# Quantifying the Echo Chamber Effect:
An Embedding Distance-based Approach

Faisal Alatawi, Paras Sheth, Huan Liu
Arizona State University
<EMAIL_ADDRESS>
###### Abstract

The rise of social media platforms has facilitated the formation of echo
chambers, which are online spaces where users predominantly encounter
viewpoints that reinforce their existing beliefs while excluding dissenting
perspectives. This phenomenon significantly hinders information dissemination
across communities and fuels societal polarization. Therefore, it is crucial
to develop methods for quantifying echo chambers. In this paper, we present
the Echo Chamber Score (ECS), a novel metric that assesses the cohesion and
separation of user communities by measuring distances between users in the
embedding space. In contrast to existing approaches, ECS is able to function
without labels for user ideologies and makes no assumptions about the
structure of the interaction graph. To facilitate measuring distances between
users, we propose EchoGAE, a self-supervised graph autoencoder-based user
embedding model that leverages users’ posts and the interaction graph to embed
them in a manner that reflects their ideological similarity. To assess the
effectiveness of ECS, we use a Twitter dataset consisting of four topics - two
polarizing and two non-polarizing. Our results showcase ECS’s effectiveness as
a tool for quantifying echo chambers and shedding light on the dynamics of
online discourse.

###### Index Terms:

Echo Chamber, Polarization, Social Media, Ideology Detection, User
Representation, Graph Auto-Encoder

##  I Introduction

In the age of digital communication, social media platforms have
revolutionized the way we disseminate and consume information. Nevertheless,
this evolution has brought about notable challenges, particularly the
emergence of echo chambers and polarization [1, 2, 3]. These phenomena are
often characterized by high levels of controversy between members of different
groups and homogeneity among members of the same group [4]. This reinforces
pre-existing beliefs [1], discourages critical thinking [5], promotes the
spread of misinformation [6, 7], and leads to societal divisions. Hence, it is
crucial to devise methods for measuring the extent and impact of echo chambers
on social media. By quantifying them, we can better understand these phenomena
and, consequently, devise strategies to mitigate echo chamber effects and
foster more balanced and nuanced discussions. Ultimately, this could
contribute to a better informed, open-minded, and empathetic society. Such
efforts are particularly crucial in today’s world, where topics such as
politics, health, economics, and environmental issues, which are susceptible
to echo chambers [8, 4], have far-reaching implications for society.

Echo chambers are contingent on two dynamics: the interaction among users and
the individual ideological leanings of these users. Numerous measures and
metrics have been developed to leverage these dynamics, either separately or
in conjunction. One such method, is to leverage the interactions graph to
compute graph-specific metrics such as modularity [9], or resort to other
techniques like random walkers [10]. However, utilizing the graph introduces a
difficulty, as a graph may exhibit modularity without necessarily being
polarized or containing an echo chamber [11, 9]. An alternate approach
involves assessing the ideological disparity between users and their adjacent
nodes within the graph, investigating correlations between a user’s ideology
and that of their neighbors [12, 1], or observing ideological deviations from
the center of an opinion scale after deploying opinion-spreading models [13,
14]. These methodologies, although insightful, are fraught with challenges.
Labeling users to ascertain their ideologies or opinions is a laborious task
that is susceptible to errors. Similarly, semi-supervised methods that depend
on weak labels also present their own unique set of complications.

In response to these issues, we introduce the Echo Chamber Score (ECS) a
metric that captures the essence of the echo chamber concepts by focusing on
the dynamic interactions both within and across different user communities.
The crux of our approach is to gauge the similarity of users with their
respective communities (i.e., cohesion) and across different communities
(i.e., separation). Here, an interaction graph can be characterized as
exhibiting an echo chamber-like structure if it exhibits a low average
distance between users of a single community (i.e., high cohesion) and a high
average distance between users across different communities (i.e., high
separation). This strategy of using the distance allows us to bypass reliance
on potentially incidental graph structures and eliminates the need to split
the graph into two separate communities, an action that erroneously assumes
inherent polarization. Further, our method uses similarity in the embedding
space as a proxy for ideological distance, thereby circumventing the arduous
and error-prone task of detecting individual users’ ideologies.

To facilitate the measurement of ideological distance, we propose EchoGAE, a
self-supervised Graph Auto-Encoder [15] (GAE) based user embedding model.
EchoGAE is designed to capture the ideological similarities among users
through their interactions and shared posts, operating on two core principles:
homophily [16], where individuals associate and interact with those similar to
themselves, and linguistic homophily [17, 18], the tendency of socially
connected users to use language in similar ways. EchoGAE leverages homophilic
interactions such as retweets, regarded as endorsements of similar ideologies
[10, 19], along with the content of user posts. Both serve as inputs to
capture and map these ideological similarities. The model architecture
comprises an encoder that positions similar nodes closely together in the
embedding space, and a decoder that uses users’ embedding to reconstruct the
graph structure in a self-supervised manner. Additionally, it utilizes
Sentence-BERT [20], a BERT-based language model, to embed tweets, thus
reflecting their semantic similarities. By uniquely combining the interaction
graph structure and linguistic information from user posts, EchoGAE generates
representations that accurately reflect ideological similarities, establishing
it as a robust tool for measuring the effects of echo chambers and
polarization.

In this research, we evaluate the ability of the Echo Chamber Score (ECS) to
measure echo chamber effects within homophilic social interaction networks.
Our experiments are based on real-life Twitter datasets related to four
topics: two polarizing and two non-polarizing. Our findings confirm that the
ECS metric accurately identifies polarized interaction graphs and quantifies
the echo chamber effect in a manner consistent with existing state-of-the-art
methods. Furthermore, ECS proves successful in determining which communities
within the interaction graph are more polarized, demonstrating its unique
ability to rank communities based on their polarization. We also verify that
EchoGAE’s user embedding effectively reflects ideological distances between
users, showcasing its capacity to detect user ideologies. To promote
reproducibility and foster further development in this field, we make our
datasets and code available to the public111https://github.com/faalatawi/echo-
chamber-score.

##  II Related Work

Echo chambers and polarization measures can be divided into two main types:
graph-based and ideology-based methods. Graph-based methods are based on the
concept of a graph representing interactions between users on a given topic.
These methods operate on the assumption that polarization can be observed
within the graph itself. For instance, the modularity of a graph, which
quantifies how well a graph can be divided into distinct communities, has been
used to measure echo chambers [9]. However, challenges arise from this
approach, as modularity and other similar methods may not accurately represent
echo chamber phenomena due to the possibility that non-polarized graphs can
also exhibit high modularity [11, 9].

To address these limitations, new methods have been developed that scrutinize
the interactions between communities within a graph. These improved methods
involve dividing the graph into two distinct communities and measuring
polarization at the boundaries between them [11]. An alternative approach
involves using the Random Walk Controversy [10] (RWC), a popular polarization
method [21, 19] that calculates the probability of a random walker starting at
one community and ending at another. Nonetheless, these methods have their own
drawbacks, such as the necessity of splitting the communities in the graph and
making an inherent assumption that the graph is already polarized. This
results in difficulties in measuring polarization that may not actually exist.

Our novel approach, the Echo Chamber Score (ECS), alleviates these issues. The
ECS does not require the division of the graph into two communities and is
capable of measuring the effects of echo chambers and polarization across any
number of communities, making it a more flexible and accurate method for
assessing polarization.

Ideology-based methods for measuring echo chambers and polarization take a
different approach, focusing on a user’s ideological leaning and the users
they interact with. Two primary approaches exist within this category: (1)
measuring the ideological distance between a user and their neighboring users
in the graph, and (2) measuring the divergence from an ideological center
after applying an opinion-spreading model.

In the first approach, the ideological leanings of all users are estimated and
then compared to their neighboring users. The fundamental idea here is that an
echo chamber is formed when users mostly interact with others who share
similar opinions [12, 1, 22]. For instance, the ideology of users can be
inferred from the hashtags they share or the content they post [12, 1]. The
polarization is then quantified by measuring the Pearson correlation between a
user’s ideological score and the average ideological score of their neighbors
[12, 1].

In the second approach, opinion-spreading models such as the Friedkin-Johnsen
or DeGroot opinion model are utilized [23, 13, 14, 24]. For instance, the
Friedkin-Johnsen model operates by updating a node’s opinion through
repeatedly averaging the opinions of its neighbors until reaching equilibrium
[13, 14]. Polarization is then measured by how much opinions at equilibrium
deviate from the average [13, 14]. Alternatively, the DeGroot opinion model is
used to construct a Polarization Index (PI) based on the probability density
distribution of individuals’ opinions [24]. A bimodal distribution would
suggest the existence of polarization, while a unimodal distribution would
indicate its absence [24].

Both these ideology-based approaches have challenges, such as the laborious
and error-prone task of estimating users’ ideological leanings from their
content or interactions. Therefore, we have opted instead for a model based on
similarity in the embedding space as a proxy for ideology, eliminating the
need for ideology estimation.

##  III Methodology

This section presents our approach to quantifying echo chambers in online
conversations. Our objective is to assess whether the discussion surrounding a
given topic exhibits polarization and whether the communities formed by users
can be characterized as echo chambers or comprise a diverse group of
individuals with varying ideologies. To achieve this, we construct a graph
$G=(V,E)$, where $V$ represents the set of social media users, and $E$
represents the edges denoting homophilic interactions, such as retweets.
Additionally, we obtain a set of communities $\Omega$ from a community
detection algorithm, where each community consists of a group of users. Our
primary aim is to measure the level of polarization within the entire graph by
computing the Echo Chamber Score (ECS) for each community. Consequently, this
section presents our novel ECS metric for quantifying echo chambers. However,
as ECS relies on user embedding, we begin by introducing our user embedding
framework, EchoGAE, which enables the representation of users based on their
ideological similarity.

###  III-A Embedding Social Media Users

The EchoGAE model (see figure 1) is essential to our methodology for
quantifying echo chambers in online conversations. Its purpose is to embed
users in a way that reflects their ideological similarity, facilitating the
calculation of the Echo Chamber Score (ECS). By placing ideologically similar
users closer in the embedding space, EchoGAE enables the measurement of
cohesion and separation of communities in the graphs, the two components of
ECS, as we will explain later in the section.

EchoGAE is an adaptation of the Graph Auto-Encoder (GAE) model [15], tailored
for user embedding based on tweets and interactions. As a self-supervised
model, EchoGAE eliminates the need for user ideological labeling. It employs
two graph convolutional layers to encode the graph into a latent
representation, which is subsequently decoded to reconstruct the graph
structure. EchoGAE aims to minimize the binary cross-entropy between the real
and reconstructed adjacency matrices.

The EchoGAE model consists of two main components: an Encoder and a Decoder.
The Encoder takes both the tweets and the graph as input to create node
embeddings, which serve as the user embeddings. The Encoder is divided into
two parts. Firstly, the tweets component utilizes Sentence-BERT [20] to embed
the user’s tweets, and the average of these tweet embeddings is taken to form
the content embeddings (In fig 1, it’s represented as the matrix
$\mathbf{X}$). Secondly, the network component leverages the adjacency matrix
($\mathbf{A}$ in fig 1) of the graph. Together, these components contribute to
the creation of nodes embeddings (or users embeddings
$\mathbf{Z}\in\mathbb{R}^{n\times d}$ where $n$ is the number of users in the
graph and $d$ is the dimension of user embedding) that capture the information
from both the users’ content and their network interactions.

The Decoder performs an inner product operation [15] on the node
representations ($\sigma(\mathbf{Z}*\mathbf{Z^{T}})$) obtained from the
Encoder, resulting in a reconstructed adjacency matrix ($\mathbf{\hat{A}}$).
Subsequently, the binary cross-entropy loss is used to train the model and
ensure accurate graph reconstruction.

Figure 1:  The EchoGAE Model comprises two primary components: an Encoder and
a Decoder. The Encoder employs both user content embeddings ($\mathbf{X}$) and
adjacency matrix ($\mathbf{A}$) to generate user embeddings ($\mathbf{Z}$).
The Decoder then reconstructs the adjacency matrix ($\mathbf{\hat{A}}$) using
the user representations.

###  III-B Measuring the Echo Chamber Effect

We introduce ECS (Echo Chamber Score), a measure for quantifying the echo
chamber and polarization effects on social media. To measure the echo chamber
effect using user embedding, we assess in-group cohesion and between-group
separation [25]. We utilize the distance in the embedding space as a proxy for
these factors, reflecting how closely related users within a community are
(cohesion) and how distinct a community is from others (separation).

Let $Z\in\mathbb{R}^{n\times d}$ represent user embeddings, where $n$ is the
number of users and $d$ is the embedding dimension. Additionally, let
$\Omega=\\{\omega_{1},\omega_{2},\dots,\omega_{M}\\}$ denote the set of
communities, where $\omega_{i}\subset V$ represents the $i^{th}$ community
consisting of users. For a user $u\in\omega$, we compute the cohesion value
($\lambda_{u}$) as the average distance between $u$ and other users in the
same community using Equation 1.

$\lambda_{u}=\frac{1}{|\omega|}\sum_{\begin{subarray}{c}v\in\omega\\\ v\neq
u\end{subarray}}dist(u,v)$ (1)

Here, $|\omega|$ denotes the number of users in the community $\omega$, and
$dist(u,v)$ represents the distance (e.g., Euclidean) between users $u$ and
$v$ in the embedding space ($Z^{(u)}$ and $Z^{(v)}$ respectively). Similarly,
we compute the separation value ($\Delta_{u}$) as the average distance between
$u$ and the nearest community other than $\omega$ using Equation 2.

$\Delta_{u}=\operatorname*{min}_{\begin{subarray}{c}\omega\in\Omega\\\
u\notin\omega\end{subarray}}\left[\frac{1}{|\omega|}\sum_{v\in\omega}dist(u,v)\right]$
(2)

To calculate the Echo Chamber Score (ECS) for a community
$\omega=\\{u_{1},u_{2},\dots,u_{N}\\}$, we use a formula inspired by the
silhouette score [26] (in the appendix we show how to derive the ECS from the
silhouette). Equation 3 produces a score between 0 and 1, with a higher score
indicating a greater likelihood of an echo chamber effect within the
community.

$ECS^{*}(\omega)=\frac{1}{|\omega|}\sum_{u\in\omega}\frac{max(\Delta_{u},\lambda_{u})+\Delta_{u}-\lambda_{u}}{2*max(\Delta_{u},\lambda_{u})}$
(3)

The Echo Chamber Score can be computed for the entire graph using Equation 4,
where $\Omega$ represents the set of communities obtained from a community
detection algorithm such as Louvain [27] or Leiden [28].

$ECS(\Omega)=\frac{1}{|\Omega|}\sum_{\omega\in\Omega}ECS^{*}(\omega)$ (4)

The Echo Chamber Score (ECS) allows for comparison across different graphs
representing various controversial topics. A higher ECS indicates a higher
degree of echo chamber within a conversation. The components of ECS can
provide additional insights, such as ranking communities based on their
polarization, using Equation 3.

Note that our approach does not assume a specific number or size of
communities and is independent of the community detection method. Moreover, it
does not require prior knowledge of users’ internal ideologies, setting it
apart from related works [10, 24].

###  III-C Estimating Users’ Ideology

Our embedding model, EchoGAE, aims to position users with similar ideological
leanings closer to each other in the embedding space. Therefore, we assume
that we can utilize the distance in the embedding space to infer users’
ideological leanings. This helps us evaluate whether EchoGAE embeds users in a
way that reflects their ideology, which is the core idea behind ECS.

After applying the EchoGAE embedding, we employ a clustering algorithm (e.g.,
KMeans) to detect two communities of users in the embedding space, denoted as
$\omega_{1}$ and $\omega_{2}$. These communities represent the pro and anti
sides of the debate, respectively. We follow similar works [10, 24] that split
the ideology spectrum into two sides.

The ideology score for each user is calculated using Equation 5. It is
determined by the difference between the average distance of the user $u$ to
other users in $\omega_{1}$ and the average distance to users in $\omega_{2}$.

$I(u)=\frac{1}{|\omega_{1}|}\sum_{\begin{subarray}{c}v\in\omega_{1}\\\ v\neq
u\end{subarray}}dist(u,v)-\frac{1}{|\omega_{2}|}\sum_{\begin{subarray}{c}v\in\omega_{2}\\\
v\neq u\end{subarray}}dist(u,v)$ (5)

Here, $dist$ represents any distance function normalized between 0 and 1. In
our implementation, we employ the Euclidean distance, but other distance
measures can be used. The ideology scores $I(u)$ range from -1 to +1.
Importantly, values of -1 and +1 do not inherently indicate ”good” or ”bad”
ideologies. In Equation 5, the order of the communities ($\omega_{1}$ and
$\omega_{2}$) affects the sign of the ideology score. If a user belongs to
$\omega_{1}$, their score is positive when $\omega_{1}$ is in the first term.
Reversing the order of communities changes the sign but not the magnitude of
the score. This introduces an additional layer of complexity in evaluating our
method, which we address in the experimental results section.

##  IV Experiments

In this section, we present the experiments we used to assess the
effectiveness of our proposed method, Echo Chamber Score (ECS), in analyzing
the echo chamber effect. To evaluate its performance and reliability, we
compare ECS with two commonly used methods, Random Walk Controversy (RWC) and
Polarization Index (PI). Additionally, we utilize ECS to analyze echo chambers
at the community level, examining the distances between users in the embedding
space to gain insights into the cohesion and separation of user communities.
Furthermore, We conduct an experiment to determine if the distances in the
embedding space can predict the ideological leaning of users. Finally, we
perform an ablation study to examine the impact of using tweets in measuring
the echo chamber effect and predicting user ideology. These experiments
provide valuable insights into the performance and applicability of ECS in
analyzing echo chambers, predicting user ideology, and assessing the role of
tweets in these measurements.

###  IV-A Datasets

To investigate the echo chamber phenomenon, we selected four topics to examine
user interactions related to these subjects. Two topics were controversial:
abortion and gun debates, while the other two were non-controversial: the SXSW
conference and the Super Bowl. The inclusion of non-controversial topics aimed
to assess our method’s performance in non-polarized settings. The datasets
used in our experiments are outlined in Table I, and we have made them
publicly available222https://github.com/faalatawi/echo-chamber-score to ensure
reproducibility and facilitate further research in the field of echo chamber
analysis and detection.

TABLE I: Our datasets. We build a retweet graph for each topic where the nodes represent users, and the edges are retweet interactions. This table shows the communities of users ($\Omega$) that we detected using the Louvain Algorithm. |  Gun |  Abortion |  Super Bowl |  SXSW
---|---|---|---|---
# Nodes |  6566 |  5087 |  5460 |  2436
# Edges |  14322 |  10572 |  8732 |  5325
# $\Omega$ |  2 |  2 |  3 |  6
$|\Omega|$ |  [3984, 2582] |  [3933, 1154] |  [5398, 39, 23] |  [1532, 717, 85, 54, 34, 14]
Keywords |  Gun, 2nd Amendment, School Shooting |  Abortion Rights, Pro-Life, Pro-Choice, RoeVWade, #LifeWins |  #SuperBowl |  SXSW
Dates |  May-Jul 2022 |  Jun 24-25, 2022 |  Feb 13-Mar 13, 2022 |  Mar 10-19, 2022
# Labeled users |  1022 |  1999 |  194 |  200

Data collection. To collect data for each topic, we identified frequently used
keywords in discussions (see Table I) and monitored the conversation. We then
gathered the retweeters of the most popular tweets associated with these
keywords. This data was used to construct a graph for each topic, where users
were represented as nodes, retweet interactions formed the edges, and users’
tweets provided node attributes. We collected up to 200 of the users’ most
recent tweets (excluding retweets) to ensure an adequate amount of user-
generated text for analysis.

The gun debate dataset was collected during the period of intense debate
following the Uvalde school shooting in Uvalde, Texas, on May 24, 2022.
Unfortunately, school shootings in the United States often ignite polarized
discussions [11] on gun violence and constitutional gun ownership rights. To
capture this discourse, we selected commonly used words from both sides of the
debate and monitored the conversation from May to July. We then selected the
top 1200 most retweeted tweets and constructed the retweet graph. The
resulting graph (shown in the lower left panel of Figure 2) exhibited two
communities, as identified by the Louvain algorithm [27], indicating the
presence of two polarized communities [10]. Similarly, we collected the
retweet graph from the abortion rights debate following the US Supreme Court
ruling on abortion that was issued on June 24, 2022, using relevant keywords.
Both the gun debate [29, 11, 1] and abortion [30, 10] have been widely studied
as topics for analyzing echo chambers and polarization.

On the other hand, for non-controversial topics, we selected the topics that
have been used to study echo chambers Super Bowl [22] and SXSW [10, 31]. The
Super Bowl is an annual sports event in the US, while the SXSW conference is
an annual event that combines music, film, and interactive media in Austin,
Texas. We followed the same data collection procedure as with the
controversial topics.

Labeling. To evaluate the embedding quality of EchoGAE in capturing
ideological similarity, we estimated users’ ideological leanings. Following
previous works that used news URLs to infer political leanings [32, 33, 34,
35], we obtained ideological labels for URLs from the non-partisan media
watchdog AllSides333https://www.allsides.com/media-bias. To assign labels to
users, we utilized the news URLs they post as indicators of their ideology,
using AllSides’ political leanings for news websites’ URLs. A user’s political
leaning is calculated as the average of the news articles they share.
AllSides’ ratings consist of five categories: left, center-left, center,
center-right, and right, to which we assigned values of -1, -0.5, 0, 0.5, and
1, respectively. It is important to note that these values indicate opposing
sides of the debate and do not inherently represent good or bad ideologies. We
only used labels for users who shared at least five links. The number of
labeled users for each dataset is specified in Table I. Notably, controversial
topics tend to have more labeled users due to the nature of user engagement
with these topics, as users are more likely to express their ideological
leanings in these topics.

###  IV-B Measuring the Echo Chamber Effect

In this experiment, our objective is to evaluate the effectiveness of our
proposed method in measuring the echo chamber effect. To accomplish this, we
compare our method with commonly used techniques for calculating polarization
and echo chamber effects. This comparison aims to demonstrate that our method
performs comparably to existing methods and produces reliable results for
measuring the echo chamber effect.

TABLE II: Measuring the echo chamber in each dataset. ECS was able to assign a higher score for polarized topics (in red) in compassion with non-polarized ones (in blue). Please view it in color. | ECS | RWC | PI
---|---|---|---
Gun | 0.714 | 0.419 | 0.314
Abortion | 0.626 | 0.512 | 0.186
Super Bowl | 0.485 | 0.273 | 0.016
SXSW | 0.465 | 0.483 | 0.002

For our experiments, we utilize two widely used baselines: Random Walk
Controversy (RWC) [10] and Polarization Index (PI) [24]. We then compare these
baselines with our proposed method, Echo Chamber Score (ECS). RWC measures the
likelihood of transitioning from one community to another in a network, where
a value close to one indicates polarization and close to zero indicates no
polarization. On the other hand, PI measures the degree of segregation within
a population by modeling the propagation of opinions based on the probability
density distribution of individuals’ opinions.

To compute RWC, we partition the graph into two communities using the FluidC
[36] algorithm. Subsequently, we calculate the probability of transitioning
from one partition to another. For PI, we employ the DeGroot opinion model
[23] with labeled users as seeds to disseminate opinions, and then we compute
the PI index for each graph. In contrast to RWC, our proposed method ECS does
not require dividing the graph into two communities. The graph may consist of
multiple communities, and any community detection method can be employed. In
this study, we use the Louvain algorithm [27] to identify the communities,
which are then used to compute ECS. Furthermore, unlike PI, our method does
not rely on any labeled users, as we utilize the embeddings obtained from
EchoGAE.

As shown in Table II, our approach effectively assigns higher scores to
controversial topics (e.g., Gun debate and Abortion) compared to non-
controversial ones, demonstrating its ability to perform on par with existing
methods. Our method aligns with PI, a highly regarded technique that employs
ideology labels to gauge polarization. PI’s approach closely approximates the
actual labels, and our method exhibits strong agreement with it, as evidenced
by a 0.99 Pearson correlation. In contrast, there are notable differences
between our method and RWC. For instance, both ECS and PI indicate that the
Gun Control debate is more polarized than the Abortion debate, which
contradicts the findings of RWC. We posit that the requirement of RWC to
partition the graph into only two communities hinders its performance. By
relaxing this requirement, our measure ECS can evaluate any number of
communities identified by various community detection algorithms.

These techniques (RWC, PI, and ECS) enable us to rank topics based on their
polarization levels, from highest to lowest. Both PI and our method (ECS)
consistently rank the topics in a similar manner. It is worth noting that our
method considers the Gun debate more polarized than the Abortion debate,
aligning with opinion polls. According to the Pew Research
Center444https://www.pewresearch.org/, in 2022, 61% of Americans supported
abortion access, while only 53% advocated for stricter gun laws. This
demonstrates greater disagreement and polarization within the Gun debate
compared to the Abortion debate.

###  IV-C Analysing the Echo Chamber Effect on Community Level

To showcase ECS’s capability in analyzing the echo chamber at a more detailed
level, we conducted an experiment to examine the insights provided by our
measure at the community level. The objective was to determine which community
within a topic exhibited a higher level of polarization. For this experiment,
we focused on the controversial topics, namely the Gun debate and Abortion,
and explored how we could investigate the interaction both between and within
communities. These topics were chosen due to the presence of echo chambers, as
identified in the previous experiment.

Upon examining the gun dataset, we observed that the debate surrounding guns
and school shootings exhibited a higher level of polarization compared to
abortion, as evidenced by an ECS score of 0.714 compared to 0.626 (see Table
II). Applying the Louvain algorithm, we identified two communities in the
interaction graph, with sizes of 3984 and 2582 nodes, respectively. Computing
the ECS* (equation 3) for each community, we obtained echo chamber scores of
0.739 and 0.676, indicating polarization and ideological homogeneity within
both communities. Notably, the larger community demonstrated a little bit
higher level of polarization.

Upon labeling a sample of ten users from each community, we discovered that
the larger community aligned with the anti-gun group, while the smaller
community represented the pro-gun group. By examining the 2D projection of the
EchoGAE embedding of users (refer to Figure 2), we observed that the blue
community (anti-gun) appeared to have similar size as the other community
(pro-gun), suggesting close levels of polarization between the two
communities. However, the anti-gun higher ECS score indicates that this group
is more homogenized than the other group, which is surprising. It is possible
that this debate around guns is not a right and left issue, and more centrists
voices are participating in the debate. This analysis would be challenging to
perform using PI or RWC techniques. However, ECS, being community-independent
and not reliant on ideology labels, enables such analysis without prior
knowledge of community divisions and ideologies.

In the abortion dataset, we identified two communities with sizes of 3933 and
1154. The ECS* scores for these communities were 0.6 and 0.69, respectively.
To gain deeper insights, we conducted a random sampling of ten users from each
community and manually examined their Twitter accounts. Our analysis revealed
that the larger community primarily consisted of supporters of abortion
rights. On the other hand, the anti-abortion community exhibited a higher
level of polarization compared to the other community. This finding aligns
with the opinion polls mentioned earlier, as the anti-abortion group tends to
hold a more fringe position compared to the pro-abortion group. Additionally,
this alignment can be observed in the abortion rights vote that took place in
Kentucky, which is considered a conservative state. During the vote, the
majority of voters rejected555https://www.pbs.org/newshour/politics/kentucky-
voters-reject-constitutional-amendment-on-abortion the proposal to restrict
abortion rights.

Figure 2: The top two panels show the 2D projection (using t-SNE [37]
algorithm) of the embedding of users in each graph after embedding them using
EchoGAE. The lower two panels are the two graphs. We used the ForceAtlas2 [38]
algorithm to plot the graphs. The colors represent the communities that were
discovered by Louvain Algorithm

###  IV-D Using Ideology Detection to Verify the Embedding Space

We assume that the distance in the embedding space could be used to predict
the political leaning of users and that users with similar ideological
leanings are closer to each other in the embedding space. If we prove that the
distance in the embedding space could be used to estimate the ideology of
users, we could then use the distance to measure the echo chamber effect, as
we rely on the distance to measure the separation (eq 2) and cohesion (eq 1)
of communities in order to gauge the echo chamber effect.

After labeling users, we then split the labeled users into training and
validation sets (10% - 90% respectively). Since our model is unsupervised, the
training set is used by the baseline model only, and we use the validation set
to validate the estimation of both models. For the baseline model, we used the
DeGroot opinion model [23], in which the user’s ideology is the average
ideology of their neighbors. After embedding users using EchoGAE, we employed
the KMeans algorithm to detect two communities of users in the embedding
space, referred to as $\omega_{1}$ and $\omega_{2}$, representing the pro and
anti sides of the debate. Lastly, we calculated the ideology score of each
user, taking into account their distances to the members of communities
$\omega_{1}$ and $\omega_{2}$ in the embedding space as shown in equation 5.

In Table III, we present our method’s outcomes for estimating ideology
compared to the baseline. The resulting ideology scores were compared to the
pseudo-scores obtained from AllSides labeling. Our analysis involved comparing
our ideology scores to those obtained from the AllSides labeling and the
baseline model using Mean Absolute Error (MAE), and Mean Squared Error (MSE).
The results shown in Table III demonstrate that our model performs comparably
to the semi-supervised baseline, even though our method is unsupervised (we do
not use any labels in our model). Furthermore, as depicted in Figure 3, a high
degree of concurrence is observed between the distributions of the predicted
and actual ideologies.

It should be noted that in Equation 5, the order of the communities (i.e.,
$\omega_{1}$ and $\omega_{2}$) influences the sign of the ideology score. For
instance, if a user belongs to $\omega_{1}$ (i.e., is more closely associated
with users in $\omega_{1}$), their ideology score would be positive if
$\omega_{1}$ appeared in the equation’s first term. However, if the order of
the communities is reversed, the score’s magnitude remains the same, but the
sign changes. Consequently, in our measurement, we tried both orders (i.e.,
$\omega_{1}$ first then it becomes the second) and report the minimum value.

Figure 3: A histogram of the predicted vs pseudo ideology scores of users for each topic. In all the topics, the estimated (predicted) distribution of ideology scores closely matches the ideology scores estimated from the URLs that user share. TABLE III: We compare the predicted ideology scores from our model vs. the score from the baseline. We see that our model is comparable to the baseline in all the topics using the two measures we have. | MAE $(\downarrow)$ | MSE $(\downarrow)$
---|---|---
| Our | Baseline | Our | Baseline
Gun | 0.291 | 0.271 | 0.135 | 0.111
Abortion | 0.323 | 0.269 | 0.178 | 0.136
Super Bowl | 0.352 | 0.253 | 0.229 | 0.147
SXSW | 0.281 | 0.185 | 0.148 | 0.065

###  IV-E Ablation Study

The primary objective of this study is to examine the impact of the components
of EchoGAE on the performance of two tasks: measuring the echo chamber effect
and predicting the ideology of users. Specifically, the study explores the
significance of using textual information, i.e., tweets, in these tasks. Table
IV presents the results obtained from this study. It demonstrates that the
model’s performance is enhanced when tweets are utilized. This finding
emphasizes the importance of linguistic similarity in measuring echo chambers
and estimating ideology. Therefore, the study suggests that investing more
resources to extract knowledge from tweets could lead to improved accuracy in
both tasks.

However, the study also observes that good results can be achieved with the
graph component alone, in situations where textual information is unavailable.
Notably, even in cases where the difference in echo chamber scores between
controversial and non-controversial topics is not substantial, the tweet-less
model still performs well by assigning higher scores to controversial topics.

In conclusion, this study provides empirical evidence supporting the
importance of incorporating textual information, such as tweets, in measuring
echo chambers and estimating ideology. Nevertheless, it also highlights that
satisfactory results can be obtained with graph-only models in the absence of
textual data.

TABLE IV: In the ablation study, tweets (T) were used to embed users,
resulting in improved performance for ECS in scoring both controversial and
non-controversial topics. Additionally, using tweets for estimating users’
ideology showed superior performance with lower MAE and MSE.

| ECS | MAE $(\downarrow)$ | MSE $(\downarrow)$
---|---|---|---
| T | No T | T | No T | T | No T
Gun | 0.714 | 0.571 | 0.289 | 0.304 | 0.132 | 0.143
Abortion | 0.626 | 0.550 | 0.321 | 0.497 | 0.176 | 0.363
Super Bowl | 0.485 | 0.438 | 0.361 | 0.428 | 0.233 | 0.315
SXSW | 0.465 | 0.464 | 0.274 | 0.421 | 0.140 | 0.286

##  V Conclusion

In this paper, we introduced Echo Chamber Score (ECS), a novel metric for
quantifying echo chambers and polarization in social media networks. ECS
leverages an embedding space to measure the cohesion and separation of user
communities, providing insights into the echo chamber effect. To enable this
measurement, we presented EchoGAE, a self-supervised user embedding model that
captures ideological similarities among users and generates accurate
embeddings.

Our evaluation of ECS on a Twitter dataset demonstrated its effectiveness in
ranking topics based on echo chamber scores and ordering communities by
polarization levels. Compared to existing metrics, ECS showcased unique
capabilities in capturing the dynamics of online discourse. Our research
contributes to understanding and quantifying echo chambers and polarization,
which could help the development of strategies to mitigate their negative
impacts and promote a more informed and open-minded society.

## References

  * [1] M. Cinelli, G. De Francisci Morales, A. Galeazzi, W. Quattrociocchi, and M. Starnini, “The echo chamber effect on social media,” _PNAS_ , vol. 118, no. 9, p. e2023301118, Mar. 2021.
  * [2] L. Terren and R. Borge-Bravo, “Echo chambers on social media: A systematic review of the literature,” _RCR_ , vol. 9, pp. 99–118, Mar. 2021.
  * [3] F. Alatawi, L. Cheng, A. Tahir, M. Karami, B. Jiang, T. Black, and H. Liu, “A survey on echo chambers on social media: Description, detection and mitigation,” Dec. 2021.
  * [4] G. Villa, G. Pasi, and M. Viviani, “Echo chamber detection and analysis,” _SNAM_ , vol. 11, no. 1, p. 78, Aug. 2021.
  * [5] S. C. Rhodes, “Filter bubbles, echo chambers, and fake news: How social media conditions individuals to be less critical of political misinformation,” _PolComm_ , vol. 39, no. 1, pp. 1–22, Jan. 2022.
  * [6] M. Del Vicario, A. Bessi, F. Zollo, F. Petroni, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, “The spreading of misinformation online,” _PNAS_ , vol. 113, no. 3, pp. 554–559, Jan. 2016.
  * [7] C. Diaz Ruiz and T. Nilsson, “Disinformation and echo chambers: How disinformation circulates on social media through identity-driven controversies,” _JPP &M_, vol. 42, no. 1, pp. 18–35, Jan. 2023.
  * [8] F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, “Modeling echo chambers and polarization dynamics in social networks,” _Physical Review Letters_ , vol. 124, no. 4, p. 048301, Jan. 2020.
  * [9] M. Wolfowicz, D. Weisburd, and B. Hasisi, “Examining the interactive effects of the filter bubble and the echo chamber on radicalization,” _JEC_ , vol. 19, no. 1, pp. 119–141, Mar. 2023.
  * [10] K. Garimella, G. D. F. Morales, A. Gionis, and M. Mathioudakis, “Quantifying controversy on social media,” _ACM Transactions on Social Computing_ , vol. 1, no. 1, pp. 1–27, Feb. 2018.
  * [11] P. Guerra, W. M. Jr, C. Cardie, and R. Kleinberg, “A measure of polarization on social media networks based on community boundaries,” _ICWSM_ , vol. 7, no. 1, pp. 215–224, 2013.
  * [12] W. Cota, S. C. Ferreira, R. Pastor-Satorras, and M. Starnini, “Quantifying echo chamber effects in information spreading over political communication networks,” _EPJ Data Science_ , vol. 8, no. 1, pp. 1–13, Dec. 2019.
  * [13] A. Matakos, E. Terzi, and P. Tsaparas, “Measuring and moderating opinion polarization in social networks,” _DMKD_ , vol. 31, no. 5, pp. 1480–1505, Sep. 2017.
  * [14] C. Musco, C. Musco, and C. E. Tsourakakis, “Minimizing polarization and disagreement in social networks,” in _WWW ’18_ , 2018, pp. 369–378.
  * [15] T. N. Kipf and M. Welling, “Variational graph auto-encoders,” 2016.
  * [16] M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a feather: Homophily in social networks,” _ARSO_ , vol. 27, no. 1, 2001.
  * [17] Y. Yang and J. Eisenstein, “Overcoming language variation in sentiment analysis with social attention,” _TACL_ , vol. 5, pp. 295–307, 2017.
  * [18] B. Kovacs and A. M. Kleinbaum, “Language-style similarity and social networks,” _Psychological Science_ , vol. 31, no. 2, pp. 202–213, 2020.
  * [19] A. Cossard, G. D. F. Morales, K. Kalimeri, Y. Mejova, D. Paolotti, and M. Starnini, “Falling into the echo chamber: The italian vaccination debate on twitter,” _ICWSM_ , vol. 14, pp. 130–140, May 2020.
  * [20] N. Reimers and I. Gurevych, “Sentence-bert: Sentence embeddings using siamese bert-networks,” Aug. 2019.
  * [21] A. Rashed, M. Kutlu, K. Darwish, T. Elsayed, and C. Bayrak, “Embeddings-based clustering for target specific stances: The case of a polarized turkey,” _ICWSM_ , vol. 15, pp. 537–548, May 2021.
  * [22] P. Barberá, J. T. Jost, J. Nagler, J. A. Tucker, and R. Bonneau, “Tweeting from left to right: Is online political communication more than an echo chamber?” _Psychol. Sci._ , vol. 26, no. 10, pp. 1531–1542, Oct. 2015.
  * [23] M. H. DeGroot, “Reaching a consensus,” _JASA_ , vol. 69, no. 345, 1974.
  * [24] A. J. Morales, J. Borondo, J. C. Losada, and R. M. Benito, “Measuring political polarization: Twitter shows the two sides of venezuela,” _Chaos_ , vol. 25, no. 3, 2015.
  * [25] H. Xiong and Z. Li, “Clustering validation measures,” in _Data Clustering_. Chapman and Hall/CRC, 2014\.
  * [26] P. J. Rousseeuw, “Silhouettes: A graphical aid to the interpretation and validation of cluster analysis,” _JCAM_ , vol. 20, pp. 53–65, Nov. 1987\.
  * [27] V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre, “Fast unfolding of communities in large networks,” _JSTAT_ , 2008.
  * [28] V. A. Traag, L. Waltman, and N. J. van Eck, “From louvain to leiden: Guaranteeing well-connected communities,” _SciRep_ , vol. 9, no. 1, 2019\.
  * [29] V. Morini, L. Pollacci, and G. Rossetti, “Toward a standard approach for echo chamber detection: Reddit case study,” _ApplSci_ , 2021.
  * [30] F. H. Calderón, L.-K. Cheng, M.-J. Lin, Y.-H. Huang, and Y.-S. Chen, “Content-based echo chamber detection on social media platforms,” in _ASONAM ’19_ , Aug. 2019, pp. 597–600.
  * [31] H. Emamgholizadeh, M. Nourizade, M. S. Tajbakhsh, M. Hashminezhad, and F. N. Esfahani, “A framework for quantifying controversy of social network debates using attributed networks: Biased random walk,” _SNAM_ , vol. 10, no. 1, p. 90, Dec. 2020.
  * [32] J. Jiang, X. Ren, and E. Ferrara, “Retweet-bert: Political leaning detection using language features and information diffusion on social networks,” Jul. 2022\.
  * [33] A. Badawy, K. Lerman, and E. Ferrara, “Who falls for online political manipulation?” in _WWW ’19_ , May 2019, pp. 162–168.
  * [34] A. Bovet and H. A. Makse, “Influence of fake news in twitter during the 2016 us presidential election,” _NatComm_ , vol. 10, no. 1, p. 7, 2019.
  * [35] E. Ferrara, H. Chang, E. Chen, G. Muric, and J. Patel, “Characterizing social media manipulation in the 2020 u.s. presidential election,” _First Monday_ , Oct. 2020.
  * [36] F. Parés, D. Garcia-Gasulla, A. Vilalta, J. Moreno, E. Ayguadé, J. Labarta, U. Cortés, and T. Suzumura, “Fluid communities: A competitive, scalable and diverse community detection algorithm,” Oct. 2017.
  * [37] L. van der Maaten and G. Hinton, “Visualizing data using t-sne,” _JMLR_ , vol. 9, no. 86, pp. 2579–2605, 2008.
  * [38] M. Jacomy, T. Venturini, S. Heymann, and M. Bastian, “Forceatlas2, a continuous graph layout algorithm for handy network visualization designed for the gephi software,” _PLOS ONE_ , vol. 9, no. 6, 2014.

[Deriving the ECS* equation]

Here we derive the ECS* equation. We start with the silhouette score [26]:

$ECS^{*}(\omega)=\frac{1}{|\omega|}\sum_{u\in\omega}\left[\frac{\Delta_{u}-\lambda_{u}}{max(\Delta_{u},\lambda_{u})}\right]$

We want to scale it from 0 to 1 instead of -1 to +1. So we have:

$ECS^{*}(\omega)=\frac{1}{|\omega|}\sum_{u\in\omega}\frac{1}{2}\left[\frac{\Delta_{u}-\lambda_{u}}{max(\Delta_{u},\lambda_{u})}+1\right]$

Multiply the terms within the square brackets by the denominator of the first
fraction, $max(\Delta_{u},\lambda_{u})$:

$ECS^{*}(\omega)=\frac{1}{|\omega|}\sum_{u\in\omega}\frac{1}{2}\left[\frac{\Delta_{u}-\lambda_{u}+max(\Delta_{u},\lambda_{u})}{max(\Delta_{u},\lambda_{u})}\right]$

Finally, rearrange the terms in the numerator, then separate the fractions to
have the ECS equation:

$ECS^{*}(\omega)=\frac{1}{|\omega|}\sum_{u\in\omega}\frac{max(\Delta_{u},\lambda_{u})+\Delta_{u}-\lambda_{u}}{2*max(\Delta_{u},\lambda_{u})}$


# Dirac clouds around dilatonic black holes

Yang Huang 1<EMAIL_ADDRESS>Hongsheng Zhang 1 corresponding author:
<EMAIL_ADDRESS>1 School of Physics and Technology, University of
Jinan, 336, West Road of Nan Xinzhuang, Jinan 250022, Shandong, China

###### Abstract

Dirac cloud is in absence in general relativity since the superradiance
mechanism fails to work for Dirac fields. For the first time we find a novel
mechanism to support Dirac clouds, which is independent on superradiance
mechanism. We study quasi bound states of Dirac particles around a charged
spherical black hole in dilatonic gravity. We find that the quasi bound states
become real bound states when the central black hole becomes extremal. We make
an intensive study of the energy spectrum of the stationary clouds for
different fine structure constant $\mu M$, and reveal the existence condition
of these clouds. Our result strongly implies that extreme dilatonic black
holes behave as elementary particles.

The black hole uniqueness theorem is a milestone in black hole physics, which
implies that numerous characters, such as multi-poles, baryon number, lepton
number, chemical composition, etc, of the progenitor will be dissipated in the
formation process of black holes except mass, electric charge, and angular
momentum in Einstein-Maxwell theory Bekenstein (1972). Since the proposition
of the uniqueness theorem, explorations of new hair of black holes never
cease. To find an analytical metric of a hairy black hole is certainly
admirable, but extremely difficult. Back to general tradition in physics, a
feasible approach is to introduce a small perturbation around a black hole,
and then to observe the evolution of the small perturbation. If the
perturbation can enter a stationary state, it becomes a cloud of the hole,
just like an electron cloud in a hydrogen atom. If one further considers the
back reaction of the cloud to the metric, one may reach a hairy black hole
Herdeiro and Radu (2014); Hong et al. (2020).

The most important and practicable road to develop a cloud in astrophysics
depends on superradiance mechanism. The superradiance mechanism is an
extension of the Klein paradox, which says that more particles will be excited
when a scalar particle scattered by a special potential. Physically, the
collision of the particle and the potential leads to pair production and the
anti particle with negative energy is absorbed by the potential. For a
rotating (Kerr) black hole, a scalar particle passing through the ergosphere,
i.e., scattered by the effective potential of the ergosphere of the hole, may
generate more particles sensed by observers at spacelike infinity. This is the
black hole superradiance mechanism. There is a complementary effect of
superradiance, which reduces the particle number around a black hole. This may
be dubbed super absorption effect, which indicates that any surrounding
particles of a black hole has a probability to permeate into the hole because
of quantum tunneling effect. If the superradiance and super absorption effects
get a fine balance, the life time of surrounding particle of a black hole can
be very long, even longer than the age of the universe. In this case, the
surrounding particles compose a cloud of a black hole Huang and Liu (2016);
Huang et al. (2017a); Hod (2012); Bernard (2016).

