelihood models, and for identifying a combination graph. We demonstrate that these quantities determine the probability of error of the true hypothesis estimator, and we introduce a notion of a global agent influence, which quantifies the individuals' contribution to learning. As a result, the suggested approach enables us to determine the most influential agents in the opinion formation process. We also describe how to apply the algorithm to Twitter data. Our experiments on both synthetic data and Twitter data illustrate that we can accurately find global influencers and learn the underlying graph.

IX. ACKNOWLEDGEMENTS
We thank our colleague Virginia Bordignon for her valuable suggestions on application to Twitter data.
The expectation above can be expanded into: Introducing the filtration F i {ζ k,j , j ≥ i, ∀k ∈ N }, which collects all observations up to time i, and conditioning over F i , we have This equality holds since E (Λ j1 |F j1 ) = Λ j1 . Using the main recursion formula (8), we can represent Λ j2 in terms of Λ j1 and observations L t with j 1 + 1 ≤ t ≤ j 2 : It follows that whereL is the expected value of L i defined in (16). Using (53), we can rewrite (51) as: (54) where the last equation holds since in steady state EΛ j1 = EΛ. The trace of (54) then becomes 1 : where the last inequality holds because A is left-stochastic, i.e. all the entries are non-negative and each column's entries add up to one. In [21,Lemma 1] it is shown that Λ can be element-wise bounded bȳ Therefore, for each considered j 1 , the following holds: and therefore, since Λ j1 reached the steady-state distribution (13). Combining the derivations above, expression (49) becomes: Taking M → ∞, the right-hand side of (59) is zero for any ε > 0. Thus, we establish convergence in probability by definition.

APPENDIX B PROOF OF LEMMA 2
Under some reasonable assumptions on the distribution of the random variables, as required by the dominated convergence theorem in mathematical analysis, it is possible to exchange the expectation and gradient operations. Thus, using (19), the gradient of (21) is given by: We first verify that the risk function J i (A) has Lipschitz gradients. Not that for any A 1 , A 2 : where step (a) follows from the following considerations. For matrices X and Y of appropriate dimensions, where X is positive definite with eigendecomposition X = U X Λ X U −1 X , the following inequality holds: Similarly, we can verify that J i (A) is strongly convex since Next we verify that E∆ i ∆ T i is positive definite and finite. For this purpose, we refer to [21,Lemma 1], which establishes that Λ (13) is element-wise bounded as follows: Then, E∆ i ∆ T i is also bounded: From Lemma 1 we know that the following convergence in probability holds: By definition of convergence in probability, for any ε > 0 and δ > 0, there exists M 0 such that for any M ≥ M 0 and with probability at least 1 − δ, the following inequality holds: Taking Thus, using definition (20) and the fact that g(X) = XX T is a continuous bounded function 2 , we get that with probability at least 1 − δ: where the last inequality holds due to Assumption 2. We now examine the relation of the minimizer of A min,i min A J i (A) to the true combination matrix A . From strong convexity, the minimizer of J i (A) is unique and satisfies: Referring to (60), we get Therefore, In turn, using (20), the recursion (8) can be modified into Subtracting (72) and (74)gives: Iterating (8), we get the following representation for Λ i : Since L i are i.i.d and using (75) and the definition of ∆ i in (20), we rewrite the expectation in (75) as: with and where we used the fact that Thus, combining (69), (75) and (77), we find the following relation between A and A min : APPENDIX C PROOF OF THEOREM 1 Using recursion (8) for Λ i and the definition of L i−1 in (18), the SGD step (23) can be rewritten as follows: Recall the deviation from the true combination matrix: Combining (81) with (82) and the definition of ∆ i in (20), and using (8), we obtain: The mean-square deviation is then given by where ρ(·) denotes the spectral radius of its matrix argument. Considering the first term: where the last inequality holds since ν ≤ κ by definitions (34) and (35). We omit the time indices in ν i and κ i since we are assuming that Λ i has reached the steady-state distribution (13), and ν and κ are defined as: and are positive with high probability by Lemma 2. Introduce the filtration F i {ζ k,j , j < i, ∀k ∈ N }, and consider the conditional expectation of the second term of (84): From [22,Lemma 1] we know that Λ i can be bounded entrywise: and the upper bound has a finite limit, lim i→∞Λi . Therefore, the second term of (87) can be upper bounded by: Thus, by the law of total expectations, the full expectation of (87) is bounded as follows: Now, we proceed with the third and fourth terms of (84), which take the following form: and which is finite under Assumption 2. With similar considerations as above, expression (91) contains moments of L i up to the fourth term. To find the expectation of the second trace term in (84), consider first the conditional expectation, and then use Assumption 3: where we use the fact that ∆ i−1 depends only on L i−M −1 , . . . , L i−1 . Finally, consider the last term of (84). Using Assumption 3 and equation (77), we obtain: Summarizing the derivations above, we can transform (84) into: with γ = O(1), c = O δ 2 /M and α = 1 − 2µν + O(µ 2 ). Thus, For small enough µ, α is strictly less than one. Hence, we obtain the following limiting MSD: APPENDIX D PROOF OF THEOREM 2 First, using recursion (8), we rewrite (18) as: Consider the mean square deviation of L i : Consider the first norm: Let us study the second term of (98) using Assumption 3 and Lemma 1. Introduce the filtration F i {ζ k,j , j ≤ i, k ∈ N }. Then, where δ in O (δ/M ) appears due to the form of Λ in (13). Now, consider the trace term in (98): where (a) holds due to Assumption 3. Using Assumption 2, the second multiplier is on the order of O (1). Proceeding with (101): where again δ in O (δ) is present due to the form of Λ in (13). Summarizing the derivations above, we can upper bound the expectation (98) as: where the last equality holds because we can represent EΛ 2 F = O(δ 2 ) since EΛ has δ multiplier as in (13). By Theorem 1, we derive the following MSD in the limit, that holds with high probability: