# Followback Clusters, Satellite Audiences, and Bridge Nodes:
Coengagement Networks for the 2020 US Election

Andrew Beers1, Joseph S. Schafer1, Ian Kennedy2, Morgan Wack3,
Emma S. Spiro4, Kate Starbird1

###### Abstract

The 2020 United States (US) presidential election was — and has continued to
be — the focus of pervasive and persistent mis- and disinformation spreading
through our media ecosystems, including social media. This event has driven
the collection and analysis of large, directed social network datasets, but
such datasets can resist intuitive understanding. In such large datasets, the
overwhelming number of nodes and edges present in typical representations
create visual artifacts, such as densely overlapping edges and tightly-packed
formations of low-degree nodes, which obscure many features of more practical
interest. We apply a method, coengagement transformations, to convert such
networks of social data into tractable images. Intuitively, this approach
allows for parameterized network visualizations that make shared audiences of
engaged viewers salient to viewers. Using the interpretative capabilities of
this method, we perform an extensive case study of the 2020 United States
presidential election on Twitter, contributing an empirical analysis of
coengagement. By creating and contrasting different networks at different
parameter sets, we define and characterize several structures in this
discourse network, including bridging accounts, satellite audiences, and
followback communities. We discuss the importance and implications of these
empirical network features in this context. In addition, we release open-
source code for creating coengagement networks from Twitter and other
structured interaction data.

## Introduction

The 2020 United States (US) presidential election was — and has continued to
be — the focus of pervasive and persistent mis- and disinformation spreading
through our media ecosystems, including social media (Center for an Informed
Public et al. 2021; Benkler et al. 2020). Efforts to understand these dynamics
have driven the collection and curation of large social media datasets, and
the subsequent production of large, directed network representations of social
interactions to make sense of them (Abilov et al. 2021; Kennedy et al. 2022).
But such network datasets can resist intuitive understanding. In large network
datasets, the overwhelming number of nodes and edges present in typical
representations create visual artifacts, such as densely overlapping edges and
tightly-packed formations of low-degree nodes, which obscure many features of
more practical interest (Nocaj, Ortmann, and Brandes 2015; Schulz and Hurter
2013) (Figure 1). In the case of the US presidential election, one feature of
particular interest is the functional level of interaction between different
political communities who, due partly to pervasive misinformation spread in
this country’s right-wing media ecosystems, no longer share a common
understanding of the election’s outcome (Pennycook and Rand 2021; Reuters
2021). Critical to understanding these inter-community interactions is
characterizing the role of platform elites, who are responsible for a
disproportionate share of election-related misinformation (Center for an
Informed Public et al. 2021).

Here, we present an extensive case study on a dataset of English-language
Twitter posts relating to the 2020 US presidential election. We take advantage
of the interpretative capabilities of coengagement networks, which are similar
to the co-citation networks widely used in bibliographic scholarship. This
dataset, totaling 585M retweets collected from September 1st, 2020 to December
18, 2020, contains tweets referencing generic English-language terms related
to voting and the election, with a focus on tweets relating to election
misinformation. In practice, this dataset contains public discourse related
not only to the presidential election, but also discourse related to the
persistent and false claims that the results of the election were fraudulent.
We create and interrogate three different coengagement networks of retweets
filtered under different parameter sets, describing via a mixed-methods
analysis how the salient features of these networks correspond to different
discourse phenomena. These phenomena include bridge nodes, users that are
retweeted by multiple and disparate audiences; satellite audiences, groups of
detached users which connect to mainstream conversations in very specific
ways; and followback clusters, unique and highly active groups of users that
incessantly retweet each other and very specific mainstream accounts. Our
analysis of followback clusters particularly shows how Twitter’s much-noted
mass account removals in the wake of the 2021 attack on the US Capitol
Building particularly affected these followback groups.

Our empirical and methodological contributions together are themselves a case
study in the proposed triangulation analysis method, where the intersection of
understandings from multiple, sometimes contradictory networks generates
greater knowledge than any one network alone (Brandes 1999; Doreian 1988). We
conclude this paper by discussing the advantages of coengagement networks over
other social network formats, the importance of triangulation as a method for
analysis of social networks, and future extensions and ethical considerations
for using such a method.

