Entropy-based detection of Twitter echo chambers — TEI-tagged view

Manuel PratellipersName, Fabio SaraccopersName, Marinella PetrocchipersName (2024date). “Entropy-based detection of Twitter echo chambers.” PNAS Nexustitle·j.

Object of study — echo chambersterm·dom in the Italian Twitter debate on Covid-19 vaccination

Research question — how to detect echo chambersterm·dom without a generally accepted method, and what their characteristics and link to disinformation are

Contribution — an unbiased entropy-based method that detects echo chambersterm·dom as the overlap of discursive and news-engagement communities

Full text — tagged

Entropy-based detection of Twitter echo chambersterm·dom Abstract Echo chambersterm·dom, i.e. clusters of users exposed to news and opinions in line with their previous beliefs, were observed in many online debates on social platforms. We propose a completely unbiased entropy-based method for detecting echo chambersterm·dom. The method is completely agnostic to the nature of the data. In the Italian Twitter debate about the Covid-19 vaccination, we find a limited presence of users in echo chambersterm·dom (about 0.35% of all users). Nevertheless, their impact on the formation of a common discourse is strong, as users in echo chambersterm·dom are responsible for nearly a third of the retweets in the original dataset. Moreover, in the case study observed, echo chambersterm·dom appear to be a receptacle for disinformative content.

Keywords: echo chambersterm·dom, misinformation, Twitter, complex networks, information theory While the concept of echo chambersterm·dom is clear, a generally accepted method for their detection is still lacking in the literature. Our study provides a general and unbiased method for detecting echo chambersterm·dom, using entropy-based null modelsterm·mth as statistical benchmarks. The rationale is to detect groups of users with similar opinions based on the significant similarity of the main content creators and, similarly, to detect groups of users engaged with the same news. A nontrivial overlap between the two groups indicates the presence of an echo chamberterm·dom. Using the Italian Twitter debate on the Covid-19 vaccination as a case study, we found that users in echo chambersterm·dom, while representing a small minority, strongly contribute to the debate, often disseminating misinformation.

Introduction In the virtual world, the tendency to seek out information that confirms existing beliefs and to interact with users who share similar opinions leads to the formation of echo chambersterm·dom, i.e. “bounded, enclosed media spaces that have the potential to both amplify the messages delivered within it and insulate them from rebuttal” ( 1–4 ). We thus have two key events in echo chamberterm·dom formation: (i) interaction between users with similar opinions; (ii) exposure of users to the same news articles.

Contributions This article studies the actual presence of echo chambersterm·dom in social networks by detecting the overlap of the two events. The detection is done by adopting an entropy-based technique. The platform considered in this study is Twitter/X. From now on, we will refer to the platform by its former name, Twitter, since the analyses were conducted before the change in the company name to X.

Recently, entropy-based null modelsterm·mth have been introduced in studies of complex networks as an unbiased benchmark capable of revealing nontrivial structures of real systems ( 5 ), and thus they represent the appropriate framework for our analysis. Figure 1 shows how we intend to assess the occurrence of the two events, and, consequently, the occurrence of the echo chamberterm·dom.

Pipeline for Echo-chamber detection. The upper path focuses on the detection of Discursive Communities ( DiCoterm·dom ), while the lower one on the detection of News Engagement Communitiesterm·dom ( NECterm·dom ). Both procedures pass through the statistical validation of empirical data with an entropy-based null modelterm·mth.

Assessing the opinions of the various accounts is not an easy task, but it can be inferred from the interaction among the various accounts. In Refs. ( 6–9 ), a method to infer the presence of a discursive communityterm·dom, i.e. a group of accounts contributing to the formation of a common discourse, was presented. It is based on verified usersterm·dom, i.e. those accounts for which Twitter has a procedure to check the identity of their owners. Verified accounts mainly belong to politicians, journalists, and celebrities; usually, they are strong creators of contents ( 6 , 7 , 9 ). Verified usersterm·dom are among the greatest contributors to the formation of a common discourse. It is possible, then, to let similarities emerge among the content created by verified usersterm·dom, based on the behavior of their common audience in terms of retweets, since retweets are considered a measure of engagement ( 10–12 ). In detail, for each pair of verified usersterm·dom, the number of common retweeters is counted. If the number is statistically significant with respect to an entropy-based benchmark, it is validated and we project a link between the verified usersterm·dom’ pair. On the monopartite network of verified usersterm·dom thus obtained, we run a community detection algorithm to extract groups of similar verified usersterm·dom (i.e. the Verified usersterm·dom’ DiCoterm·dom s in Fig. 1 ). Then, the various communities of verified usersterm·dom are labeled in terms of the users who belong to them (since the users are verified, it is possible, for example, to derive their political leanings and test a posteriori the resulting communities). At this point, the labels are extended to unverified users using a label propagationterm·mth algorithm ( 13 ) on the entire retweet networkterm·key—thus encompassing both verified and unverified users.