A point worth emphasizing is that the superradiance mechanism works only for
Boson. Because of Pauli’s exclusion principle, Fermion never undergoes
superradiance process. While the super absorption mechanism still works for a
Fermion, and has no complementary mechanism to balance it. The conclusion is
that Dirac particles cannot develop a cloud around a black hole since they
inevitably infiltrate into the hole, and disappear soon. Fermions are
fundamental bricks of the Universe. It is quite significative to find novel
mechanism to develop a Dirac cloud for a black hole.

We will show that a stationary Dirac cloud around a GMGHS (Gibbons-Maeda-
Garfinkle-Horowitz-Strominger) black hole is able to survive when the hole
becomes an extreme one. The GMGHS solution in dilatonic gravity is an
influential solution, which is originally presented as a black hole in low
energy heterotic string theory Gibbons and ichi Maeda (1988); Garfinkle et al.
(1991). Dilatonic gravity (scalar-tensor theory) is a historic and alive
theory of modified gravities, which can be traced back to Kaluza-Klein
compactification and Dirac’s large number hypothesis. Many different modified
gravities reduce to dilatonic gravity at low energy limit.

The dilatonic black hole we consider here is described by the GMGHS metric,

$ds^{2}=-Fdt^{2}+F^{-1}dr^{2}+r^{2}G\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right),$ (1)

with $F(r)=1-r_{+}/r$ and $G(r)=1-r_{-}/r$, where $r_{+}=2M$, $r_{-}=Q^{2}/M$.
The black hole mass and charge are given by $M$, and $Q$, respectively. It is
useful to parameterize the black hole charge by
$q=Q/Q_{\text{max}}=1-e^{-\eta}$. The black hole becomes extremal when
$\eta\rightarrow\infty$.

We first consider the quasibound states of massive Dirac fields in GMGHS
spacetimes. The equation of a massive Dirac field in curved spacetime is
written as

$\left[\gamma^{\alpha}\left(\partial_{\alpha}-\Gamma_{\alpha}\right)-\mu\right]\bm{\Psi}=0,$
(2)

where $\mu$, $\bm{\Psi}$, and $\Gamma_{\alpha}$ are the mass, the Dirac-four
spinor, and spinor connection matrices, respectively. A suitable choice of
matrices $\gamma^{\mu}$ obeying the relation
$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}_{\text{dilaton}}$
are given by $\gamma^{\mu}=e^{\mu}_{a}\hat{\gamma}^{a}$, where $e^{\mu}_{a}$
are tetrad fields and matrices $\hat{\gamma}^{a}$ are defined by the flat
space Dirac matrices $\tilde{\gamma}^{a}$, i.e.,
$\hat{\gamma}^{0}=i\tilde{\gamma}^{0}$,
$\hat{\gamma}^{1}=i\tilde{\gamma}^{3}$,
$\hat{\gamma}^{2}=i\tilde{\gamma}^{1}$, and
$\hat{\gamma}^{3}=i\tilde{\gamma}^{2}$, with

$\tilde{\gamma}^{0}=\left(\begin{array}[]{c c}0&I_{2}\\\
I_{2}&0\end{array}\right),\;\;\;\tilde{\gamma}^{j}=\left(\begin{array}[]{c
c}0&\sigma_{j}\\\ -\sigma_{j}&0\end{array}\right).$ (3)

The spinor connection matrices are written as

$\Gamma_{\alpha}=-\frac{1}{4}g_{\mu\nu}e^{\mu}_{a}e^{\nu}_{b;\alpha}\hat{\gamma}^{a}\hat{\gamma}^{b}.$
(4)

Using the metric (1), we find the explicit form of the components of
$\Gamma_{\alpha}$

$\displaystyle\Gamma_{t}$
$\displaystyle=\frac{F^{\prime}}{4}\hat{\gamma}^{0}\hat{\gamma}^{1},\;\;\;\;\;\;\;\Gamma_{r}=0,$
(5) $\displaystyle\Gamma_{\vartheta}$
$\displaystyle=W\;\hat{\gamma}^{1}\hat{\gamma}^{2},$
$\displaystyle\Gamma_{\varphi}$
$\displaystyle=W\sin\vartheta\;\hat{\gamma}^{1}\hat{\gamma}^{3}+\frac{\cos\vartheta}{2}\hat{\gamma}^{2}\hat{\gamma}^{3},$

where

$W(r)=\frac{r}{4}\sqrt{\frac{F}{G}}\left(\frac{2}{r}+\frac{d}{dr}\right)G.$
(6)

Defining $\bm{\Phi}=rG^{1/2}F^{1/4}\left(\sin\vartheta\right)^{1/2}\bm{\Psi}$,
we can rewrite the Dirac equation as Brill and Wheeler (1957)

$i\partial_{t}\bm{\Phi}=H\bm{\Phi},$ (7)

where

$\displaystyle H$
$\displaystyle=i\hat{\gamma}^{0}\hat{\gamma}^{1}F\partial_{r}+\frac{1}{r}\sqrt{\frac{F}{G}}\hat{\gamma}^{1}K-i\mu\sqrt{F}\hat{\gamma}^{0},\;\;\text{and}$
(8) $\displaystyle K$
$\displaystyle=i\hat{\gamma}^{1}\hat{\gamma}^{0}\hat{\gamma}^{2}\partial_{\vartheta}+i\hat{\gamma}^{1}\hat{\gamma}^{0}\hat{\gamma}^{3}\csc\vartheta\partial_{\varphi}.$

The angular operator $K$ is Hermitian and commutes with $H$, $[K,H]=0$.
Therefore, operators $H$ and $K$ have simultaneous eigenfunctions, whose
$\mu$-th component can be separated by

$\Phi_{\mu}=R_{\mu}(r)\Theta_{\mu}(\vartheta,\varphi)e^{-i\omega t}.$ (9)

The angular part $\Theta_{\mu}(\vartheta,\varphi)$ is determined by the
requirement $K\bm{\Phi}=\lambda\bm{\Phi}$, where $\lambda$ is constant. The
orbital angular momentum and spin angular momentum are no longer conserved
quantities, respectively. $\lambda$ denotes the quantum number of the total
angular momentum. Boundary conditions at $\vartheta=0,$ and $\pi$ determine a
spectrum of non-zero integers of $\lambda$, i.e., $\lambda=\pm 1,\pm 2,\cdots$
Brill and Wheeler (1957).

Substituting the ansatz (9) into Eq.(7), and replacing the operator $K$ by the
number $\lambda$, we obtain $H\bm{\Phi}=\omega\bm{\Phi}$. After replacing the
operator $K$ by number $\lambda$, only two matrices $\hat{\gamma}^{0}$ and
$\hat{\gamma}^{1}$ remain explicitly in $H$. The radial factor can therefore
be represented by a two component spinor

$R=\left(\begin{array}[]{c}R_{+}\\\ R_{-}\end{array}\right).$ (10)

In this representation, the radial equations are given by

$\mathscr{D}_{\pm}R_{\pm}=\mathscr{C}_{\mp}R_{\mp},$ (11)

where $\mathscr{D}_{\pm}=F\partial_{r}\pm i\omega$ and

$\mathscr{C}_{\pm}=\frac{1}{r}\sqrt{\frac{F}{G}}\left(\lambda\pm i\mu
r\sqrt{G}\right).$ (12)

It is very clear that the above two equations reduce to the Dirac equation of
the case of Schwarzschild when $G=1$ Lasenby et al. (2005). One can eliminate
one of the two dependent variables to obtain a single second order ordinary
differential equation for the other variable.

Generally, quasibound states are purely ingoing at the event horizon and
vanish at spatial infinity. Our aim is to compute the quasibound states of
massive Dirac fields in non-extreme GMGHS spacetime. Close to $r=r_{+}$, the
ingoing wave solutions of Eqs.(11) are given by

$R_{\pm}\sim\mathcal{T}_{\pm}(r-r_{+})^{\frac{1\mp 1}{4}}e^{-i\omega r_{*}},$
(13)

where $r_{*}=\int dr/F$ is the tortoise coordinate, and $\mathcal{T}_{\pm}$
are the transmission coefficients. At infinity $r\rightarrow\infty$, the
vanishing solutions are given by

$R_{\pm}\sim r^{\chi}e^{kr},$ (14)

where $k=-\sqrt{\mu^{2}-\omega^{2}}$, and $\chi=M(\mu^{2}-2\omega^{2})/k$.
With these boundary conditions, we solve the second order equation of $R_{+}$
derived from Eqs.(11) to obtain the quasibound state frequencies. To get high
accurate eigen-frequency of the quasibound state, the spectral method is used
to solve this eigenvalue problem.

For bounded modes with $\omega<\mu$, the field cannot propagate to spatial
infinity. However, the system is still nonconservative, since the field will
leaks into the black hole through the horizon. Therefore, the quasibound state
frequencies are in general complex, i.e., $\omega=\omega_{R}+i\omega_{I}$. We
find that $\omega_{I}$ is always negative for Dirac fields in non-extreme
GMGHS spacetime, which excludes the existence of stationary states in this
case. Physically, negative $\omega_{I}$ implies that the Dirac particles have
probabilities to permeate into the black hole.

