# Quantifying the structure of controversial discussions with unsupervised
methods: a look into the Twitter climate change conversation

Blas Kolic  Institute for New Economic Thinking at the Oxford Martin School,
University of Oxford  Mathematical Institute, University of Oxford  Fabián
Aguirre-López  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France
Sergio Hernández-Williams  Sinnia, Mexico City, Mexico  Guillermo Garduño-
Hernández  Sinnia, Mexico City, Mexico

###### Abstract

Social media provides an essential platform for shaping and sharing opinions
and consuming information in a decentralized way. However, users often
interact with and are exposed to information mostly aligned with their
beliefs, creating a positive feedback mechanism that reinforces those beliefs
and excludes contrasting ones. In this paper, we study such mechanisms by
analyzing the social network dynamics of controversial Twitter discussions
using unsupervised methods that demand little computational power.
Specifically, we focus on the retweet networks of the climate change
conversation during 2019, when important climate social movements flourished.
We find echo chambers of climate believers and climate skeptics that we
identify based solely on the retweeting patterns of Twitter users. In
particular, we study the information sources, or chambers, consumed by the
audience of the leading users of the conversation. Users with similar
(contrasting) ideological positions show significantly high (low)-overlapping
chambers, resulting in a bimodal overlap distribution. Further, we uncover the
ideological position of previously unobserved high-impact users based on how
many audience members fall into either echo chamber. We uncover the ideology
of more than half of the retweeting population as either climate believers or
skeptics and find that the cross-group communication is small. Moreover, we
find that, while the echo chamber structures are consistent throughout the
year, most users inside the echo chambers change from one week to the next,
suggesting that they are a stable emergent property of the Twittersphere.
Interestingly, we detect a significant decrease in polarization and a
significant increase on the number of climate skeptics that coincides with the
main #FridaysForFuture strikes during September, 2019.

##  1 Introduction

Social media is crucial for information consumption and public opinion
formation [14, 23, 22, 7]. It leverages communication channels between one-to-
many and many-to-many in a decentralized way by enabling its users to choose
whom to follow and interact with, thus democratizing information access and
spreading [4, 11]. However, the decentralized nature of social media promotes
that users consume information mostly aligned with their beliefs and interact
with like-minded individuals and communication channels. Authors have studied
these behaviors under the sociological frameworks of confirmation bias [48]
and homophily [35]. Additionally, social media platforms are designed to
maximize engagement time, with recommendation algorithms trained to show
content similar to what users typically consume [20]. To a certain extent, the
aforementioned mechanisms explain an empirical observation about controversial
topics in social media: public discourse is polarized. Authors have studied
polarization for several relevant topics, ranging from climate change [15, 49,
12], to politics [23, 7, 40], to Covid-19 [19, 30].

According to Caves’ Encyclopedia of the City [10], social polarization is the
segregation within a society that emerges from several socio-economic factors
that differentiate social groups. In the context of public discourse in social
networks, echo chambers reflect a parallel mechanism to social polarization.
An echo chamber emerges, according to Bruns et al. [9], when a group of actors
chooses to preferentially connect with each other, with the exclusion of
outsiders. However, over which preferences would such actors choose to
connect? Garimella et al. [22] state that opinions or beliefs stay inside
communities created by like-minded people who reinforce and endorse each
other’s opinions. Therefore, agents prefer to connect to other actors with
similar opinions and exclude those with contrasting ones, suggesting that
homophily is a driving force in the creation of echo chambers [24, 35]. An
actor inside an echo chamber will mostly receive information coherent with her
beliefs, so she will reject any new information that does not align with those
beliefs, i.e., actors in an echo chamber experience a confirmation bias. This
bias creates a positive feedback mechanism that reinforces the echo chambers
in the light of new information [43].

Echo chambers inhibit communication across groups with different ideologies.
Thus, even when a scientific consensus about a topic is reached, it will
hardly permeate all the echo chambers of a social group, as happens with
climate change. According to the latest Intergovernmental Panel on Climate
Change (IPCC) report [1], the cumulative scientific evidence is unequivocal:
Climate change threatens human well-being and planetary health. Despite this,
a significant fraction of the population in many countries express some form
of doubt or skepticism about whether trends in climate change are human-made,
if these trends will bring harmful consequences to society, or both [38, 31].
Such skepticism may be linked with factors such as the spread of
misinformation online [45], nationalist and individualist identities [31], or,
particularly in the United States, conservatism and conspiratorial beliefs
[28]. The issue of climate change is not only polarizing but also a matter
with deep social, economic, and ecological impact. Shifts in public opinion
might lead to significant behavioral and political changes that could face
strong opposition from specific population sectors.

Several authors have studied the climate opinion landscape in social media,
finding, in most cases, that a group of climate believers, i.e., actors that
support the science about climate change, and a group of climate skeptics
coexist and segregate into online echo chambers [49, 46, 29, 12]. Online
social media has been a crucial driver of pro-climate movements, such as the
“Fridays for Future” movement started by Greta Thunberg [44, 32]. Using
Twitter data, Williams et al. [49] constructed several interaction networks
from relevant climate-related hashtags and manually labeled the most active
users as believers, skeptics, or neutral. They measured high levels of
homophily in the followers and retweet networks, suggesting that the
conversation is polarized. With a similar approach, Jang and Hart [29] found a
polarized Twitter climate change conversation in the United States between
Republicans and Democrats. They claim that “climate change” and “global
warming” are meaningful query filters to study the climate conversation, with
the latter being more common among climate skeptics. Chen et al. [12] studied
Finland’s Twitter climate change conversation by creating interaction networks
from retweets. They found that the climate conversation was subject to
partisan sorting and aligned with the universalist-communitarian dimension of
European politics. Xia et al. [50] found that viral climate topics around the
2019 Nobel peace prize spread in different groups, enhancing in-group
connections and repulsing out-group engagement, suggesting the presence of
echo chambers.

In this paper, we study the structure of the Twitter climate change
conversation during 2019, when the “Fridays for Future” and “Extinction
Rebellion” social movements flourished. We introduce unsupervised methods to
1) identify the leading users of the conversation throughout the year, 2)
measure the ideological similarity between leading users by examining the
many-to-many communication channels of the audience of a leading user, 3)
determine the ideology of the leading users using their ideological
similarities, and 4) present an operational definition of echo chamber based
solely on the structure of the Twitter interaction networks for which we
classify more than half of the total retweeting population. We construct these
methods under the assumptions that retweeting is a good proxy for endorsement
[23, 2], and that information flows in the opposite direction of retweets.
Moreover, we acknowledge that Twitter conversations are highly heterogeneous
in that a tiny number of users explain the majority of the retweets produced
by the population in a given dataset [25]. These methods build upon other
works that provide an operational definition to detect echo chambers [14] and
create a model to infer the ideological position of active users on Twitter
[2].

We organize the remainder of this paper as follows: In section 2, we identify
the leading users of the climate conversation, introduce the chamber
associated with a leading user, and identify two polarized ideological groups
based on the similarity of the chambers of the leading users using
unsupervised methods. In section 3, we introduce a definition of echo chamber
and classify most of the users in the retweet network as either climate
believers or climate skeptics. We inspect the properties of the echo chambers
and discuss the communication within and across groups. Finally, in section 4,
we conclude and suggest future research directions. We code all the methods
and analysis of this work in Python, and we make the code freely available at
https://github.com/blas-ko/Twitter_chambers.

##  2 The chambers of the Twitter conversation

Twitter users often consume information from various sources. They typically
follow, audit, and interact with broadcasters that represent the one-to-many
communication channels of the Twittersphere and with other lower-impact users
that represent the many-to-many communication channels [33, 26]. In this
paper, we introduce quantitative methods that leverage the impact of the
broadcasters, here called leading users, and the set of lower-impact users,
here called chambers, of a given audience. Later, we introduce a quantitative
definition of echo chamber based solely on the structure of the Twitter
interaction networks. We focus in the climate change Twitter conversation
during 2019, where, given the previous research [49, 46, 29, 12], we expect to
find echo chambers of climate believers and climate skeptics.

Our aim is to construct a set of observables that characterizes the dynamics
and structure of the conversation in an _unsupervised_ way. Thus, we avoid
missing certain features that could be ignored by our biases. Most
importantly, the methods we introduce in this paper are applicable to other
datasets where we do not have clear expectation of what the structure is.

In this section, we define a leading user and its associated chamber and
derive methods that identify polarized ideological groups in an unsupervised
way. We validate a posteriori that such ideological groups correspond to those
who believe in anthropogenic climate change and those who are skeptic about
it.

###  2.1 Data

Alongside the company Sinnia111http://www.sinnia.com/en/, we create a dataset
with $41.8$ million climate-related tweets from $8.7$ million users spanning
1st March to 1st December 2019, totalling $39$ weeks. We collect all the
tweets that satisfy any of the following queries: “climate change”, “global
warming”, “climatechange” and “globalwarming”. We choose these queries
following the results from prior studies [47, 49, 29]. Of the total volume of
tweets, $73\%$ corresponds to _retweets_. We therefore focus on the retweets
only, assuming that they provide significant information of what users are
looking at or interested in.

###  2.2 High-impact and leading users

Discussions in Twitter are highly heterogeneous [41]. This means that a huge
fraction of the information consumed and spread through Twitter interaction
networks is generated by just a handful of users. However, these networks
often exhibit community-like structures where the density of interactions is
higher for users with similar features. Throughout this paper, we assume that
retweets are a good proxy for endorsement [2, 23, 12], so that we may study
the retweet interaction network as a social network of endorsements. Thus, we
study the dynamics of the retweet network, $\mathbf{W}^{t}$, which we
construct following Beguerisse et al. [3] as

$\displaystyle W_{ij}^{t}=$ $\displaystyle\textrm{ \\# of retweets of user $i$
originally posted by $j$ in a given week $t$},$ (1) $\displaystyle w_{i}^{t}=$
$\displaystyle\sum_{j}W_{ji}^{t}=\textrm{\\# of retweets of tweets posted by
user $i$ in week $t$},$ (2)

where $i$ and $j$ denote users from the complete set of retweeting users,
$\mathcal{U}$, and $t$ denotes time, where we choose the _week_ as our
temporal resolution.

In this context, we will refer to $w_{i}^{t}$ as the impact of user $i$ during
week $t$. From a networks perspective, the impact is the same as the weighted
in-degree of a given node. Using the impact vector,
$\mathbf{w}^{t}=(w_{i}^{t})$, we can quantify how unequal the discussion in
Twitter is by treating $w_{i}^{t}$ as a measure of the wealth of user $i$ and
computing $\mathbf{w}^{t}$’s Gini index. The Gini index would be $0$ if every
user received the same number of retweets and $1$ if only one user is
responsible for every retweet.

The Gini index gives us a global picture of the impact heterogeneity, but we
can also use $\mathbf{w}^{t}$ to identify important users in the conversation.
We define the set of $N$ _high-impact users_ in week $t$ as the $N$ most
retweeted users during that week, i.e.,

$\displaystyle\mathcal{I}(t)=\\{i\in\mathcal{U}\mid\textrm{rank}(w_{i}^{t})\leq
N\\}.$ (3)

Importantly, $\mathcal{I}(t)$ is a dynamic set, as $rank(w_{i}^{t})$ typically
changes weekly for all users. This contrasts with previous works, [25], where
they fix the set of high-impact users a priori.

The set $\mathcal{I}(t)$ may vary significantly throughout the weeks, so, to
characterize the most important actors in the conversation, we introduce the
persistence, $\Delta_{i}$, as the number of weeks where $i$ is a high-impact
user. The persistence lets us identify the high-impact users that are relevant
in longer time windows. We consider users with high impact and high
persistence as _leading users_ because they function as one-to-many
information channels that are relevant to focused audiences for extended
periods [33]. We thus define the set of $M$ _leading users_ as

$\displaystyle\mathcal{I}^{\Delta}(t)=\\{i\in\mathcal{I}(t)\mid\textrm{rank}(\Delta_{i})\leq
M\\},$ (4)

that have an associated vector of impact,

$\displaystyle\mathbf{w}_{\mathcal{I}^{\Delta}}^{t}=(w_{i}^{t})_{i\in\mathcal{I}^{\Delta}(t)}.$
(5)

Figure 1: Impact dynamics of the high-impact users normalized to each week. We
identify green users as climate believers, red users as climate skeptics, and
gray users with other sources such as news media and artists. In blue, we show
the total volume of retweets for the given week, where we identify two
significant peaks: one on the 3rd week of of August, and the other on the 3rd
week of the September, 2019.

When we apply these definitions to our Twitter dataset on climate change, we
observe that almost $50\%$ of the weekly retweets are given to the top $N=50$
users. This corresponds to a massive inequality, as the vast majority of
interactions are produced by just a handful of tweets. We observe an average
Gini index of $0.89$ ($\pm 0.02$)222Unless otherwise stated, quantities with
uncertainty bars represent the standard deviation around the mean across
weeks., meaning that we should be able to understand a significant fraction of
the Twitter conversation just by looking at its high-impact users.

We take the $M=50$ leading users from the $N=50$ high-impact weekly users.
While the set of leading users, $\mathcal{I}^{\Delta}(t)$, is dynamic, in the
analyses that follow we focus on the complete set of leading users,
$\mathcal{I}^{\Delta}=\cup_{t}\mathcal{I}^{\Delta}(t)$ where
$\lvert\mathcal{I}^{\Delta}\rvert=50$. As a first approach to understanding
the set of leading users, we manually assign each of them a category about
climate change. We identify three main ideological currents: climate believers
that support scientific consensus about anthropogenic climate change, climate
skeptics that doubt that the trends in climate are anthropogenic or that such
trends will bring harmful consequences to society, and news media channels and
entertainers. In Fig. 1, we show the impact dynamics of the leading users
_per_ week. Although we observe an overall dominance of the climate believers
(green) throughout the year, we observe a clear irruption of the climate
skeptics (red) in the second half of the year, with a specially pronounced
peak in the third week of September, 2019. Additionally, in Appendix D, we
present Table 1 which enlists the $M=50$ leading users and their main
features.

###  2.3 The chamber: quantifying leading users’ similarity

In the previous section, we proposed a way to identify the (small) set of
leading users responsible for most of the Twitter discussion. We manually
labelled them based on their ideological position about climate change by
inspecting their Twitter bios, Wikipedia pages and related news stories.
Manual labelling is often time consuming and requires external sources to
retrieve information about the users. In this section, we introduce an
unsupervised approach for identifying ideological (dis)similarities between
leading users. This approach relies on analyzing the many-to-many
communication channels of an audience, i.e., the chain of interactions
stemming from members of the audience, with a behavior similar to viral
spreading [33].

Each leading user has an associated audience who endorses her, where the
content posted by the leading user flows in the direction of her audience,
creating a one-to-many communication channel, similar to broadcasting [26].
Simultaneously, the audience consumes and endorses posts of other Twitter
users, forming a many-to-many communication channel, which we call the chamber
of such an audience. The audience and the chamber associated with a leading
user are our main objects of study, and we introduce them precisely in what
follows.

######  Definition 2.1 (Audience).

Given a user $i\in\mathcal{U}$, her audience at week $t$,
$\mathcal{A}_{i}^{t}$, is the set of users that have retweeted $i$ during $t$:

$\mathcal{A}_{i}^{t}=\\{j\in\mathcal{U}\mid W_{ji}^{t}>0\\}.$ (6)

The set $\mathcal{A}_{i}^{t}$ indicates who endorsed $i$ during week $t$.
Thus, the ideologies of the members of $\mathcal{A}_{i}^{t}$ are coherent to
that of $i$’s. Moreover, the audience might interact with and endorse other
lower-impact sources, creating an information flow from these sources to the
audience. We call these collection of sources the chamber of the audience
associated with the leading user $i$.

######  Definition 2.2 (Chamber).

Given an audience, $\mathcal{A}_{i}^{t}$, associated with a user
$i\in\mathcal{U}$, the _chamber_ of $\mathcal{A}_{i}^{t}$ during week $t$,
$\mathcal{C}^{t}_{i}$, is the set of users retweeted by the audience of $i$
excluding other high-impact users:

$\mathcal{C}^{t}_{i}=\\{j\in\mathcal{U}\mid W_{kj}^{t}>0\textrm{ for
}k\in\mathcal{A}_{i}^{t},\ k\not\in\mathcal{I}(t)\\}.$ (7)

Figure 2: Schematic diagram of the audience and the chamber for some user
$i\in\mathcal{I}^{\Delta}(t)$. The audience, $\mathcal{A}_{i}^{t}$, is the set
of users that retweet $i$ while the chamber, $\mathcal{C}_{i}^{t}$, is the set
of users retweeted by every member of the audience excluding users from
$\mathcal{I}(t)$, i.e. the sources of information of the audience other than
the high-impact users.

The audience is a collection of information consumers, while the chamber is a
collection of sources. See Fig. 2, where we show a schematic representation of
the audience and the chamber associated with a leading user. A leading user
relates to its chamber through its audience, so we expect that the members of
the chamber have a similar ideology to that of the leading user. We consider
the (dynamic) audiences and chamber associated with the leading users,
$\mathcal{I}^{\Delta}$, throughout the year.

The chambers transmit information from the sources to the audiences. Thus, we
can estimate the information flowing between two audiences associated with the
leading users $i$ and $j$ by comparing how similar their chambers are.
Following from our assumption that retweets indicate endorsement, this
similarity signals some notion of ideological distance between of $i$ and $j$.
Thus, we can compare the ideology two leading users by looking at their
chamber overlaps, which we introduce in what follows.

######  Definition 2.3 (Chamber overlap).

Given two leading users $i$ and $j$ with respective chambers
$\mathcal{C}_{i}^{t}$ and $\mathcal{C}_{j}^{t}$, the chamber overlap,
$q_{ij}^{t}$, during week $t$ is the Jaccard similarity between
$\mathcal{C}_{i}^{t}$ and $\mathcal{C}_{j}^{t}$:

$\displaystyle
q_{ij}^{t}=\frac{\left|\mathcal{C}_{i}^{t}\cap\mathcal{C}_{j}^{t}\right|}{\left|\mathcal{C}_{i}^{t}\cup\mathcal{C}_{j}^{t}\right|}=\frac{\textrm{\\#
of users in common of both chambers}}{\textrm{total \\# of users of both
chambers}}.$ (8)

The overlap between the chambers gives us a proxy of the ideological
similarity between $i$ and $j$ without determining their ideological positions
explicitly, meaning that $q_{ij}^{t}$ is a relative measure between the two
users. We could compare the audience overlap using the audiences
$\mathcal{A}_{i}^{t}$ instead of the chambers in Eq. (8), but we prefer using
the chambers instead for mathematical and conceptual considerations (See
Appendix C for details).

Following from our analysis from Section 2.2, we expect for the climate change
discussion to exhibit a clear separation between climate believers and climate
skeptics [49, 15]. We quantify such a separation by looking at the aggregate
chamber overlap distribution over the year –i.e., we consider $p(q)$ for
$q\in(q_{ij}^{t})_{t,i<j}$. If the separation exists, the chamber overlap
between pairs of users from the same ideological group should be significantly
larger than pairs from different groups.

Figure 3: Aggregate chamber overlap distributions for the empirical and null
model networks. We construct the aggregate empirical overlap distribution
(blue, left y-axis) by concatenating the overlap pairs, $q_{ij}^{t}$, for
every week $t$ on the dataset. We observe a bimodal structure characterized by
low-overlap peak, $q_{off}=0.04\pm 0.02$, and a high-overlap peak,
$q_{in}=0.23\pm 0.08$ (see the main text for details on how we separate the
peaks). We compute the null-model chamber overlap distribution (orange, right
y-axis) using Eq. (22) over the empirical degree sequences for every week. Eq.
(22) computes the expected overlap between pairs $(i,j)$ of users of degree
$k_{i}$ and $k_{j}$, respectively, based on the configuration model (see
Appendix A for details). We observe that the configuration model predicts a
unimodal overlap distribution characterized by $q_{null}=0.22\pm 0.03$ without
any noticeable skew.

In Fig. 3, we show the overlap distribution, $p(q)$, aggregated over the whole
period, where we observe a clear bimodal structure with a sharp peak at
$q_{off}=0.04\pm 0.02$ and a more spread-out peak at $q_{in}=0.23\pm 0.08$. We
approximate the position and spread of the peaks using a Gaussian kernel
density estimator with an optimal bandwidth parameter using Scott’s rule [42]
and separating the peaks according to the minimum between them. The observed
bimodal structure is not trivial. To claim this, we compare the empirical
structure with the expected chamber overlap distribution of the configuration
null model. The configuration model consists of random graph ensembles where
the edges are statistically independent, but the in-degree sequence is, on
average, the same than that of the empirical network (see Appendix A for
details). For the null model, we find that the expected chamber overlap
distribution is unimodal with a well-defined peak. We show the empirical and
the expected null-model chamber overlap distributions in Fig. 3. We find that
the peak of the null model coincides with the in-block empirical one but has a
lower spread ($q_{null}=0.22\pm 0.03$). In Appendix A, we derive an explicit
expression for the expected chamber overlap, $\left\langle
q_{ij}\right\rangle$, between users $i$ and $j$ for a given degree sequence
$\bm{k}$.

The overlap distribution indicates that the ideologies on the climate
conversation are polarized. However, it does not identify which users are from
what group nor how many groups are there. In what follows, we describe how to
obtain the ideological position of the users based on their chamber overlaps,
$q_{ij}^{t}$, and compare it with quantitatively with the manual labelling.

###  2.4 Classifying leading users by chamber overlap

In this section, we create a partition of the set of _all_ leading users,
$\mathcal{I}^{\Delta}$, based on the chamber overlap distribution, $p(q)$,
which exhibits a bimodal structure (See Fig. 3). A partition corresponds to a
collection of disjoint subsets of $\mathcal{I}^{\Delta}$,
$\mathcal{P}=\\{P_{\alpha}\\}_{\alpha\in I}$ with
$P_{\alpha}\subseteq\mathcal{I}^{\Delta}$ and $\cup_{\alpha\in
I}\mathcal{P}_{\alpha}=\mathcal{I}^{\Delta}$. In general, we could create a
partition with an arbitrary number of clusters, but, given that the Twitter
climate discussion exhibits a natural ideological division into climate
believers and climate skeptics, we expect to find a separation of
$\mathcal{I}^{\Delta}$ into two clusters.

In order to create a partition of the leading users, we first consider the
(temporal) chamber overlaps matrices between all the pairs of leading users,
$\bm{Q}^{t}=(q_{ij}^{t})_{i,j\in\mathcal{I}^{\Delta}(t)}$. By treating
$\bm{Q}^{t}$ as weighted, undirected, adjacency matrices, we can dispose of a
plethora of community detection algorithms to detect communities in networks.
Thus, we classify the leading users in an unsupervised way by first
considering the aggregate chamber overlap matrix ,
$\bm{Q}:=\left\langle\bm{Q}^{t}\right\rangle_{t}$, where
$\left\langle\cdot\right\rangle_{t}$ denotes average over $t$, and then
partitioning $\bm{Q}$ into two groups using the spectral clustering algorithm
described by Mohar et al. [36] (see Appendix B for details). We choose a
spectral clustering algorithm because 1) it naturally partitions the network
into two distinct groups, and 2) it ranks the nodes in the network (here, the
leading users) according to how well-separated they are from the out-group. In
general, we could consider other community detection algorithms [6, 39, 37] if
we need to partition the network in more than two groups.

(a) Aggregate chamber overlap matrix

(b) Aggregate similarity network

Figure 4: Aggregate chamber overlaps between the leading users. a) Aggregate
chamber overlap matrix, $\bm{Q}$, of every leading user pair (see main text
for details). White pixels represent leading users that were never present
simultaneously during the same week. We order the users in $\bm{Q}$ according
to the rank we obtain with the unsupervised spectral clustering algorithm (see
Appendix B for details). b) Weighted similarity network constructed from
$\bm{Q}$. We color the nodes according to the partition obtained unsupervised
spectral clustering. We identify a group of climate believers (green) and a
group of climate skeptics (red) according to the users profiles within those
groups.