Once again, the use of the retweet networkterm·key for label propagationterm·mth is motivated by the fact that there is evidence that users belonging to communities in a retweet networkterm·key share similar views ( 10–12 ). In the following, such communities will be called discursive communities (or DiCoterm·dom), and their detection is sketched in the top path of Fig. 1 . Discursive communities embrace those users who contribute to the formation of a common discourse .

Regarding the exposure to the same news articles, we approach its assessment by analyzing the ties between the users and the URLs present in their tweets and retweets. The bottom path of Fig. 1 shows the approach. The idea of leveraging the bipartite networkterm·mth of users and URLs was already considered in Ref. ( 14 ) for Facebook: in the present case, we translate the idea therein to Twitter. Again, the procedure goes through a comparison between observations and an entropy-based benchmark: if two users tweeted (or retweeted) the same URLs significantly more than the benchmark, we conclude that the two users share the same information dietterm·dom in a statistically significant way. We can thus identify groups of users sharing the same URLs. In the following, user communities that passed the validation are called news engagement communitiesterm·dom of users, for short user NECterm·dom s. User NECsterm·dom contextualize the second event: exposure of users to the same news articles.

Now, we were able to identify groups of users exposed to the same news articles (user NECterm·dom ) and groups of users who share a common discourse (DiCoterm·dom). Users who share a group of the first type and a group of the second type form an echo chamberterm·dom, provided they interact with each other. The interaction for us is that of retweets since retweets are considered as a form of endorsement to the content created by others ( 6 , 10–12 ). Verifying user interactions is an important step because accounts belonging to the same user NECterm·dom may either not belong to the same DiCoterm·dom or, even in the case where they are in the same discursive communityterm·dom, may not interact with each other. In this sense, only users who (i) belong to the same user NECterm·dom and (ii) belong to the same DiCoterm·dom and (iii) are connected, even indirectly, through retweets (i.e. they form a weakly connected component in the retweet networkterm·key) can be said to represent an echo chamberterm·dom.

As a case study for evaluating the presence of echo chambersterm·dom, we consider the online debate on Twitter regarding the Covid-19 vaccination campaign. Surprisingly, compared to numerous examples found in the literature, we find a limited presence of echo chambersterm·dom in the analyzed dataset, mainly due to the small dimensions of users’ NECsterm·dom. Although the detected echo chambersterm·dom are composed of a small number of users with respect to the total number of active users, they play a significant role in terms of retweet interactions, i.e. the echo chambersterm·dom that emerged in the case study of the Covid-19 vaccination debate have a significant impact on the creation of a common and cohesive discourse that is not devoid of disinformation. Furthermore, users who belong to such echo chambersterm·dom show the same ideas and opinions after years.

Summarizing, the main contribution of this article is a novel unbiased method for echo chamberterm·dom detection. The procedure is based on the very definition of echo chambersterm·dom and involves the application of an entropy-based null modelterm·mth to discard signals assimilated to noise.

Research questions Keeping in mind that our ultimate goal is to observe if and when discursive communities and NECsterm·dom of users overlap, thus forming echo chambersterm·dom, we organize the structure of the article to answer the following research questions (RQs): RQ.1: What are the characteristics of the discursive communities (DiCosterm·dom) and of the news engagement communitiesterm·dom of users (users’ NECsterm·dom)? Are there users in common?; RQ.2: What is the relation between the emergent echo chambersterm·dom and the presence of disinformation, if any?

Results Dataset Our dataset consists of ∼1.87 M public posts in Italian and 136 k users; nearly ∼220 k tweets contain URLs. We relied on the Twitter’s streaming API and data were collected from 2021 September 1st to September 24th. The data collection was keyword-based and related to the COVID-19 vaccination online debate. The keywords are compatible with chronicles regarding the vaccination debate in Italy at time of data collection. We remind the reader that the Twitter’s streaming API returns any tweet containing those terms in the text of the tweet, as well as in its metadata. It is worth noting that it is not always necessary to have each permutation of a specific keyword in the tracking list. For example, the keyword “COVID” would return tweets that contain also both “COVID19” and “COVID-19”. The keywords for the data collection are in Section S1 .

Data were downloaded using the Twitter API access for researchers (available at the time of the download) and were analyzed following X/Twitter’s policy (available at the following url: https://twitter.com/en/privacy ).

Discursive communities Figure 2 (A) describes the characteristics of the main discursive communities (DiCosterm·dom) that emerge from the data. We recall that it is possible to assign labels to verified accounts, as the identity of their owner has been certified by the platform. Starting from the original dataset, we run the community detection algorithm ( 15 ) on the validated network of verified usersterm·dom and the label propagationterm·mth algorithm ( 13 ) on the network of retweets of the different communities. In our case study, two main discursive communities emerge, associated with political parties and Italian newspapers. Specifically, most of the users who are part of a DiCoterm·dom belong either to the Itav-pd-media community ( ∼ 34.7 % ; the community includes journalists and exponents of the Italian parties Italia Viva and Democratic Party) or to the Fdi-l-media community ( ∼ 26.6 % ; the community includes journalists and exponents of the Italian parties Fratelli D’Italia and Lega). About 2.1 % of users belong to smaller DiCosterm·dom, while ∼ 36.7 % of users do not belong to any DiCoterm·dom. The Fdi-l-media community posted the most new content ( 64.3 % ), although it represents about a quarter of all users in our dataset. The Itav-pd-media community is responsible for 19.7 % of the new content, while the remaining 15.5 % is posted by users who do not belong to any particular community. In terms of retweets, Fdi-l-media is by far the most active community with 77.6 % of the retweets.

Characterization of the main DiCosterm·dom in terms of the number of users, tweets, and retweets. The charts in A) consider all tweets and retweets, the charts in B) only consider tweets and retweets that contain URLs.

Figure 2 (B) characterizes DiCosterm·dom by focusing only on posts containing URLs. In general, the observations made for the doughnut charts in panel (A) still hold, with the exception that almost half of the users who post tweets with URLs belong to the Fdi-l-media community ( 48.7 % ).

NECsterm·dom of users Table 1 shows that of all users who have published at least one post with a URL ( ∼ 33 k ), only 566 are part of a user NECterm·dom, which is less than 2 % . Accounts in user NECsterm·dom are proportionally much more active in publishing URLs than users not validated by our procedure (67.7 vs 5.90 URLs per account).

Users in user NECterm·dom.

Figure 3 (A) shows how the 566 users cluster into different user NECsterm·dom, while Figure 3 (B) provides a statistical view of the 566 users associated with the user NECterm·dom. On the right, the top doughnut chart illustrates the largest communities based on the number of users. Each of these prominent user NECsterm·dom (IDs 0, 1, 2, 3, 4, 5, and 6) accounts for at least 95 % of the total user population within this type of community. Furthermore, the lower doughnut chart shows that these communities have the highest frequency of tweets containing URLs. Communities 1, 2, 3, 4, and 5 collectively account for over 78 % of the total URL traffic generated by all user NECterm·dom communities. An analogous analysis of the URL NECsterm·dom, i.e. the community detected on the validated projection on the layer of the URLs can be found in Section S2 .

User NECsterm·dom. A) Network representation of user NECsterm·dom. B) Top: percentage (and number) of user NECterm·dom users belonging to each group. B) Percentage (and number) of URLs disseminated by users belonging to the various user NECsterm·dom.

Echo chambersterm·dom Our analysis shows that all but 1 of the 566 users in the user NECsterm·dom are also part of the same discursive communityterm·dom, i.e. Fdi-l-media . This is the discursive communityterm·dom with users affiliated with political parties Fratelli D’Italia and Lega, and news outlets showing similar leanings. However, the fact that all users in the user NECsterm·dom belong to the same DiCoterm·dom only tells us that users with similar “information dietsterm·dom” contribute to the formation of the same discourse, but not that they influence each other and reinforce the opinions of their siblings. In other words, users who refer to the same news sources may never meet on the platform. In fact, the information about who interacts with whom is not used to detect user NECsterm·dom.

As mentioned in the Introduction section, users in an echo chamberterm·dom are users who share a common discourse, are exposed to the same news sources, and are exposed to the same opinions. Being exposed to the same opinions, translated to Twitter, means that they retweet each other. In this sense, if users in the same user NECterm·dom form a (weakly) connected component in the same DiCo-induced subgraph of the retweet networkterm·key (i.e. if there is a flow of influence in the retweet networkterm·key that is restricted to nodes in the same discursive communityterm·dom), they form an echo chamberterm·dom.

The analysis of the weakly connected component shows that 92 users do not belong to it. This leaves 473 users trapped in echo chambersterm·dom. In particular, all users in user NECsterm·dom 8, 9, and 10 did not retweet others in the same user NECterm·dom on the topic under analysis. Regarding the other user NECsterm·dom, we observe that for each of them, most of the nodes form echo chambersterm·dom. In the following, echo chambersterm·dom inherit the ID of their user NECterm·dom. Some echo chambersterm·dom are relatively large: for example, those induced by user NECsterm·dom 1 and 2 contain more than 100 nodes.

To study how much users in echo chambersterm·dom are connected, we use the undirected clustering coefficientterm·mth: ignoring the direction of the edges, it captures the observed frequency of interactions between the neighbors of each node ( 16 ).

We compare the clustering coefficientsterm·key of the echo chambersterm·dom with the one measured on the Largest Weakly Connected Componentterm·mth (LWCCterm·mth) of the retweet networkterm·key restricted to users in the Fdi-l-media DiCoterm·dom. In this way, we have a benchmark that captures the main contribution to the discourse to which the echo chambersterm·dom belong. The clustering coefficientterm·key associated with users in echo chambersterm·dom is more than three times as high as that for other users within the LWCCterm·mth [ 0.56 compared to 0.16 , Fig. 4 (A)]. We then examine the average clustering coefficientterm·key within each echo chamberterm·dom. Figure 4 shows that the average clustering coefficientsterm·key of echo chambersterm·dom 2, 4, and 11 are greater than 0.6.

A) Average clustering coefficientterm·key measured on the LWCCterm·mth of the retweet networkterm·key restricted to users of Fdi-l-media and measured on all users belonging to echo chambersterm·dom. B) Average clustering coefficientterm·key calculated on each echo chamberterm·dom. Each echo chamberterm·dom inherits the ID and the color from its user NECterm·dom. The number of users in the echo chamberterm·dom is shown at the top of each bar.

High values of the clustering coefficientterm·key imply that accounts are highly connected and frequently retweet each other. Therefore, we can conclude that their endorsement activity contributes to the reinforcement of their opinions. Such a conclusion is confirmed by a manual examination of the content shared by users in echo chambersterm·dom after almost 2 years. At the time of the data collection, the opinions of the users were strongly against the Covid-19 vaccination. After 2 years, the positions of the users there still adhere to conspiracy theories and have become particularly extreme.

To provide a concrete example, we focus on the content shared in echo chamberterm·dom 4. In practice, we first manually extract the main narratives from the news shared within echo chamberterm·dom 4, focusing on the users with the highest number of followers at the time of data collection. Then, still focusing on the users with the highest number of followers, we analyze whether there are signals of these narratives in their most recent posts (as of 2023 June 7) and which narratives they currently support. In echo chamberterm·dom 4, there are about 1.7k unique news that have been shared about 7.3k times in total. First, we exclude the news with connection errors at the time of this analysis (1k shares) and those that have been shared less than 10 times. Then, we analyze the resulting news narratives, which amount to ∼ 3.3 k shares and 146 unique URLs from 51 different domains. By classifying only these 146 news stories, we cover about ∼ 45 % of the total URL traffic within echo chamberterm·dom 4. Table 2 shows the narratives’ distribution and their descriptions: the main eight narratives are all against vaccination and government regulations.

Narratives’ descriptions in echo chamberterm·dom 4.

Table 3 shows the narratives supported by the users in echo chamberterm·dom 4 with the most followers, almost 2 years after data collection (2023 June 7). Users hold extreme views on current controversial issues such as the war in Ukraine, migrants, and LGBT issues. Remarkably, conspiracy theories about vaccines are still present in their narratives.

Main narratives supported in recent posts (as of 2023 June 7) by users in echo chamberterm·dom 4 with the most followers.

Echo chambersterm·dom, their role in the common discourse and the plague of misinformation Figure 5 shows the flow of retweets within an echo chamberterm·dom and between different echo chambersterm·dom.

Retweet networkterm·key for Fdi-l-media DiCoterm·dom, aggregated with respect to echo chambersterm·dom. Node − 1 represents users who do not belong to an echo chamberterm·dom. Edges indicate the number of retweets between different user groups; weights less than 1 k have been filtered out.

Node − 1 represents all nodes in the DiCoterm·dom that are not part of an echo chamberterm·dom, and an arrow indicates that tweets published by the source group are retweeted by a certain number of users in the target group. Self-loops represent retweet activity within the same group. The values on the edges indicate the number of retweets associated with each interaction. Although the echo chambersterm·dom are composed of a small number of users (on the order of 10 2 , compared to the total number of DiCoterm·dom users, on the order of 10 4 ), they contribute significantly to the DiCoterm·dom’s retweet activity. Echo chambersterm·dom are involved in generating about 288k retweets, while users not in echo chambersterm·dom generate about 569k retweets. More specifically, echo chambersterm·dom 2 and 3 are mainly composed of popular users (in terms of received retweets), while others are mainly composed of retweeting users (0, 1, 4).

To quantify the presence of misinformation in echo chambersterm·dom, we have tagged URLs in our dataset that point to news sites. The labels are those that the NewsGuardterm·key journalistic organization has assigned to online media outlets. a The use of the labels has been licensed to the authors of this article. More details about the reputability measure implemented in the present manuscript can be found in Section S3 .

Figure 6 shows the number of URLs pointing to news from publishers that NewsGuardterm·key classifies as “Trustworthy” (T), “Not Trustworthy” (N), and “Unclassified” (UNC) for the entire dataset and for each type of user community. If the same URL is shared multiple times by users in the same group, this multiplicity is taken into account in the analysis.

Number of distinct URLs pointing to news publishers tagged as “Trustworthy” (T), “Not trustworthy” (N), or “Unclassified” (UNC) for the entire dataset and for each type of users’ community (DiCosterm·dom, user NECsterm·dom, echo chambersterm·dom.) The first observation is that the differences between user NECsterm·dom and echo chambersterm·dom are negligible. Second, DiCosterm·dom cover almost the entire volume of both T and N traffic. Remarkably, while the ratio between untrusted and trusted URLs is around 0.5 for the entire dataset, the ratio is almost reversed for echo chambersterm·dom: the frequency of N news sources is almost twice that of T news sources.

Section S4 will show even more alarming results regarding the spread of disinformation in echo chambersterm·dom. We do not show these results in the main text due to space limitations, but i) the probability that a link shared by a user in an echo chamberterm·dom refers to an untrustworthy news source is 0.377, compared to 0.129 for users outside echo chambersterm·dom; ii) the probability that a link shared by a user in an echo chamberterm·dom refers to a trustworthy news source is 0.232, compared to 0.379 for users outside echo chambersterm·dom.

Conclusion In this article, we propose a novel unbiased method to detect echo chambersterm·dom. The method is mainly based on two observations. First, echo chambersterm·dom form when users interact with others who share similar opinions and refer to the same news. Second, a proper null model should be implemented to detect a true signal. This necessity has recently been highlighted in the literature on online social networks and has been shown to be particularly important for the detection of nontrivial phenomena ( 6–8 , 17–19 ). Our echo chamberterm·dom detection method is based on the validation of observed structures by comparison with a proper maximum entropy null model; the maximization of entropy guarantees the unbiased nature of the benchmark.

We tested our procedure on a dataset containing the Italian Twitter debate on Covid-19 vaccination: we found that our procedure detects a low presence of echo chambersterm·dom (just under 0.35% of all users in our dataset belong to an echo chamberterm·dom). All the echo chambersterm·dom we detected are part of the same discursive communityterm·dom, i.e. a community of users with similar political positions. Even if their dimension in terms of the number of users is limited, their impact on the shared discourse is remarkable: echo chambersterm·dom are responsible for almost a third of the retweets in their discursive communities.

The methodology can be extended to other online social networks. In fact, it is based on (i) the analysis of the activity of accounts that share URLs to news sources and (ii) the detection of discursive communities. While the extension of the former to other online social networks is straightforward, the latter may be more problematic: in the present case, we used the activity of verified usersterm·dom, who are among the main content creators in Twitter ( 6 ), but not all social platforms have such certification. Nevertheless, when analyzing other platforms, we can still focus on users who are particularly active in creating new content, such as influential users as defined in ( 20 ).

Not unlike other studies, our study has some limitations, which we believe do not affect our final conclusions. First, it may be argued that the validation procedure is quite strict: the validation of multiple P -values leads to the validation of extreme events. While this is true, it is the only way to eliminate random noise from the system and analyze the true signal (see Sections S5 and S6 for more details). Finally, the main idea of echo chambersterm·dom is that users follow accounts with similar ideas, while in the present study only the retweet networkterm·key is used, not the information about friendships. Still, the retweet networkterm·key captures the effective interactions with interesting content as perceived by different users, whether it comes from friends or is suggested by the platform itself: focusing only on friendship will not fully capture the effect of the platform’s recommendation algorithm.