Figure 1: Eigen frequencies of quasibound state, with $\lambda=-1$, as
functions of $\eta$ for different values of $\mu M$. The left panel shows the
imaginary parts, whereas the right panel shows the real part.

Figure 1 shows the typical numerical results of the quasibound state
frequencies as functions of $\eta$. We see that both $\omega_{R}$ and
$\omega_{I}$ are sensitive to the values of $\eta$, or the black hole charge.
In particular, when $\eta\gg 1$, $\omega_{R}$ tends to constant, and
$|\omega_{I}|$ decays to zero exponentially. This result is in analogy with
the case of a massive scalar field in non-extreme GMGHS spacetime Huang and
Zhang (2021). It was shown that in the extreme limit $\eta\rightarrow\infty$,
the potential barrier becomes infinitely wide, and the tunneling effect could
be completely suppressed. Therefore, real bound states with $\omega_{I}=0$ may
exist for massive fields in extreme GMGHS spacetime.

Figure 2: Dotted curves denote the the imaginary part of eigen frequencies of
Dirac particles in the case of RN, while solid curves for the case of GMGHS.
We set $\lambda=-1$.

As a comparison, in Fig.2, we plot the imaginary part of eigen frequencies of
quasibound state of Dirac particles on the back ground of a Reissner-Nordstrom
black hole, which is the corresponding one of GMGHS in general relativity.
Note that we have parameterized the RN charge in the same way, i.e.,
$Q_{\text{RN}}/Q_{\text{RN, max}}=1-e^{-\eta}$. From Fig 2, it is clear that
Dirac particles permeate into an extreme RN hole since the imaginary part of
the eigen frequencies go to constant when the hole becomes extremal, while the
permeating process will cease for an extreme GMGHS hole.

With the above observations, we have every reason to conjecture that an
extreme GMGHS black hole permits real bound state of Dirac particles. To
investigate the existence of real bound states of massive Dirac field, we
solve the Dirac equation in extreme GMGHS spacetime, in which $F=G=1-r_{+}/r$,
and $\mathscr{C}_{\pm}=(\lambda\pm i\mu r\sqrt{F})/r$, see Eq.(12). First, we
eliminate $R_{-}$ in Eq.(11) to obtain the second order equation of $R_{+}$,
and introduce a new dimensionless coordinate

$x\equiv\frac{\sqrt{(r-r_{+})\mu}}{\sqrt{(r-r_{+})\mu}+1}\in\left[0,1\right].$
(15)

The black hole horizon is reached when $x=0$, whereas the infinity is denoted
by $x=1$. Then, the second order equation of $R_{+}$ can be written as

$\left(\frac{d^{2}}{dx^{2}}+\mathcal{C}_{1}\frac{d}{dx}+\mathcal{C}_{2}\right)R_{+}=0,$
(16)

where the form of $\mathcal{C}_{i}$ are too cumbersome to be presented here.
One can also eliminate $R_{+}$ to obtain the second order equation of $R_{-}$,
but the final result remains unchanged.

To obtain the proper boundary conditions, we first investigate the asymptotic
behavior of the Dirac equation in the background of an extreme hole. At
infinity, the asymptotic solution of Eq.(16) is given by Eq.(14). However, the
ingoing waves at the horizon (13) are no longer asymptotic solutions of the
radial equation. Instead, the exact regular solutions (with
$M\omega\leq|\lambda|/2$) at this boundary are now given by

$R_{+}\sim x^{2\alpha},$ (17)

where $\alpha=\sqrt{\lambda^{2}-4M^{2}\omega^{2}}$. We seek the solutions of
Eq.(16) obeying boundary conditions (14) and (17). This is an eigenvalue
problem of $\omega$. We apply the spectral method together with the Newton-
Raphson method Dias et al. (2016). This enables us to determine the eigenvalue
of $\omega$ and the corresponding radial function simultaneously. After fixing
$\lambda$, we find a series of solutions by sweeping the $\mu$ space. The
numerical results are presented in Figs. 3-6.

Figure 3: Frequency of the ground states as functions of $\mu M$, for
different values of $\lambda$. Figure 4: Binding energy of the node-less
($n=0$) states for different values of $\lambda$. The solid curves are given
by Eq.(18).

In Fig.3, we plot the frequency of the ground states for different values of
$\lambda$. As expected, these frequencies are purely real, and they are
exactly the extreme limit of the quasibound state frequencies in the non-
extreme case. Thus, the solutions we have found are real bound states, as they
will not grow or decay with time. Thus the particles in bound state develop a
stationary cloud over the black hole horizon.

More interestingly, when $\mu M\ll 1$, the real bound state we have found has
a hydrogen-like spectrum Ternov et al. (1980); Huang et al. (2017b)

$\omega\approx\mu\left(1-\frac{\mu^{2}M^{2}}{2\tilde{n}^{2}}\right),\;\;\;\tilde{n}=n+\ell+1,$
(18)

where $n=0,1,2,\cdots$, is the node number of the radial functions, whereas
$\ell=|\lambda+1/2|-1/2=0,1,2\cdots$, is the orbital angular momentum. The
energy level is determined by the principle quantum number
$\tilde{n}=1,2,3,\cdots$. The consistency between Eq.(18) and numerical
results is presented in Fig. 4.

Except for ground state, we further explore the excited bound state. Figure 5
shows the bound state frequencies of the first two excited states. A common
feature of Figs. 3 and 5 is that some of the curves have endpoints on the
right side, which implies that no bound states exist when $\mu M$ is greater
than some critical value. In fact, we find that approaching to the critical
value of $\mu M$, parameter $\alpha$ defined in Eq.(17) tends to zero, as
such, the regularity of the radial function at the horizon becomes delicate.

Figure 5: Comparison of the bound state frequencies of the first two excited
states, with $\lambda=-1$. The two points mark the states for $\mu M=0.4$,
whose radial functions are presented in Fig. 6.

In Fig. 6, we show the radial functions of the bound states marked in Fig. 5.
Although the eigenvalue of $\omega$ is purely real, the corresponding radial
functions are in general complex. As expected, we see that all the radial
functions, both the real and imaginary parts, are smooth and finite in the
whole range of $x$. Thus, we conclude that the extreme GMGHS can support
stationary Dirac clouds.

Figure 6: Radial profiles of the bound states marked in Fig. 5.

We present a detailed analysis of the mechanism for this Dirac cloud. We have
found that extremal GMGHS black hole can also support stationary scalar clouds
in its exterior spacetime Huang and Zhang (2021). The mechanism for Dirac
clouds in GMGHS black hole spacetime is similar to the scalar field. First, we
illustrate the mechanism for the scalar clouds. After separation of variables,
the radial function of the scalar field is given by

$\frac{d^{2}\psi}{dr^{2}_{*}}+\left[\omega^{2}-V_{l}(r)\right]\psi=0,$ (19)

with $r_{*}=\int dr/F$ the tortoise coordinate, and the effective potential
given by

$V_{l}(r)=\frac{F(r)}{G(r)}\left[\frac{F^{\prime}(r)}{r}+\frac{l(l+1)}{r^{2}}+\mu^{2}G(r)\right]-\frac{2M^{2}q^{2}}{r^{4}}\frac{F(r)}{G(r)^{2}}\left[1+\frac{q^{2}}{2}\left(1-\frac{6M}{r}\right)\right].$
(20)

We plot the potential as functions of $r_{*}$ in Fig. 7. In the left panel, we
see that the width of the potential barrier increases monotonously with the
increase of $\eta$. In the extreme limit $\eta\rightarrow\infty$, the width of
the potential barrier increases without bound, whereas the height tends to
$\omega^{2}_{0}\equiv(2l+1)^{2}/16M^{2}$. In this case, the potential (20) can
be written as

$V_{l}(r)=\frac{l(l+1)}{r^{2}}+\frac{M}{r^{3}}\left(2-\frac{3M}{r}\right)+F(r)\mu^{2}.$
(21)

One may check that $V_{l}(r\rightarrow 2M)=\omega^{2}_{0}$. This implies that
a low frequency mode with $\omega<\omega_{0}$ coming from infinity will be
totally reflected back to infinity by the potential barrier. If the scalar
field has a nonzero mass ($\mu\neq 0$), a potential well appears at
$r_{*}/M\approx 0$, as is shown in the right panel of the of fig. 7. Since
both sides of the potential well is infinitely wide, real bound states with
$\omega_{I}=0$ does exist. Clearly, the bound state frequencies are purely
real, and must obey $\omega<\text{min}\left\\{\omega_{0},\mu\right\\}$.

Figure 7: Effective potential (with $l=1$) of the radial equation of massive
scalar field in GMGHS spacetime. In the left panel, we fix the mass $\mu=0$,
and compare the potential for different values of $\eta$. The dotted curves
correspond to $\eta=0,1,2,\cdots,14$, and the solid curves correspond to
$\eta=15,20,25,30$. In the right panel, we show the effective potential for
$\eta=\infty$, and the mass of the scalar field is $\mu M=0.7$.