## Background

### Mis/Disinformation, Platform Elites, and US Presidential Elections

Researchers have demonstrated the critical value of networks and network
visualizations in efforts to identify key actors in political mis- and
disinformation campaigns (Starbird, Arif, and Wilson 2019; Starbird 2017).
Even within work in this domain, however, the terms mis- and disinformation
themselves have been variously defined (Jack 2017), sometimes eschewed in
favor of the broader term “influence operations” (Wanless and Pamment 2019),
and sometimes even criticized as a contemporary moral panic (Mejia,
Beckermann, and Sullivan 2018; Carlson 2020). For this paper, we simply define
mis- and disinformation of interest as the unintentional and intentional
spread of false or misleading claims that the results of the 2020 US
presidential election were fraudulent. While previous research on
disinformation in the 2016 US presidential election focused on foreign
interference (Lukito 2020), recent analyses of disinformation in the 2020 US
presidential election have focused on domestic right-wing campaigns
coordinated by elites on social media and beyond (Center for an Informed
Public et al. 2021; Benkler et al. 2020). Recent research has shown how
platform elites vary between different political groupings in the US, and have
highlighted their role in spreading misinformation during the COVID-19
pandemic (Gallagher et al. 2021).

### Challenges in Visualizing Social Networks

Network visualizations of large social data can provide valuable insight into
the structure of online conversations, and these visualizations have become
increasingly popular as representations of computational social science’s
promise (Foucault Welles and Meirelles 2015). A common goal in social network
visualization is to highlight influential nodes and characterize the
relationships they hold with one another (Arif, Stewart, and Starbird 2018;
Stewart and Spiro 2021; Freelon, McIlwain, and Clark 2016). The simplest
approach is to visualize the network in its observable entirety, with nodes
representing user accounts and edges representing interactions between
accounts. However, the large size of social media datasets, now often
numbering in the millions or billions of nodes and edges so defined, can be
intractable to visualize and render the exercise of doing so meaningless.
Large social networks often encode multiple and seemingly contradictory
dynamics at different scales, further exacerbating the difficulty in
faithfully representing these phenomena to scholarly peers and the lay public
(Jacomy 2021). These representational ambiguities can become especially
misleading in the case of social data, where the lay-public often has strong
priors about what to expect from the social world (Foucault Welles and
Meirelles 2015).

Figure 1: Directed retweet networks with typical node layout algorithms. Here,
each node is an account, an edge represents a directed retweet, and edge
weights represent the number of retweets. Edges with fewer than 50 retweets
have been filtered out to aid visualization. The following layout algorithms
as implemented in the software package Gephi are used from left to right:
ForceAtlas2, YifanHu, OpenOrd. The bottom row contains close-ups of the same
figures in the top row, highlighting dense node formations which obscure
cluster interpretability. Dataset derived from Twitter data on the 2020
presidential election (size = 142K nodes, 424K edges).

A common solution to this problem of large graphs is to heuristically filter
unimportant nodes (e.g. with low node degree) or edges (e.g. with low edge
weight) until the visualization reaches a tractable size (Ham and Wattenberg
2008; Dianati 2016). Identifying those unimportant nodes and edges is a
significant challenge, as the concept of importance is highly contingent on
the interpretative aims of the researcher, and individually unimportant nodes
may yet in the aggregate encode relevant structural information. Furthermore,
there may be no single definition of node importance which addresses the full
spectrum of phenomena represented by a social network. One innovation of
particular relevance is the co-citation network developed in bibliographic
network science, in which two published articles or authors are connected in a
new work if other articles cite both of them together (Small 1973). Importance
filtering based on frequency of interaction in co-citation networks is
frequently implemented, and its effects on apparent resulting clusters have
been analyzed in (Shaw Jr 1985). Co-citation networks have typically focused
on scientific literature, although others have used similar principles in
relation to the hyperlink structure of the web, most notably with Kleinberg’s
concept of online hubs and authorities in search engine retrieval (Kleinberg
1999).

Here, we extend the concept of co-citation networks to social interaction data
in what we call the coengagement network. Like co-citation networks, this
method significantly reduces the number of nodes visualized in large datasets,
while encoding information from missing nodes in visible edges that can reveal
significant relationships. Additionally, this method is also tunable, meaning
that researchers can produce different visualizations according to different
notions of node importance as defined by two interpretable parameters. A
primary contribution is the application of co-citation principles to social
data representing users, rather than documents, interacting with one another
online.

## Coengagement Networks

Coengagement networks are closely related to the method of projection in
bipartite graphs. In a typical bipartite projection, a network with two types
of nodes and no within-group connections (such as a network composed of
researchers and the papers that they author (Newman 2001)) is projected into
one primary node type, with the remaining node type being collapsed into edge
representations. This operation can productively reduce the number of nodes
and edges to be visualized and analyzed, and preserves information about a
primary node type while still retaining structural information from the
projected node type. While we would not expect engagement networks among
social media users to be bipartite, we aim to take advantage of the visual and
analytical properties of bipartite projections, and therefore propose a
transformation of non-bipartite graphs into bipartite graphs via the
duplication of each node into two types: engaging and receiving nodes. We then
project the resulting bipartite network such that engaging nodes are collapsed
into edges between receiving nodes. In the context of users on Twitter, for
example, this method privileges users with large audiences engaging in
retweeting, commenting, liking, or even viewing, and defines relationships
between users in terms of sharing similarly engaged audiences.

Figure 2: Schematic representation of coengagement visualizations. Panel 1) of
this figure shows a schematic for transforming A) a unipartite directed graph
into B) a bipartite directed graph, and then into C) a coengagement network.
The blue node is linked to the yellow node because of their shared engaging
node in red, and the blue node is linked to the green node because of their
shared engaging node in yellow. Panel 2) shows the effect of the node
filtering parameters on an example graph. Engaging nodes, colored grey, are
sized by their average out-degree. Under different combinations of the
filtering parameters n and s, different edges will result.

Formally, we define a directed graph G = (V, E, w) with vertices V, edges E,
and edge weights w. In the current case study, we interpret G as a collection
of Twitter users (V) retweeting other users (E), with edge weights w defined
as the total number of retweets from one user to another. We then define a new
graph G′ with vertices V′ , which contains duplicate sets of vertices VS and
VR that send and receive retweets, respectively. We similarly define a new set
of undirected, weighted edges E′ in G′, where directed edges from Vi to Vj are
represented as undirected edges from VSi to VRj, with the same weights w′. In
effect, each original user has a vertex representing the instances in which
they retweet others, and a separate vertex representing when they are
retweeted. We then define a projection of G′ as X, such that vertices in X are
the interaction-receiving vertices VR, and edges in X are defined such that
the edge weight between any two vertices i, j in X is the number of vertices
in VS that have defined edges to both VRi and VRj. This final vertex set in
the projected graph represents users that are retweeted and draws edges
between them when they are jointly retweeted by at least one other user. We
finally define two edge filtering parameters n and s on the resulting graph X.
Specifically, an edge between two users in X is defined if at least n other
users have retweeted both users at least s times each. The parameter n
represents a minimum diversity of users retweeting two users, while s
represents the minimum volume of retweeting a user must do to be considered in
n.

The n and s parameters, which in practice control the number and distribution
of edges in a coengagement network, are a powerful tool for targeting specific
visualizations. These edge filtering parameters allow researchers to shape the
output of their networks along two important, yet distinct, qualitative
dimensions by modifying the distribution of edges between users. When
researchers filter with a higher n value, influential users are related only
if they attract engagement from large, diverse audiences, a typical goal in
influential user analysis. When researchers filter with a higher s value,
nodes are instead related by audiences that frequently retweet their content
with a dataset, which can reveal dedicated rather than transient audiences.
Different ratios of n to s can reveal other relationships: low n with high s
can make highly active and coordinated audiences more salient, whereas low s
with high n make more salient those infrequent instances in which content is
shared widely across different communities.

In what follows, we present a series of case studies using a dataset of tweets
related to the 2020 US presidential election. Specifically, we use a dataset
of 585M Twitter posts containing substrings related to voting and the election
(‘vote’, ‘voting’, ‘mail’, ‘ballot’, ‘poll’, and ‘election.’), define nodes as
the authors of these posts, and (unprojected) edges as retweets from one user
to another. We do not include quote tweets. We show that depending on the
choice of the parameters n and s, different clusters of influential nodes can
be distinguished, and different forms of qualitative analysis can be applied.
In doing so, we demonstrate the practical implications of coengagment networks
and their interpretation. Empirically, these case studies offer a unique look
at engagement during the 2020 presidential election.

In the first case study, we choose a very high value of n and s = 1 to create
a network that shows a broadly two-part structure to Twitter discussions
around the presidential election, aligned with pro-Trump and pro-Biden
accounts. The low s parameter highlights transient instances of high-volume
crossover between these two groups but does not necessarily represent
sustained engagement across these groups. We then choose a parameter set with
much lower n and slightly higher s, to illustrate how a third pro-socialist
grouping becomes salient at different audience sizes, and how some crossover
nodes do not hold sustained engagement. We end with a third case study at very
high s and very low n, to highlight two new followback communities that become
salient when active, sustained engagement is prioritized over large audiences.

## Findings

### Case 1: Bridging Between Clusters at High Audience Sizes

Figure 3: Case 1, election discourse for large audiences (s=1, n = 10,000). A
coengagement visualization of retweet relationships in a collection of tweets
related to the US presidential election. Each node represents a Twitter
account, and two nodes are linked if at least 10,000 users retweeted them both
nodes at least once. Edges are undirected and weighted according to how many
users retweeted both nodes. Nodes are sized according to their weighted
degree, i.e. the sum of the weights of their incoming edges. Highlighted nodes
represent nodes with connections to both pro-Trump and pro-Biden clusters.
Figure 4: Fig. 4. Example accounts with bipartisan engagement in Case 1. Plots
of partisan retweets of four different bridging accounts labeled in Figure 3.
Tweet dates are marked by the first time they were retweeted in our dataset.
“Pro-Trump Only” partisan retweets indicate users who only tweeted accounts
labeled in the pro-Trump cluster in Figure 3. “Pro-Biden Only” signifies the
same totals for the pro-Biden cluster. Retweets of bridging nodes themselves
are excluded in both retweet totals.

In Case 1, we generate a coengagement network where node relationships are
defined by low restrictions on retweet frequency (s = 1), but high
restrictions on total retweet volume (n = 10,000, Figure 3). In this and
future visualizations, we use the ForceAtlas2 (Jacomy et al. 2014)
visualization algorithm as implemented in the application Gephi (Bastian,
Heymann, and Jacomy 2009). Intuitively, this means that two nodes are
connected if at least 10,000 users retweeted both at least once during the
period of this dataset, and nodes with more users retweeting both of them are
more tightly linked together. Visualizing this projection reveals two tightly
interconnected communities of influential users discussing the US presidential
election, which we term and briefly describe as pro-Trump, and pro-Biden
clusters. The pro-Trump cluster is anchored around Donald Trump’s account, and
includes an array of pro-Trump political activists, political organizations,
politicians, media outlets, journalists, anonymous and self-identified online
influencers, activists from the antifeminist “manosphere” online culture, and
conspiracy-based QAnon communities. The pro-Biden cluster includes an array of
politicians, journalists, media outlets, and online influencers, some of which
self-identify as pro-Biden or liberal, and others of which, such as the
television network CNN, identify as non-partisan but rejected false pro-Trump
claims of election fraud.

Overall, 2,499 nodes are generated in this graph, with 1,385 nodes in the pro-
Trump cluster (55%) and 1,114 nodes in the pro-Biden cluster (45%). However,
this set of retweets may be biased towards pro-Trump accounts due to our focus
on terms related to election misinformation disproportionately spread by pro-
Trump accounts.

We use the “pro-” descriptor to describe clusters as a whole, but some
individual members of these clusters may identify themselves otherwise. We
also use the terms pro-Trump and pro-Biden, rather than Republican and
Democrat, to illustrate the extent to which traditional US party alignments
are contradicted in the membership of these clusters. The Lincoln Project, a
Republican advocacy group that supports conservative causes, is one of the
most prominent accounts in the pro-Biden cluster, while many accounts in the
pro-Trump cluster have ambivalent stances towards the Republican party outside
of Trump. Their membership in a community is contingent not on their core
beliefs or associations, but on their behavior in the dataset we observe,
namely their posting behavior in the months before and after the presidential
election. In datasets with the same users but different topics of discussion,
the membership of individual users may change.

Audiences rarely retweet across clusters in large numbers, and when they do,
they tend to retweet a select few cluster-spanning nodes. Almost all (98%)
cross-cluster connections route through only eight nodes, labelled in Figure
3. These bridging nodes serve different functions in this discourse
environment, and connect people in different ways (Figure 4). The most common
form of bridging node was one where the account generally created two types of
tweets, one that appealed to pro-Trump audiences, and one that appealed to
pro-Biden audiences. The most transparent examples of these accounts were
those that tracked polling results (@PpollingResults, @Politics_Polls,
@AP_Politics, @NBCNews, @APPolitics), where those results and polls that
favored Biden were retweeted by pro-Biden accounts, and those that favored
Trump were retweeted by pro-Trump accounts. However, this form of apparent
bridging also occurred when media accounts reported in neutral tones on events
that fed preexisting pro-Trump and pro-Biden narratives respectively. The
journalist account @joshgerstein was separately retweeted by both clusters for
neutrally reporting on Trump’s attempts to contest the election results, with
apparent pro-Trump legal judgments being more retweeted by pro-Trump users and
their subsequent legal refutations more retweeted by pro-Biden users. The
alternating quality of these nodes complicates the notion that they bridge
communities, as their tweets are most often disproportionately shown to only
one community at a time.

A special form of this alternating bridging occurred with the account
@FrankLuntz, which posted updates on polls and predictions about the outcome
of the presidential election. Before election day, this account was sometimes
critical of Biden and released some predictions favorable to Trump, which led
it to garner a slightly right-leaning cumulative audience. After the election,
this account was resolute in affirming Biden’s victory in the face of false
pro-Trump claims of election fraud, then earning a growing pro-Biden audience.
This account displays the sensitivity of such analyses to dataset selection,
as a pre-election discourse analysis would likely place the account firmly in
the pro-Trump cluster, whereas a post-election discourse in the pro-Biden, and
when combined, firmly between. It was rare when accounts created posts that
consistently appealed to audiences in both clusters. Some individual posts had
equal appeal across clusters, such as when polling accounts released vote
tallies tied at nearly 50% in critical states. Other posts that had a similar
appeal were simply neutral statements of fact about recent news relating to
the presidential election, which were made particularly often by the account
for the online news organization The Hill (@thehill). While pro-Biden leaning,
the Hill’s account was one of the only accounts to consistently find
engagement from both pro-Biden and pro-Trump accounts across this election
time period.

The last significant point of crossover between the pro-Trump and pro-Biden
clusters is the account for former president Trump itself (@realDonaldTrump).
This circumstance reveals that though we imply for much of this analysis that
retweets constitute endorsements, they do not always behave as such. Many pro-
Biden accounts may be retweeting Trump’s account simply because his tweets are
often consequential in and of themselves, even when they are clearly opposed
to Biden’s election to the presidency. In several instances, pro-Biden users
likely retweeted Trump’s tweets sarcastically, such as when pro-Biden users
disproportionately retweeted an old tweet from 2012 reading: “Scary
thought–@JoeBiden is a heartbeat away from the Presidency.”

### Case 2: Third Party and Satellite Audiences at Lower Audience Sizes

Figure 5: Case 2, Election discourse for medium audience sizes (s = 5, n =
100). A coengagement visualization of retweet relationships in a collection of
tweets related to the US presidential election. Each node represents a Twitter
account, and two nodes are linked if at least 100 users retweeted them both
nodes at least five times. Edges are undirected and weighted according to how
many users retweeted both nodes. Nodes are sized according to their weighted
degree, i.e. the sum of the weights of their incoming edges. Nodes bridging
between the pro-socialist and other clusters are labelled, as well as what we
term satellite audiences: clusters of nodes with low-degree representing
audiences with only tangential connection to mainstream US election discourse.