Methods Network analysis methods Recently, De Clerck et al. stressed the importance of using proper statistical benchmarks for the analyses of Online Social Networks ( 18 , 19 ): in fact, such systems are affected by strong noise and detecting genuine signals is fundamental in order to drive the proper conclusions. In fact, our procedure for the detection of echo chambersterm·dom is based on the statistical validation of different co-occurrence networks. Co-occurrences are implicitly based on a bipartite structure: if we count, for instance, the number of URLs that have been shared by both a pair of users, we are implicitly projecting the information contained in a bipartite networkterm·mth in which layers represent users and URLs on the layer of users. Therefore, including the bipartite information in the analysis of the observed co-occurrences provides a more accurate benchmark.

A general framework for providing unbiased benchmarks for the analysis of complex networks was recently proposed in the literature ( 5 ), inspired by the derivation of Statistical Physics from Information Theory by Jaynes ( 21 ). The main idea is to first create an ensemble of all graphs having the same number of nodes as in real systems. We can then define the Shannon entropy associated with the ensemble: in order to have a maximally random benchmark, we maximize the Shannon entropy, constraining some defining quantities about the system. In this sense, by comparing the real network with our null model, all observations that cannot be explained by the constraints can be captured. Constraints can be global, as the total number of links, or local, as the degree sequenceterm·mth, i.e. the number of connections per node.

In the following, we will first introduce the Bipartite Configuration Modelterm·mth ( BiCMterm·mth , ( 22 )), i.e. the application of the procedure described above to bipartite networksterm·mth in which the degree sequencesterm·mth are the constraints. Then we will describe the validation procedure for co-occurrences, proposed in Ref. ( 23 ). Both the BiCMterm·mth and the validation procedure used in the present manuscript were performed using the bicmterm·mth b python module included in NEMtropy; c the methods used to solve BiCMterm·mth system of equations implemented in NEMtropy and bicmterm·mth can be found in Ref. ( 24 ).

Formalism In a bipartite networkterm·mth, nodes are divided into two sets, called layers and links exist only between nodes belonging to different layers. Given a bipartite networkterm·mth G Bi , let us call its layers ⊤ and ⊥, respectively, and N ⊤ and N ⊥ their dimensions. Then, a bipartite binary network is completely described by its biadjacency matrix B , i.e. a N ⊤ × N ⊥ rectangular matrix whose generic entry b i α is either 1 or 0 if there exists a link connecting node i ∈ ⊤ and α ∈ ⊥ or not. The degree of a generic node i ∈ ⊤ ( α ∈ ⊥ ) is simply k i = ∑ α ∈ ⊥ b i α ( h α = ∑ i ∈ ⊤ b i α ). In the following, quantities related to real networks will be indicated with an asterisk *.

Bipartite configuration modelterm·mth Let us call G Bi the ensemble of graphs of the BiCMterm·mth in which each representative graph G Bi ∈ G Bi is a N ⊤ * × N ⊥ * bipartite networkterm·mth. d We define the Shannon entropy associated with the system as We can perform a constrained maximization of the Shannon entropy using the methods of Lagrangian multipliers, the constraints being the degree sequencesterm·mth of both layers, i.e. k i , ∀ i ∈ ⊤ , and h α , ∀ α ∈ ⊥ . In this way, we will achieve a benchmark that is maximally random, but, in which the average degree sequencesterm·mth are equal to the ones observed in the real system. Therefore, by observing deviations from the null model we will detect all structures of the real system that cannot be simply explained by the constraints. Such a procedure can be achieved through the maximization of the function S ′ defined as where S is the Shannon entropy defined in Eq. 1 and β , θ i , and η α are the Lagrangian multipliers associated, respectively, to the normalization of the probability, to the degree sequenceterm·mth on layer ⊤ and to the degree sequenceterm·mth on layer ⊥. The maximization of S ′ returns a probability per graph that can be written in terms of independent probabilities per link ( 25 ): Equation 2 is just formal since we do not know the numerical value of Lagrangian multipliers θ i and η α . This can be obtained through the maximization of the likelihood of observing the real system ( 26 , 27 ). It can be shown that maximizing the likelihood is equivalent to set: In Section S5 , the interested reader can find a detailed description of how to use the Bipartite Configuration Modelterm·mth as a statistical benchmark to validate the co-occurrences observed in the real network.