In Fig. 4(a), we present the aggregate chamber overlap matrix, $\bm{Q}$, where
we sort the users according to the rank given by the spectral clustering
algorithm (see Fig. 9 in Appendix B). We observe a clear block structure that
separates the climate skeptics (top left) from the climate believers (bottom
right). Notice that the in-block overlaps are often two-digit percentages: a
large fraction of the chamber is shared for users in the same community.
Moreover, in Fig. 4(b) we show the resulting network constructed with
$\bm{Q}$, where the node colors represent the partition found by the spectral
clustering algorithm. We observe two well-separated groups that correspond to
the climate believers and climate skeptics discussed before. We also observe
four satellite nodes, namely, BhadDhad, narendramodi, sunfloweraidi, and
LilNasX that have a significantly lower overlap than the rest of the in-group
user pairs. In a few cases, there are cross-chamber overlaps that are higher
than the average cross-chamber overlap, such as in the pair
cathmckenna(believer)-RealSaavedra(skeptic). However, such cross-chamber
overlaps are much smaller than the within-chamber overlaps.

####  2.4.1 Polarization dynamics of the leading users

The previous analysis suggests a big ideological polarization in the climate-
related conversation on Twitter, which is a known fact for conversations about
climate change [49, 29, 12]. However, our results are valuable in that 1) we
detect these groups in an unsupervised way, and 2) we quantify the relative
ideological similarity between leading users using the chamber overlaps, where
we found significantly high (low) overlaps for pairs of users in the same
(other) group. However, we have not yet quantified polarization, and the
analysis so far has been static.

We define the polarization between two groups $P_{\alpha}$ and $P_{\beta}$ as
the difference between the probability that an edge exists within a group and
the probability that an edge exist across groups. To do so, we consider the
adaptive E-I index, following Chen et al. [12] and Bruns [9], as

$\Phi(t|P_{\alpha},P_{\beta})=\frac{n_{\alpha\alpha}^{t}+n_{\beta\beta}^{t}-(n_{\alpha\beta}^{t}+n_{\beta\alpha}^{t})}{n_{\alpha\alpha}^{t}+n_{\beta\beta}^{t}+(n_{\alpha\beta}^{t}+n_{\beta\alpha}^{t})},$
(9)

where $n_{\alpha\beta}^{t}=\sum_{i\in P_{\alpha},j\in P_{\beta}}q_{ij}^{t}$ is
the total strength of the edges going from $P_{\alpha}$ to $P_{\beta}$.

Our measure of polarization is bounded such that $\Phi\in[-1,1]$, where
$\Phi=1$ when all the connections happen within groups (total assortativity),
$\Phi=-1$ when all connections happen between groups (total disassortativity),
and $\Phi=0$ when the connections between groups equal the connection within
groups (no assortativity).

Figure 5: Polarization dynamics for the weekly chamber overlap matrices,
$\bm{Q}^{t}$, of the leading users (see Eq. (8)) for the supervised partition
of climate believers and skeptics (green,
$\langle\Phi_{\text{sup}}\rangle=0.83$), the unsupervised partition using
spectral clustering (orange, $\langle\Phi_{\text{unsup}}\rangle=0.84$) and the
average over an ensemble of randomly reshuffled labels over the same networks
(black, $\langle\Phi_{\text{null}}\rangle=0.23$). The red marker indicates a
week without leading climate skeptics.

In Fig. 5, we show the polarization dynamics of the leading users for both the
supervised and unsupervised partitions of climate believers and skeptics (see
Fig. 1). As a null model benchmark, we compute the polarization dynamics of
the average over an ensemble of the same networks with randomly reshuffled
labels.

We find a high polarization of $\Phi_{\text{sup}}=0.83\pm 0.11$ and
$\Phi_{\text{unsup}}=0.84\pm 0.11$ for the supervised and unsupervised
partitions, respectively, meaning that users with similar ideologies have
similar (climate-related) chambers. Moreover, we show that the polarization
dynamics between the supervised and unsupervised approaches are almost
identical: even when we tried to separate the news media sources from the
climate believers, the unsupervised clustering algorithm puts them in the same
group. We find an average polarization of $\Phi_{\text{null}}=0.23$ for $100$
null model realizations of randomly reshuffling the labels. Such a low value
indicates that neither the degree sequence nor the size of each ideological
group can explain the high polarization of the empirical network. We find no
significant temporal trends in the polarization dynamics but observe that
polarization decreases during the 3rd and 4th week of September, which
coincides with the #FridaysForFuture biggest strikes organized by Greta
Thunberg [32, 44]. We provide a deeper discussion about this and other signals
that coincide with the strikes later in Section 3.1.

##  3 From chambers to echo chambers

Previously, we introduced the chamber as the many-to-many information sources
of the audience of a leading user. In the case of the climate change retweet
network, we found that the overlap distribution between all chamber pairs is
bimodal (see Fig. 3), suggesting that such communication channels are divided,
polarizing the climate change retweet network. Thus, we classified the
chambers –following an unsupervised clustering approach– according to their
overlap similarities and found two well-separated groups which we then
identified as climate believers and climate skeptics.

While many authors have studied polarization and echo chambers in social
networks, there is not yet a consensus of what an echo chamber is. However,
some characteristics of echo chambers are transversal to all its definitions:
homophilic interactions drive their formation [50, 24, 14], where actors in
the system choose to preferentially connect with each other with the exclusion
of outsiders [9], and attitudes and beliefs stay inside groups of like-minded
people [22]. In particular for interaction networks in social media, we assume
that information flows through the edges of the network and that bimodal
structures indicate the presence of echo chambers [14].

Under this framework, the leading users associated with an ideological group
(e.g. climate skeptics), together with their audiences and their chambers,
_approximate_ an echo chamber. Our rationale is that information flows through
the opposite direction of retweets, as we schematize in Fig. 2, and such
information flows mostly between high-overlapping chambers. The overlap
between two chambers indicates the proportion of their common users.
Therefore, a multimodal overlap distribution with high-overlap modes and a
low-overlap mode (close to 0) suggests that users choose to preferentially
connect with each other (high-overlap modes) and exclude outsiders (low-
overlap mode). If a network exhibits such characteristics, then the set of
leading users should be easily partitioned as described in the previous
section, so we thus introduce a definition of an echo chamber.

######  Definition 3.1 (Echo chamber).

Given a partition of the leading users, $\mathcal{I}^{\Delta}=\cup_{x\in
I}\mathcal{P}_{x}$, obtained by clustering the chamber overlap matrix,
$\bm{Q}=(q_{ij})_{ij\in\mathcal{I}^{\Delta}}$, with the distribution $p(q)$
containing a well-defined peak near $q=0$ and other well-defined peak(s) well-
separated from $q=0$, the echo chamber of the group $\mathcal{P}_{x}$ is the
union of the leading users, chambers, and audiences associated with
$\mathcal{P}_{x}$. Mathematically,

$\mathcal{E}_{\mathcal{P}_{x}}=\bigcup\limits_{i\in\mathcal{P}_{x}}\left(\\{i\\}\cup\mathcal{A}_{i}\cup\mathcal{C}_{i}\right)$
(10)

where $\mathcal{A}_{i}$ is the audience of $i$ and $\mathcal{C}_{i}$ her
chamber.

An echo chamber is an ill-defined concept, so we remark that Def. 3.1 is an
approximation that works well for retweet networks with multimodal overlap
distributions, $p(q)$. In this work, we only consider a partition into two
groups, but we can generalize this using other clustering algorithms.