Top Election Accounts by Cross-Cluster Connections (Case 2)
---
Biden-Trump | Biden-socialist | Trump-socialist
Account Name |  | Share of
---
Cross-Cluster
Connections
(n = 2,515)
Account Name |  | Share of
---
Cross-Cluster
Connections
(n = 299)
Account Name |  | Share of
---
Cross-Cluster
Connections
(n = 34)
thehill | 24% | Proudsocialist | 16% | ggreenwald | 79%
PpollingNumbers | 18% | BernieSanders | 13% | jimmy_dore | 9%
Politics_Polls | 11% | davidsirota | 12% | TulsiGabbard | 6%
realDonaldTrump | 7% | AOC | 11% | aaronjmate | 6%
AP_Politics | 6% | briebriejoy | 10% |  |
threadreaderapp | 5% | KyleKulinski | 7% |  |
Garrett_Archer | 5% | IlhanMN | 6% |  |
spectatorindex | 4% | ryangrim | 5% |  |
FrankLuntz | 3% | peterdaou | 5% |  |
DecisionDeskHQ | 3% | RBReich | 3% |  |

Table 1: Bridging connections between pro-Biden, pro-Trump and pro-socialist
clusters. The top 10 Twitter accounts in terms of total cross-cluster edges
for each two-cluster pairing in Case 2. These accounts share audiences of at
least 100 users retweeting at least 5 times each across election discourse
clusters.

In Case 1, a high n parameter demonstrated the nodes and graph structure of
accounts with a relatively high volume of retweets over this dataset. While
revealing the relatively few shared points of reference between pro-Biden and
pro-Trump audiences, groups of users with smaller retweet bases are not
visible in this graph. To visualize these communities, we generate the same
data at a dataset with lower n, instead filtering by s to keep the size of the
node and edge set tractable. Specifically, we generate a network in which
links are defined when two nodes are shared by 100 users four times (n = 100,
s = 5, Figure 5). After removing non-US clusters, this reveals a third group
of nodes that we term the pro-socialist cluster. The pro-socialist cluster,
much smaller than either the pro-Biden or pro-Trump clusters, is centered
around multiple political activists, journalists, and influencers associated
with US democratic socialist candidates and causes. We note that, like the
pro-Biden and pro-Trump cluster, that this cluster does not contain all pro-
socialist accounts, and some of its members may not identify as such. Indeed,
popular democratic socialist presidential candidate Bernie Sanders is located
in the pro-Biden cluster rather than pro-socialist cluster, likely due to his
widespread popularity and continued public support for Biden during this
election.

At a lower volume of retweeting, more bridges appear between the pro-Biden and
pro-Trump clusters, and new bridges are generated between the pro-socialist
and other clusters. Notably, the relative importance of some bridges changes
under the new requirement for repeated engagement (s = 5). For example, the
account for reporter @joshgerstein has no cross-cutting connections in this
graph, despite being responsible for 7% of all connections in the first graph.
This discrepancy is likely caused by how much of this account’s pro-Trump
retweet engagement comes from only two tweets, both, describing in a neutral
tone, updates on pro-Trump attempts to legally invalidate the results of the
election. This change demonstrates the effect of the s parameter, which can be
tuned upward to select nodes for sustained engagement over a dataset, rather
than widespread but momentary engagement in critical posts. The opposite
effect can also be seen in those new nodes that appear as significant bridges.
The spam media account @spectatorindex becomes a bridging node in Case 2, due
to its sustained but low level of engagement, as do circumstantially important
accounts like the governor of Arizona’s (@dougducey), whose state was the
center of many false election fraud claims.

Bridges between the pro-socialist and other clusters illustrate the different
modes of inter-cluster commonality that can exist between different user
clusters in this graph (Table 1). Between the pro-Biden and pro-socialist
clusters, there are many nodes that draw consistent engagement, including
popular democratic socialist politicians Bernie Sanders, Alexandria Ocasio-
Cortez, and Ilhan Omar, and an array of writers, podcast hosts and other
online influencers associated with the socialist movement. Between the pro-
Trump and pro-socialist clusters, however, there is little engagement, and all
of it is mediated through three nodes. Most prominently among these nodes is
Glenn Greenwald, a journalist who combined left-leaning views on topics such
as surveillance with frequent engagement with right-wing media and criticism
of mainstream media.