Discursive communities As stated above and described in detail in Section S5 , the BiCMterm·mth described above can be used as a statistical benchmark to highlight groups of users contributing to the formation of the same discourse. On Twitter, this translates to groups of users endorsing similar content. In Ref. ( 6 ) a procedure was proposed, later refined in Refs. ( 8 , 9 , 17 ). The rationale is to consider who are the creators of content and how to capture similarities among them. It has been observed in several studies that verified accounts, i.e. the ones for which the Twitter platform checked the identity of their owners—at least in the pre-Musk era—are strong creators of content ( 6 , 17 ). It is possible, then, to infer how similar they are perceived by the “general” public of unverified users by using a bipartite representation: if verified and unverified users are the two layers of a bipartite networkterm·mth in which the (undirected) links represent retweets, e we can validate the projection on the layer of verified usersterm·dom. In this way, we will detect nontrivial similarities in the common audience of unverified users: otherwise stated, if a couple of verified usersterm·dom are retweeted by the same (nonverified) users, they are probably sharing similar positions. In the monopartite validated projection of verified usersterm·dom, communities were detected using an optimized version of Louvain algorithmterm·mth ( 15 ): since Louvainterm·key is known to be node-order dependent ( 28 ), the order of the nodes is shuffled 1,000 and the configuration displaying the greatest value of the modularity is chosen. The labels of the communities found through the community detection are then propagated in the retweet networkterm·key: in fact, it is an old result that Twitter users endorse content created by others much more with retweets than with mentions ( 10–12 ). Since in many cases, some strong creators of content are not verified (and therefore run the risk of not getting a label), we run the label propagationterm·mth algorithm on the undirected version of the retweet networkterm·key: the rationale is that not only the sources give an indication of the user orientation but also her audience. Otherwise stated, if the majority of a user audience has a clear orientation, it is presumable that the considered user also has the same one. For propagating labels, we implemented the procedure proposed in Ref. ( 13 ). This algorithm assigns the unverified user the label associated with the majority of its neighbors in the retweet networkterm·key. If an unverified user has no verified usersterm·dom as direct neighbors, it will be assigned the label associated with the majority of unverified neighbors that have already been labeled. This continues iteratively until it converges.

The interested reader can find in Section S6 a comparison between DiCosterm·dom obtained using validated or nonvalidated projections. In summary, it has been observed that politicians are particularly clustered in the validated network ( 6–9 , 17 , 29–31 ). Therefore, detecting community therein is particularly efficient in finding discursive communities about political subjects. On the contrary, the discursive communities calculated on the nonvalidated projection are much noisier.

News engagement communitiesterm·dom URL manipulation To detect similarities in users’ endorsement of pieces of news, we first need to pre-process the URLs contained in the various tweets. Sharing a compact version of a URL allows for the sharing of long URLs in tweets while maintaining the maximum character limit. For our analyses, we translated all shortened links into their original long versions. This enabled us to (i) read the top-level domain of the news source to assign a nutrition label using NewsGuardterm·key and (ii) use the long links as unique identifiers for each shared news item in our network models.

NECterm·dom communities In order to find users sharing similar “information dietsterm·dom”, i.e. engaging with the same URLs, we used the same approach as in Ref. ( 14 ). We first represented users sharing (either via tweets or retweets) URLs as a bipartite networkterm·mth of users and URLs. Then, we projected the information contained therein on the layer of users and finally validated the projection using the procedure described above. As mentioned in Section 2.3, the fraction of validated nodes, in this case, is extremely limited, i.e. nearly 1.71 % , signaling that most of the users’ endorsement to URLs (and so pieces of news) is compatible with the random noise. Again, in order to find communities of users in the validated projection network, the reshuffled version of Louvainterm·key was used.

Literature review The detection of echo chambersterm·dom has generally been approached in the literature by starting with online content whose nature is known a priori. By analyzing the social accounts that interact with specific content, e.g. through likes, shares, retweets, and comments, it has been shown how information related to specific narratives attracts particular communities. As an example, the work in ( 1 ), by Del Vicario et al., focuses on public Facebook pages divided into two groups: conspiracy theories and science news (conspiracy theories are “the pages that disseminate alternative, controversial information, often without supporting evidence” ( 1 )). The results show that users are divided into homogeneous clusters: by analyzing the accounts that share news about science and conspiracies, they are bound by friendship ties in the network. To quote the authors: “different contents generate different echo chambersterm·dom, characterized by the high level of homogeneity inside them”.