In obtaining the bound state solutions, we choose boundary conditions as
follows. At infinity, the potential tends to $\mu^{2}$, and the most general
solution of the radial equation (19) is given by

$\psi=C_{1}e^{-\sqrt{\mu^{2}-\omega^{2}}r_{*}}+C_{2}e^{\sqrt{\mu^{2}-\omega^{2}}r_{*}}.$
(22)

For bound states, we have $C_{2}=0$. At the horizon, $V_{l}(r\rightarrow
2M)=\omega^{2}_{0}$, and the most general solution is

$\psi=B_{1}e^{-\sqrt{\omega_{0}^{2}-\omega^{2}}r_{*}}+B_{2}e^{\sqrt{\omega_{0}^{2}-\omega^{2}}r_{*}}.$
(23)

Close to the horizon $r_{*}\rightarrow-\infty$, we have $B_{1}=0$ for bound
states. Thus, the near horizon behavior of the radial function is

$\psi(r\rightarrow 2M)\sim
e^{\sqrt{\omega_{0}^{2}-\omega^{2}}r_{*}}\sim(r-2M)^{\beta},$ (24)

where

$\beta=2M\sqrt{\omega_{0}^{2}-\omega^{2}}=\sqrt{\left(l+\frac{1}{2}\right)^{2}-4M^{2}\omega^{2}}.$
(25)

Clearly, for bound states with $\omega<\omega_{0}$, $\beta$ is purely real,
and the radial function is vanishing at the horizon. Note that Eq.(24) does
not represent a purely ingoing wave at the horizon for $\omega<\omega_{0}$. As
discussed above, when the GMGHS black hole is extremal, the potential does not
vanish at the horizon, and the ingoing wave is no longer an asymptotic
solution of the radial equation. Imposing vanishing boundary conditions on
both sides, we find the eigenfrequencies of the scalar cloud

$\omega=\mu\left[1-\frac{\mu^{2}M^{2}}{2\tilde{n}^{2}}+\mu^{4}M^{4}\left(\frac{15}{8\tilde{n}^{4}}-\frac{2}{\tilde{n}^{3}L}\right)+\mathcal{O}\left(\mu^{6}M^{6}\right)\right],$
(26)

where $L=l+1/2$, and $\tilde{n}=n+l+1$. The frequency is purely real, and is
exactly the extreme limit of the quasibound state frequency in the non-extreme
case Huang and Zhang (2021).

In summary, the existence of scalar clouds around the extremal GMGHS black
hole is resulting from a non vanishing potential at the horizon. Such
solutions does not exist in an extremal RN black hole spacetime, since the
effective potential is always vanishing in the RN case.

Let us now discuss the mechanism for Dirac clouds. We first eliminate $R_{-}$
to obtain the second order equation of $R_{+}$. Then, by introducing a new
radial function $H_{+}=\mathscr{C}^{1/2}_{-}R_{+}$, we rewrite the second
order equation in the Schrödinger-like form

$\frac{d^{2}}{dr^{2}_{*}}H_{+}+\left(\omega^{2}-V_{+}\right)H_{+}=0,$ (27)

where

$V_{+}=\mathscr{C}_{+}\mathscr{C}_{-}+\frac{3}{4}\left(\frac{\mathscr{C}^{\prime}_{-}}{\mathscr{C}_{-}}\right)^{2}-\frac{\mathscr{C}^{\prime\prime}_{-}-2i\omega\mathscr{C}^{\prime}_{-}}{2\mathscr{C}_{-}},$
(28)

and $\mathscr{C}_{\pm}$ are defined in Eq.(12). Here, prime denotes the
derivative with respect to $r_{*}$. Note that the effective potential of the
radial equation is not real. For this reason, we cannot exhibit the potential
well as before. However, the potential behaves similarly at at the horizon,
i.e., when the GMGHS becomes extremal, the potential does not vanish at
$r=r_{+}$. We find that $V_{+}(r\rightarrow
r_{+})=\lambda^{2}/4M^{2}=(2j+1)^{2}/16M^{2}$, for $\eta=\infty$ ($q=1$),
where $j$ is the total angular momentum. Therefore, the most general solution
of the radial equation (27) at $r=r_{+}$ can be written as

$H_{+}=B_{1}e^{-\alpha r_{*}}+B_{2}e^{\alpha r_{*}},$ (29)

where

$\alpha=\sqrt{\lambda^{2}-4M^{2}\omega^{2}}=\sqrt{\left(j+\frac{1}{2}\right)^{2}-4M^{2}\omega^{2}}.$
(30)

One can see that this is very similar to Eq.(25) for scalar clouds. Again, we
have $B_{1}=0$ for bound states. Thus, the radial function of the bound state
obeys the boundary condition

$R_{+}\sim e^{\alpha r_{*}}\sim(r-2M)^{\alpha}\sim x^{2\alpha},$ (31)

at the horizon [See Eq.(17)]. Although the mechanism for Dirac clouds is not
as obvious as that of the scalar clouds, it is a natural extension of the
latter one.

Now we make a concise summary of this Letter. To find a cloud is a key step to
find the hair of a black hole. Generally, the existence of cloud depends on
superradiance mechanism, while for Dirac particles such a mechanism does not
work. We demonstrate that Dirac cloud exists over an extreme GMGHS black hole.
We first investigate the quasi bound states of Dirac particles around a GMGHS
black hole. Then we show that the quasi bound states will be real bound ones
when the hole becomes extremal. We then directly solve the Dirac equation in
the background of an exactly extreme GMGHS hole, and find bound states of the
Dirac particles. These bound states are stationary, and in particular, they
are exactly the extreme limit of the quasibound states in non-extreme
backgrounds.

The other interesting point related to our results is the proposition that
extreme dilatonic black hole behaves like an elementary particle. The entropy
of extreme GMGHS black hole is zero, which implies that it may has no internal
freedoms, or its internal freedom can never detected by an observer outside
the horizon. This property leads to a conjecture such a hole may behave follow
an elementary particle. Our results demonstrate that an external Dirac cloud
around a GMGHS black hole never deacy, which presents intense evidence that
extreme dilatonic black holes behave as elementary particles Holzhey and
Wilczek (1992).

###### Acknowledgements.

This work is supported by the National Key Research and Development Program of
China (No. 2020YFC2201400), as well as Shandong Province Natural Science
Foundation under grant No. ZR201709220395.

## References

  * Bekenstein (1972) J. D. Bekenstein, Phys. Rev. Lett. 28, 452 (1972), URL https://link.aps.org/doi/10.1103/PhysRevLett.28.452.
  * Herdeiro and Radu (2014) C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. 112, 221101 (2014), URL https://link.aps.org/doi/10.1103/PhysRevLett.112.221101.
  * Hong et al. (2020) J.-P. Hong, M. Suzuki, and M. Yamada, Phys. Rev. Lett. 125, 111104 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.111104.
  * Huang and Liu (2016) Y. Huang and D.-J. Liu, Phys. Rev. D 94, 064030 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.94.064030.
  * Huang et al. (2017a) Y. Huang, D.-J. Liu, X.-H. Zhai, and X.-Z. Li, Classical and Quantum Gravity 34, 155002 (2017a), URL https://doi.org/10.1088/1361-6382/aa7964.
  * Hod (2012) S. Hod, Phys. Rev. D 86, 104026 (2012), URL https://link.aps.org/doi/10.1103/PhysRevD.86.104026.
  * Bernard (2016) C. Bernard, Phys. Rev. D 94, 085007 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.94.085007.
  * Gibbons and ichi Maeda (1988) G. Gibbons and K. ichi Maeda, Nuclear Physics B 298, 741 (1988), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321388900065.
  * Garfinkle et al. (1991) D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D 43, 3140 (1991), URL https://link.aps.org/doi/10.1103/PhysRevD.43.3140.
  * Brill and Wheeler (1957) D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. 29, 465 (1957), URL https://link.aps.org/doi/10.1103/RevModPhys.29.465.
  * Lasenby et al. (2005) A. Lasenby, C. Doran, J. Pritchard, A. Caceres, and S. Dolan, Phys. Rev. D 72, 105014 (2005), URL https://link.aps.org/doi/10.1103/PhysRevD.72.105014.
  * Huang and Zhang (2021) Y. Huang and H. Zhang, Phys. Rev. D 103, 044062 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.103.044062.
  * Dias et al. (2016) O. J. C. Dias, J. E. Santos, and B. Way, Class. Quant. Grav. 33, 133001 (2016), eprint 1510.02804.
  * Ternov et al. (1980) I. M. Ternov, A. B. Gaina, and G. A. Chizhov, Sov. Phys. J. 23, 695 (1980).
  * Huang et al. (2017b) Y. Huang, D.-J. Liu, X.-h. Zhai, and X.-z. Li, Phys. Rev. D 96, 065002 (2017b), URL https://link.aps.org/doi/10.1103/PhysRevD.96.065002.
  * Holzhey and Wilczek (1992) C. F. Holzhey and F. Wilczek, Nuclear Physics B 380, 447 (1992), ISSN 0550-3213, URL https://www.sciencedirect.com/science/article/pii/0550321392902549.