Structurally, the climate change retweet network is well-separated, so we
claim that we can observe ideological echo chambers. We construct two dynamic
echo chambers –per week– based on the leading users,
$\mathcal{I}^{\Delta}(t)$, and our partition of climate believers and
skeptics. The believers echo chamber, $\mathcal{E}_{B}$, contains $14.8\pm
7.8\%$ of the total users per week, while the skeptics echo chamber,
$\mathcal{E}_{S}$, contains $2.6\pm 2.1\%$. Together, they cover $17.4\pm
9.9\%$ of the whole retweeting population per week. These echo chambers may
have some overlapping users by construction, i.e., users that are classified
both as believers and skeptics. However, we find that only $0.3\pm 0.1\%$
users are in both echo chambers, indicating that the cross-communication
between them is orders of magnitude lower333It might be the case that if we
inspected the reply network or the followers network instead of the retweet
network, the intensity of cross-communication between echo chambers could be
higher..

###  3.1 Augmented echo chambers

In the last section, we obtained two echo chambers, $\mathcal{E}_{B}$ and
$\mathcal{E}_{S}$, that, combined, cover a small minority ($17.4\pm 9.9\%$) of
the whole retweeting population because we only consider the leading users,
$\mathcal{I}^{\Delta}(t)$, instead of the set of high-impact users,
$\mathcal{I}(t)$. However, we can inspect the audiences of the remaining high-
impact users and evaluate if they have a clear ideological position based on
$\mathcal{E}_{B}$ and $\mathcal{E}_{S}$. To do so, we design an ideology
score, similar to [18], that determines if the audience of a high-impact user
is biased towards being believers, skeptics, or neither of the both. We
compute the ideology score, $s_{i}$, for each high-impact user $i$ as follows

$s_{i}=\frac{n_{B}^{i}-n_{S}^{i}}{n_{B}^{i}+n_{S}^{i}},$ (11)

where $n_{X}^{i}=|\mathcal{A}_{i}\cap\mathcal{E}_{X}|$ is the number of users
in the audience of user $i$ that are also in the echo chamber
$\mathcal{E}_{X}$. The score $s_{i}$ is bounded between $-1$, when all the
users in $\mathcal{A}_{i}$ are climate skeptics, and $1$ when all of them are
climate believers.

If the ideology score of a high-impact user is big in magnitude, we may
associate her and her audience to an ideological group based on the sign of
$s_{i}$. In particular, we recognize as climate believers to those high-impact
users whose scores are close to $1$ and as climate skeptics to those whose
scores are close to $-1$. Thus, we can augment the echo chambers to consider
the audiences of high-impact users with high-magnitude scores as we describe
in the following definition.

Figure 6: Ideology scores distribution for the high-impact users,
$\mathcal{I}$, where we find a clear bimodal structure (see Eq. (11)). We
choose a threshold $\eta=0.5$ for which users with $s_{i}>\eta$ (green) are
assigned as climate believers and those with $-s_{i}>\eta$ (red) are assigned
as climate skeptics. Users with intermediate scores ($s_{i}\in[-\eta,\eta]$)
are not included in the augmented echo chambers.

######  Definition 3.2 (Augmented echo chamber).

Given the echo chambers $\mathcal{E}_{B}$ and $\mathcal{E}_{S}$ and the
collection of high-impact users for which $|s_{i}|\geq\eta$, with $\eta$ a
user-defined threshold, we define their corresponding augmented echo chambers,
$\mathcal{E}_{B}^{a}$ and $\mathcal{E}_{S}^{a}$, as follows

$\displaystyle\mathcal{E}_{B}^{a}$
$\displaystyle=\bigcup\limits_{i:s_{i}\geq\eta}\left(\\{i\\}\cup\mathcal{A}_{i}\right)\cup\mathcal{E}_{B},$
(12) $\displaystyle\mathcal{E}_{S}^{a}$
$\displaystyle=\bigcup\limits_{i:-s_{i}\geq\eta}\left(\\{i\\}\cup\mathcal{A}_{i}\right)\cup\mathcal{E}_{S},$
(13)

where $\mathcal{A}_{i}$ is the audience of $i$.

In Fig. 6, we show the ideology score distribution for the remaining high-
impact users, i.e., high-impact users who are not leading users. We find that
the distribution is clearly bimodal, where the mode corresponding to climate
believers (right) is significantly higher than the mode corresponding to
climate skeptics (left). We choose $\eta=0.5$ as our threshold to decide
whether a high-impact user is included in the augmented echo chamber or not.
Using this method, of the $1360$ remaining high-impact users across all weeks,
we classified $280$ as climate skeptics, $1078$ as climate believers, while
$73$ users remain unclassified. In addition to augmenting the number of users
in the echo chambers, this method enables us to discover the ideological
position of high-impact users in an unsupervised way. For instance, we
identify users like @RollingStone ($s_{i}=0.989$) or @Greenpeace
($s_{i}=0.995$) as climate believers while users like @DonaldJTrumpJr
($s_{i}=-0.93$) or @GovMikeHuckabee ($s_{i}=-0.948$) as climate skeptics
solely based on their scores. While we do not analyze users with scores below
the threshold $\eta$ thoroughly, we noticed that low-scoring users either stay
neutral with respect to climate change, e.g., @SkyNews ($s_{i}=0.17$), or are
not a strong part of their content agenda, e.g., @Imamofpeace ($s_{i}=0.03$).

Figure 7: Size of the augmented echo chambers in logarithmic scale for every
week in the dataset. We show, in green and red respectively, the number of
believers and skeptics, while in brown we show the number of users in the
intersection. We omit a data point in June, 2019 because the skeptics echo
chamber, $\mathcal{E}_{S}$, was empty during that week. In black, we show the
total number of users in the retweet network for that week. The inset plot
shows the size ratio of skeptics against believers, where we observe a
significant increase during late September, 2019, which we suspect was
triggered by the Fridays for Future biggest strike [32, 44] (vertical pink
interval).

The augmented echo chambers contain $53.1\pm 9.8\%$ of the total retweeting
population, with the error bars representing the standard deviation across
weeks. This result is consistent with our initial observation that the top
$N=50$ high-impact users cover near the $50\%$ of the total retweeting
population. Moreover, this augmentation increases the sizes of the original
echo chambers by a factor of $3.5$. More specifically, the believers augmented
echo chamber is bigger than its original counterpart by a factor of $3.7\pm
1.6$, the skeptics by a factor of $2.6\pm 1.2$, and the users in the
intersection only by a factor of $1.4\pm 0.4$.

In Fig. 7, we present the sizes of the augmented echo chambers, the number of
users in their intersection and the total number of retweeting users each
week. For most of the year, these sizes are stable, with the believers
oscillating around $2.4\times 10^{5}$ users and the users in the intersection
oscillating around $1500$ users. In contrast, the number of climate skeptics
rises significantly from before the Fridays for Future September strikes [32,
44], oscillating from around $2.4\times 10^{4}$ users, to oscillating around
$5.6\times 10^{4}$ users after the strikes. In the inset of the figure, we
show the ratio between the number of skeptics against the number of believers,
where we find a consistently higher ratio after the strikes, which we find
counter intuitive in that the strikes were pro-climate events.

We believe that the correlation between the climate strikes and the increase
of the skeptics’ impact is not random. Besides such an increase, we already
observed in Figure 5 a significant drop in polarization at the dates of the
strikes that was not caused by changes in chamber sizes. There are several
explanations for such a correlation with size and polarization, ranging from
an endogenous social reaction from the climate skeptics to a coordinated
response using bots [13], where the bots massively retweeted the skeptic and
believers leading users during the dates of the strikes. While we cannot
establish its causes with this analysis, we find it remarkable that we can
observe such a signal from several angles purely from our unsupervised
construction of echo chambers.

###  3.2 Temporal flow of users within ideological groups

Most of the statistical features in our analysis are stable over time, ranging
from the bimodal structure of the overlap distribution, to the number of users
per week (with the exception of the peak in September that coincides with the
Fridays for Future strikes), to the Gini index of the users’ impact. Yet,
Twitter is very dynamic with users entering and leaving the conversation as
different topics emerge and decay over time [16], as is the case of the
climate change conversation [17].

Figure 8: Auto-overlap similarity decay for each augmented echo chamber as a
function of the time difference (in weeks) between them. The solid lines with
markers indicate the median overlap similarity for each week while the spread
correspond to the $25-75$ quantile spread.