Homogeneity is not only about friendship but also about emotional approach and response to debunking attempts. Zollo et al., in ( 32 ), note how users polarized on conspiracies express more negative feelings in their comments than users polarized on science news. Work in ( 4 ) confirms how the echo chamberterm·dom paradigm goes hand in hand with the phenomenon of confirmation bias—the tendency of users to seek out, favor, and interpret information in line with their thoughts ( 33 , 34 ), while ignoring or downplaying evidence that contradicts their beliefs: interactions with debunking posts (i.e. posts that provide fact-checked information on specific topics) are overwhelmingly from science-biased or nonbiased users.

Interestingly for the purposes of this article, other studies have instead analyzed the dynamics of information exposure by looking at the news URLs present in the posts. This is the case, for example, of the work by Weaver et al. ( 35 ), which constructs the network of densely connected news articles. It starts from the number of news URLs shared by each user to arrive at the weighted network of news URLs, where the weights between two URLs indicate how many users have re-shared the URLs. Leveraging a state-of-the-art community detection algorithm, communities of co-shared news items are found, distinct in terms of political leaning (i.e. left-leaning and right-leaning). Guarino et al., in ( 14 ), consider public Facebook pages without knowing a priori the kind/quality/reputability of their content. Focusing on the activity of users sharing links to pieces of online news, the authors construct the bipartite networkterm·mth of users/shared URLs and apply the Bipartite Configuration Modelterm·mth (BiCMterm·mth) to project the bipartite networkterm·mth at the two levels, the user level and the URL level.

Supplementary Material Acknowledgments The authors thank Emanuele Brugnoli, Stefano Guarino, Walter Quattrociocchi, and Fabiana Zollo for insightful discussions. The work of F.S. is part of the joint initiative CREF-SONY.

Notes Contributor Information Manuel PratellipersName, IMT School For Advanced Studies Lucca, Piazza San Francesco 19, Lucca 55100, Italy; Istituto di Informatica e Telematica, CNR, via G. Moruzzi 1, Pisa 56124, Italy.

Fabio SaraccopersName, “Enrico Fermi” Research Center, Via Panisperna 89A, Rome 00184, Italy; IMT School For Advanced Studies Lucca, Piazza San Francesco 19, Lucca 55100, Italy; Institute for Applied Computing “Mauro Picone”, CNR, Via dei Taurini 19, Rome 00185, Italy.

Marinella PetrocchipersName, Istituto di Informatica e Telematica, CNR, via G. Moruzzi 1, Pisa 56124, Italy; IMT School For Advanced Studies Lucca, Piazza San Francesco 19, Lucca 55100, Italy.

Supplementary Material Supplementary material is available at PNAS Nexus online.

Funding Work partially supported by project SERICS (PE00000014) under the NRRP MUR program funded by the EU - NGEU; by the Integrated Activity Project TOFFEe (TOols for Fighting FakEs) https://toffee.imtlucca.it/ ; by the IIT-CNR funded Project re-DESIRE (DissEmination of ScIentific REsults 2.0); by the project “CODE—Coupling Opinion Dynamics with Epidemics”, funded under NRRP Mission 4 “Education and Research” - Component C2 - Investment 1.1 - Next Generation EU “Fund for National Research Program and Projects of Singificant National Interest” PRIN 2022 PNRR, grant code P2022AKRZ9.

Author Contributions All the authors designed the research, wrote, and reviewed the manuscript. M.Pr. performed the research and analyzed data.

Previous presentation Some of the results were presented at NetSci 2023, Wien, on 2023 July 14th.

Preprints An earlier version of this work is available at https://doi.org/10.48550/arXiv.2308.01750 .

Data Availability The X/Twitter dataset is available at https://doi.org/10.6084/m9.figshare.25460962.v1 The dataset is completely anonymized, in agreement with X/Twitter’s policies, https://twitter.com/en/privacy . The association between the trustworthiness labels and the news sources that supports the findings of this study is proprietary Newsguardterm·key data.

References Associated Data Supplementary Materials Data Availability Statement The X/Twitter dataset is available at https://doi.org/10.6084/m9.figshare.25460962.v1 The dataset is completely anonymized, in agreement with X/Twitter’s policies, https://twitter.com/en/privacy . The association between the trustworthiness labels and the news sources that supports the findings of this study is proprietary Newsguardterm·key data.