Given these contrasting facts, we test the hypotheses of whether the overall
conversation is sustained by the same people over the year or there is a big
flux of users at the micro level that behave consistently at the macro level.
To discriminate between these two explanations, we measure the flux of users
within augmented echo chambers for different time windows. A small flux of
users over time would favor the first hypothesis while a significantly high
flux would favor the second one. We quantify such a flux by computing the
auto-overlap of an augmented echo chamber $\mathcal{E}_{X}^{a}(t)$ of ideology
group $X$ between weeks $t$ and $\tau$ as follows

$q_{X}^{t\tau}=\frac{\left|\mathcal{E}_{X}^{a}(t)\cap\mathcal{E}_{X}^{a}(\tau)\right|}{\left|\mathcal{E}_{X}^{a}(t)\cup\mathcal{E}_{X}^{a}(\tau)\right|}\
.$ (14)

This measure is similar to the chamber overlap of Eq. (8), but instead of
comparing the chamber associated with two different leading users, we compare
the augmented echo chambers of the same ideological group at different times.
A small value of $q_{X}^{t\tau}$ would correspond to a high flux of users and
viceversa.

In Fig. 8, we show the auto-overlap of each augmented echo chamber as a
function of the number of weeks between them. We present such auto-overlaps
for the believers and the skeptics separately, and, in both cases, we find
that the number of common users in the chambers decay monotonically with the
number of weeks passed. We find that a higher proportion of believers stay in
their echo chambers compared to the skeptics. Remarkably, we find that only
$9.7\pm 1.7\%$ of the believers and $8.1\pm 2.4\%$ of the skeptics remain in
the augmented echo chamber from one week to the next, i.e., at $t-\tau=1$,
indicating a massive flow of users in the Twitter conversation within both
ideological groups. This result thus supports our second hypothesis: at the
micro level, a lot of users enter and leave the conversation from week to
week, but, at the macro level, the structural properties of the interaction
networks are stable.

##  4 Conclusion

In this paper, we identified the echo chambers of the Twitter climate change
conversation using unsupervised methods. Acknowledging that typically in
Twitter most retweets are given to just a handful of users [25], and assuming
that retweets are a good proxy for endorsement [2, 23, 12], we analyzed the
temporal structure of the climate-related retweet networks and measured the
ideological similarities between leading users. We defined a leading user as
that who is among the most retweeted users for several weeks throughout the
conversation.

We introduced the notion of a leading user’s chamber, which is the set of
users retweeted by her audience. The chambers act as the many-to-many
information sources associated with a leading user’s audience [33], and we
used them to measure the ideological similarities between every pair of
leading users by computing their chamber overlap. We found that the chamber
overlap distribution is clearly bimodal with a low-overlap peak and a
significantly higher-overlap peak. Thus, we classified the leading users into
two groups using an unsupervised spectral clustering algorithm [36] and
recognized one group as the climate believers and the other as the climate
skeptics, validating previous findings [49, 46, 12].

Based on the bimodal structure of the chamber overlap distribution, we defined
an echo chamber as the union of the same group’s leading users with their
associated audiences and chambers. Such a bimodality implies several features
that characterize echo chambers: homophilic interactions drive their formation
[50, 24, 14], actors inside them choose to preferentially connect with the
exclusion of outsiders [9], and attitudes and beliefs stay inside groups of
like-minded people [22]. We identified, on average per week, $15\%$ of the
total retweeting population as climate believers, $3\%$ as climate skeptics,
and only $0.3\%$ of users classified as both believers and skeptics,
suggesting that the cross-communication between echo chambers is negligible.

Furthermore, we designed an ideology score that uncovers the ideological
position of any high-impact user, even when she has not been observed
previously. This score depends on the proportion of her audience that belongs
to either of the echo chambers, and we found that the ideology score
distribution over all the high-impact users is bimodal, with modes at opposite
extremes of the ideological spectrum. This finding reinforces our claim that
the users inside the echo chambers are mostly like-minded. We validated the
ideological positions of some of the high-impact users discovered with the
ideology score. For instance, we uncovered the global campaigning network
Green Peace (@Greenpeace) as a climate believer, the Republican governor Mike
Huckabee (@GovMikeHuckabee) as a climate skeptic, and the British news channel
Sky News (@SkyNews) as a neutral user.

Using the uncovered ideologies, we augmented our original echo chambers by
combining them with the audiences of the high-impact users that share the same
ideological position. Thus, the augmented echo chambers constitute more than
half of the weekly retweeting population ($46.2\%$ climate believers, $6.5\%$
climate skeptics, and only $0.3\%$ of users classified in both ideological
groups). We observed that, in most cases, the number of users in each echo
chamber is roughly stable throughout the year. However, we found a strong
positive correlation between the dates of the main #FridaysForFuture strikes
and the skeptics’ echo chamber sizes but not with the believers’. We find it
remarkable that we could identify peak activity of certain parts of the
population by using completely unsupervised methods. Moreover, we measured the
flux of users within echo chambers as a function of time, where we found that
most ($>80\%$) users leave their echo chambers from one week to the next. This
result suggests that the stable properties of the echo chambers are an
emergent feature of the system and not imposed by a fixed set of users.

We see several directions for future development of this method. First, we
only assessed two ideological positions in our analysis, whereas Twitter
discussions may have an arbitrary number of communities. Thus, we could
consider more general community detection algorithms, ranging from inference
methods able to fix the number of communities [37] to methods that maximize
some notion of community and result in optimal partitions with an unsupervised
number of clusters [6, 39]. With more than two ideologies, we should
generalize our definition ideology score to a multi-class setting by, e.g.,
measuring the proportion of users in each echo chamber in a higher-dimensional
simplex. Second, we do not distinguish between the nature of the Twitter
users, so we did not analyze the different effects that validated accounts,
non-human accounts, and bots may bring into the conversation. Previous large-
scale studies of Twitter data have demonstrated the influence bots can have on
the exposure of human accounts to emotional content [5], and how they can
coordinate and distort the discussion in controversial conversations [13, 8].
Thus, we could incorporate bot-detection methods and create a taxonomy of
users to analyze their effects.

We intentionally focused only on who and not on what the Twitter users
consumed to perform our classification. Such approach is limited with respect
of supervised approaches. Analyzing the tweets’ content would give us more
information about the ideological position of its users and help us reinforce
the existence of echo chambers [14] and uncover relevant topics of discussions
within and between ideological groups [7]. Nevertheless, we see our results as
a proof of concept that the echo chamber structure can be observed clearly
_even without_ looking at the content of tweets. While the ideology of
individual users posses a non-negligible degree of uncertainty, at the
aggregate level, we clearly observed presence of distinct ideologies each with
distinct structural features. In addition, we believe it is important to
perform an initial analysis before biasing with external information. Such an
initial unsupervised approach could also be useful in indicating starting
points of subsequent more complicated analyses.

Overall, we believe our work highlights the importance and usefulness of
exploiting the structural information present in social media networks,
especially when looking at controversial conversations. Our methodology is
computationally cheap, readily usable for other Twitter datasets, and does not
suffer from the selection bias of supervised approaches. Furthermore, we
showed that if the conversation is polarized enough, we can identify echo
chambers just by looking at the handful of users that produce the most
important tweets, which is easier than deploying clustering algorithms on the
whole network. From a social point of view, this condition shows that the
climate-related Twitter conversation –and possibly most conversations in
Twitter [25]– has very low complexity, meaning that we can identify large
scale structures within the conversation just from the activity of the few
leading users.

## Acknowledgments

BK acknowledges funding from the Conacyt-SENER: Sustentabilidad Energética
scholarship. FAL is supported by a grant from the Simons foundation (grant No.
454941, S. Franz). The authors would like to thank Luca Mungo, Santiago
Martínez Balvanera, Alexandre Bovet, and Doyne Farmer for proofreading the
manuscript and providing very valuable feedback. BK is greateful with the INET
Complexity Economics group for stimulating discussions.

## Code Availability

All the code for this work is in Python and is freely available at
https://github.com/blas-ko/Twitter_chambers.

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##  Appendix A Soft configuration model

In order to estimate the overlap distribution in a random network we analyze
the degree sequence $\bm{k}$ of the empirical retweet network (with adjacency
matrix $\bm{A}$) using the configuration model [[27], Chapter 7] with a soft
constraint on the in-degree

$\displaystyle
p(\mathbf{A}|\mathbf{k})=\prod_{i=1}^{N}\prod_{j\not=i}\left[\frac{k_{i}}{N}\delta_{A_{ji},1}+\left(1-\frac{k_{i}}{N}\right)\delta_{A_{ji},0}\right].$
(15)

This choice of randomness results in networks with a locally tree like
structure, more realistic choices can be proposed but they are in general
harder to control when trying to match with real networks, [34, 21]. The
relevant observables are also harder to compute analytically, we therefore
choose to work with the locally tree like structure of the configuration
model.

We can then approximate the expected overlap of the configuration model as
follows

$\displaystyle\left\langle
q_{ij}^{\mathcal{C}}\right\rangle=\left\langle\frac{\sum_{\ell}\mathbb{I}\left[\ell\in\mathcal{C}_{i}\vee\ell\in\mathcal{C}_{j}\right]}{\sum_{\ell}\mathbb{I}\left[\ell\in\mathcal{C}_{i}\wedge\ell\in\mathcal{C}_{j}\right]}\right\rangle\approx\frac{\sum_{\ell}\textrm{Prob}\left[\ell\in\mathcal{C}_{i}\right]\textrm{Prob}\left[\ell\in\mathcal{C}_{j}\right]}{\sum_{\ell}\textrm{Prob}\left[\ell\in\mathcal{C}_{i}\right]+\textrm{Prob}\left[\ell\in\mathcal{C}_{j}\right]},$
(16)

which is the sum of joint probabilities of a user $\ell$ being in both
$\mathcal{C}_{j})$ and $\mathcal{C}_{i}$, assuming independence. We need to
calculate this probability, which for the configuration model it is simply

$\displaystyle\textrm{Prob}\left[\ell\in\mathcal{C}_{i}\right]=1-\textrm{Prob}\left[\sum_{j=1}^{N}A_{ji}A_{j\ell}=0\right]=1-\left(1-\frac{k_{i}k_{\ell}}{N^{2}}\right)^{N},$
(17)

where $\textrm{Prob}\left[A_{ji}A_{j\ell}=0\right]$ is the probability of user
$j$ retweeting $i$ and not $\ell$ or viceversa.

We can do a similar analysis for the overlap similarity between audiences:

$\displaystyle\left\langle
q^{\mathcal{A}}_{ij}\right\rangle=\left\langle\frac{\sum_{\ell}\mathbb{I}\left[\ell\in\mathcal{A}_{i}\vee\ell\in\mathcal{A}_{j}\right]}{\sum_{\ell}\mathbb{I}\left[\ell\in\mathcal{A}_{i}\wedge\ell\in\mathcal{A}_{j}\right]}\right\rangle\approx\frac{\sum_{\ell}\textrm{Prob}\left[\ell\in\mathcal{A}_{i}\right]\textrm{Prob}\left[\ell\in\mathcal{A}_{j}\right]}{\sum_{\ell}\textrm{Prob}\left[\ell\in\mathcal{A}_{i}\right]+\textrm{Prob}\left[\ell\in\mathcal{A}_{j}\right]},$
(18)

where

$\displaystyle\textrm{Prob}\left[\ell\in\mathcal{A}_{i}\right]=\frac{k_{i}}{N}$
(19)

Substituting Eq. (17) into $q_{ij}^{\mathcal{C}}$ and Eq. (19) into
$q_{ij}^{\mathcal{A}}$ we obtain

$\displaystyle\left\langle q_{ij}^{\mathcal{C}}\right\rangle$
$\displaystyle=\frac{\left\langle\left[1-\left(1-\frac{k_{i}k}{N^{2}}\right)^{N}\right]\left[1-\left(1-\frac{k_{j}k}{N^{2}}\right)^{N}\right]\right\rangle_{p(k)}}{\left\langle
2-\left(1-\frac{k_{i}k}{N^{2}}\right)^{N}-\left(1-\frac{k_{j}k}{N^{2}}\right)^{N}\right\rangle_{p(k)}}$
(20) $\displaystyle\left\langle q_{ij}^{\mathcal{A}}\right\rangle$
$\displaystyle=\frac{k_{i}k_{j}}{N(k_{i}+k_{j})},$ (21)

so if $k_{max}\ll N$ and $\left\langle k^{2}\right\rangle>\left\langle
k\right\rangle$, then it follows that

$\displaystyle\left\langle
q_{ij}^{\mathcal{C}}\right\rangle\approx\frac{k_{i}k_{j}}{N(k_{i}+k_{j})}\frac{\left\langle
k^{2}\right\rangle}{\left\langle
k\right\rangle}>\frac{k_{i}k_{j}}{N(k_{i}+k_{j})}=\left\langle
q_{ij}^{\mathcal{A}}\right\rangle.$ (22)

In most social networks we have that the degree variance, $\left\langle
k^{2}\right\rangle$, is much larger than the average degree, $\left\langle
k\right\rangle$. Therefore, we expect the chamber overlap to be significantly
larger than the audience overlap.

##  Appendix B Unsupervised clustering

We take an spectral clustering approach to obtain the communities of the
leading users. We consider the similarity Laplacian matrix,
$\mathbf{L}=\mathbf{D}-\mathbf{Q}$, where $\mathbf{Q}$ is the overlap
similarity matrix aggregated over all the weeks in the dataset, and
$\mathbf{D}$ is a diagonal matrix with the degree sequence of $\bm{Q}$ in the
diagonal, i.e., we have that $\mathbf{D}_{ii}=\sum_{j}q_{ij}$ and $D_{ij}=0$
for $i\neq j$.

By the spectral properties of $\bm{L}$, we know that connected networks have a
smallest eigenvalue of $0$ followed by positive eigenvalues. The second
smallest eigenvalue $\lambda_{2}>0$, also called the algebraic connectivity,
reflects how well-connected the network is. Its associated eigenvector,
$\bm{u}_{2}$, also called the Fiedler vector, is typically used to partition
networks into two non-trivial communities that minimize that cut size, i.e.,
it minimizes the sum of weights between the groups [36].

In the case of the Laplacian of the aggregate overlap similarity matrix, the
second smallest eigenvector separates the satellite users (namely, BhadDhad,
narendramodi, sunfloweraidi and LilNasX) from the rest of the network. While
the separation associated the second eigenvalue separates the network in
almost independent connected components, it does not capture the groups
seuggested by the bimodality of the overlap distribution. Instead, we take the
eigenvector associated with the third smallest eigenvalue, $\lambda_{3}$, to
partition the leading users into two groups. We identify the entries of
$\bm{u}_{3}$ that are greater than zero as climate believers while the entries
smaller than zero as climate skeptics. In Fig. 9, we show the individual
entries of $\bm{u}_{3}$, where each entry corresponds to a leading user.

Figure 9: Eigenvector associated with $\lambda_{3}$, the third leading
eigenvalue, sorted by magnitude. We identify between climate believers and
skeptics with the sign of their corresponding component in
$\mathbf{u}_{\lambda_{3}}$.

##  Appendix C Comparing chambers and audiences

Many recent studies about polarization on social media have used a first-
neighbors approach to characterize the structure of the interaction networks
[14, 30, 12, 49]. In Twitter, retweets have been used extensively as a proxy
for endorsement [2, 23], so retweet networks have been used to study the
assortative structure of several conversations. In this work’s terminology,
these studies have studied the audience of a set of users to quantify
polarization, echo chambers, and other nontrivial social structures. We argue
that the chamber is, under certain conditions, a more robust tool to quantify
such social structures, both conceptually and mathematically. The difference
being that looking at the chamber corresponds to looking at the information
sources of the audience, a second-neighbors approach.

(a) Cumulative distribution function

(b) Audience-ordered sizes

Figure 10: Chamber and audience sizes. a) Cumulative distribution function. We
construct each distribution by concatenating the sizes of the chambers
(purple) (audiences (blue)) of the leading users, $\mathcal{I}^{\Delta}(t)$,
for every week $t$ on the dataset. The audiences size distribution ($s=4668\pm
11472$) has a significantly longer right-tail than that of the chambers
($s=2425\pm 1682$). b) Audience-sized ordered chamber (purple) and audience
(blue) sizes from smallest to largest for the leading users for all weeks. We
find a small correlation of $\rho=0.26$ between audiences and chambers sizes.

From a mathematical perspective, for the null model the expected overlap
between the audiences of two high-impact users is significantly lower than the
expected overlap between chambers (see Appendix A for details). Which suggests
that typically the chamber overlap should have a higher signal to noise ratio.
Moreover, in the case of the climate change retweet network the distribution
of chamber sizes has a significantly lower spread than that of the audience,
as we show in Fig. 10(a). In other words, many chambers are of roughly the
same size while the audiences sizes vary a lot. The overlap similarity is
informative whenever the size of the two sets compared are of the same order,
so using the chamber gives us more reliable results than the audience. For two
arbitrary sets $A$ and $B$ where $|A|\ll|B|$, the Jaccard overlap $|A\cap
B|/|A\cup B|\approx|A|/|B|$, which indicates the relative size of $A$ with
respect to $B$ and not their overlap.

Besides mathematical arguments, we argue that the audience and the chamber are
fundamentally different objects. By construction, one would expect that the
chamber is tightly coupled to the audience because the audience determines the
chamber. For instance, if a user has a big audience, we expect her to have a
big chamber as well. Surprisingly, the correlation correlation between
audience and chamber sizes is very low ($\rho=0.26$), suggesting that they are
loosely coupled (see Fig. 10(b)).

Figure 11: Aggregate chamber and subchamber overlap distributions. We
construct the aggregate chamber (blue) (subchamber (orange)) distribution by
concatenating the overlap pairs, $q_{ij}^{t}$, for every week $t$ on the
dataset – i.e., $\bm{q}=(q_{ij}^{t})_{t,i<j}$. A subchamber differs from a
chamber in that we remove the overlapping users of the audience before
constructing between the subchamber (see main text for details on their
construction). While the first peak of the subchamber distribution
$q_{off}^{s}=0.03\pm 0.02$ is almost identical to that of the chamber
distribution ($q_{off}=0.04\pm 0.02$), the second peak, $q_{in}=0.16\pm 0.5$
is lower and has a smaller spread ($q_{in}=0.23\pm 0.08$). However, both peaks
are clearly bimodal and have similar cutoff values.

We further analyze the behavior of the chambers by removing the coupling of
their audiences. To do so, when comparing high-impact users $i$ and $j$, we
remove the common audience members of $i$ and $j$ and construct their chambers
without them. In Fig. 11, we compare the overlap similarity distributions with
and without removing the common audience members. Both are bimodal. We observe
that the expected overlap of the second peak decreases but is still
significantly high. Moreover, both distributions share the same bimodal
structure with a similar cutoff, indicating that we only removed redundant
information by removing the coupling caused by the audiences. This suggests
that the chamber is more robust to missing information.

Finally, we argue that the audience is a collection of information consumers,
whereas a chamber is a collection of information sources. On the one hand,
every member of the audience consumes information from the leading user in a
traditional one-to-many fashion. On the other hand, the audience consumes
information from the chamber in a many-to-many fashion, where the ideological
coherence comes from considering the chamber as a whole and not its individual
members. Thus, the overlap between chambers reflects the similarity between
the many-to-many information channels of the audiences, giving us a proxy of
the ideological (dis)similarities between leading users/leading users.

##  Appendix D Leading users description

We show in Table 1 the list of leading users and their main characteristics.

leading user | Persistence | Total impact | Median impact | Chamber size | Ideology (spectral) | Ideology (manual)
---|---|---|---|---|---|---
BernieSanders | 24 | 410442 | 5093 | 35811 | believers | believers
LilNasX | 10 | 320133 | 10301 | 6583 | believers | other
PaulEDawson | 39 | 282040 | 7349 | 88025 | believers | believers
AOC | 23 | 243209 | 7481 | 40302 | believers | believers
CNN | 30 | 158068 | 3077 | 38677 | believers | other
ewarren | 23 | 116018 | 3519 | 29140 | believers | believers
MikeHudema | 29 | 112690 | 3345 | 49874 | believers | believers
sunfloweraidil | 7 | 102276 | 3951 | 1989 | believers | believers
SenSanders | 19 | 94747 | 2936 | 26297 | believers | believers
RepAdamSchiff | 8 | 92173 | 9855 | 21765 | believers | believers
ajplus | 21 | 78273 | 3179 | 20646 | believers | other
sunrisemvmt | 8 | 77320 | 1802 | 11992 | believers | believers
GeraldKutney | 31 | 73966 | 2180 | 51325 | believers | believers
KamalaHarris | 12 | 69900 | 4365 | 16269 | believers | believers
IlhanMN | 7 | 69250 | 4806 | 13716 | believers | believers
JimMFelton | 9 | 66397 | 5102 | 12995 | believers | believers
GretaThunberg | 9 | 64787 | 5480 | 20537 | believers | believers
BetoORourke | 12 | 59042 | 2455 | 16247 | believers | believers
nowthisnews | 17 | 58796 | 3094 | 18299 | believers | other
nytimes | 15 | 49493 | 2174 | 19417 | believers | other
JayInslee | 16 | 44600 | 2594 | 20705 | believers | believers
narendramodi | 7 | 43984 | 4203 | 3669 | believers | believers
JoeBiden | 9 | 40170 | 3084 | 8333 | believers | believers
ProudResister | 9 | 38873 | 3048 | 12921 | believers | believers
brianschatz | 10 | 36951 | 2959 | 10740 | believers | believers
mmpadellan | 7 | 34335 | 4047 | 13277 | believers | believers
jeremycorbyn | 7 | 33573 | 2569 | 11454 | believers | believers
UNFCCC | 16 | 32000 | 2047 | 24574 | believers | believers
SenSchumer | 9 | 31598 | 3098 | 10875 | believers | believers
JustinTrudeau | 8 | 30146 | 3582 | 10281 | believers | believers
ProfStrachan | 14 | 28860 | 1786 | 26876 | believers | believers
billmckibben | 8 | 26331 | 2398 | 16595 | believers | believers
314action | 11 | 25086 | 2215 | 8853 | believers | believers
tictoc | 7 | 22500 | 2588 | 10495 | believers | other
cathmckenna | 10 | 22239 | 1762 | 15957 | believers | believers
BillGates | 8 | 19071 | 2400 | 4872 | believers | believers
MichaelEMann | 9 | 17947 | 1966 | 18596 | believers | believers
TIME | 7 | 16815 | 2039 | 7665 | believers | other
MSNBC | 7 | 14651 | 1882 | 8401 | believers | other
PrisonPlanet | 15 | 154928 | 5180 | 29046 | skeptics | skeptics
DineshDSouza | 13 | 75806 | 4515 | 14267 | skeptics | skeptics
charliekirk11 | 7 | 65267 | 8427 | 16081 | skeptics | skeptics
BhadDhad | 7 | 60552 | 2111 | 2352 | skeptics | other
chuckwoolery | 9 | 55050 | 3691 | 13988 | skeptics | skeptics
RealSaavedra | 9 | 53977 | 3109 | 11506 | skeptics | skeptics
catturd2 | 7 | 53477 | 3783 | 14776 | skeptics | skeptics
prageru | 8 | 32300 | 2825 | 14277 | skeptics | skeptics
manny_ottawa | 12 | 28981 | 2256 | 8308 | skeptics | skeptics
JWSpry | 13 | 21588 | 1690 | 16439 | skeptics | skeptics
mitchellvii | 8 | 17876 | 1914 | 5769 | skeptics | skeptics

Table 1: Description of the $M=50$ leading users defined in Eq. (4) in terms
of their persistence, $\Delta_{i}$, their cumulative impact,
$\sum_{t}w_{i}^{t}$, their median impact, $Med(w_{i}^{t})_{t}$, their
aggregate chamber size, $|\cup_{t}\mathcal{C}_{i}^{t}|$, and their ideology
based on manual labelling and the spectral clustering algorithm described in
Appendix B. Users are ordered by ideology according to the spectral clustering
algorithm and from largest to smallest impacts.