This second case study introduces a visualization feature which we had
previously aimed to eliminate: tightly-packed clusters of low-degree nodes, in
this case one-degree nodes mostly connected only to @realDonaldTrump. We note
that while in ordinary graphs such nodes are usually uninteresting, in
coengagement visualizations the relative isolation of these nodes reveals an
important function in the election discourse environment. Particularly, many
of these one-degree nodes are from communities plausibly isolated from
mainstream US political discourse, but still displaying, for example, a
contextual support for Trump or Biden. We term edges stemming from these nodes
as satellite audiences, with inspiration from (Squires 2002).

These nodes include accounts from other countries and/or in other languages,
such as high-follower right-wing accounts writing for Japanese or Brazilian
audiences. Such accounts are unlikely to interact with the majority of
English-language right-wing accounts prominent in this graph, but may retweet
Trump as a signal of nominal allegiance to his movement. Other low-degree
nodes may originate in popular English-language, US-based communities that
focus on topics usually unrelated to electoral politics. For example, the low-
degree account for pop musician Ariana Grande, one of the most followed
accounts on Twitter, connects only to Biden and fellow musician Lady Gaga,
signaling a possible separation between entertainment-focused audiences and
mainstream election-focused audiences.

### Case 3: Followback Clusters at High-Frequency Engagement

Implicitly in the previous cases, structure is mostly determined by the number
of users choosing to retweet two different accounts. However, as s increases
and n decreases, structure is increasingly determined by repeated interactions
by relatively small groups of users, which makes the actions of well-
coordinated groups more salient. To illustrate this, we generate a graph where
links are defined by 25 users retweeting two nodes at least 25 times each over
the course of the dataset (n = 25, s = 25, Figure 6).

Figure 6: Case 3, Election discourse for small but active audiences (s = 25, n
= 25). A coengagement visualization of retweet relationships in a collection
of tweets related to the US presidential election. Each node represents a
Twitter account, and two nodes are linked if at least 25 users retweeted them
both nodes at least 25 times. Edges are undirected and weighted according to
how many users retweeted both nodes. Nodes are sized according to their
weighted degree, i.e. the sum of the weights of their incoming edges.
Followback clusters, comprising nodes which retweet and follow other accounts
to a relatively extreme degree, are labeled (Pro-Biden Followback Cluster,
Pro-Trump Followback Cluster, “Trump’s Italians”). Figure 7: Follower to
following ratios for Case 3 clusters. Scatter plots of the number of followers
and the number of accounts followed for each Twitter account visualized in
Figure 6 (n = 25, s = 25). Follower and following totals are counted at the
time of the latest tweet recorded in the dataset. Plots are separated and
colored according to cluster membership, and both axes are on logarithmic
scale.

In this case, three new clusters emerge with ties to the existing pro-Trump
and pro-Biden clusters. We term the new clusters in this graph followback
communities, due to their unique method of gaining followers and using
Twitter. Accounts in these communities attempt to gain followers by mass-
following other accounts in expectation of reciprocal follows, and sometimes
explicitly coordinate with other accounts to expose themselves to a wider
audience of potential followers. Because Twitter limits the number of users an
account can follow by that account’s current follower number, these accounts
can often be distinguished from others by their nearly 1:1 ratio between
followers and following totals (Figure 7). In addition to this follower
manipulation practice, these communities have other unique behaviors compared
to the clusters previously identified. Their median retweet total is much
higher than that of the previously-identified clusters, and retweets are a
much higher percentage of their total tweeting behavior. Their frequent
retweeting likely propels their visibility in this visualization.
Qualitatively, their behavior is also different from other users on the
platform. They engage in retweet “trains,’ in which they make posts tagging
members of their own community and then retweet these posts incessantly in an
attempt to garner more followers for all participants (Gallagher 2020). They
are almost entirely pseudonymous, with screen names and profile information
often detached from any offline presence. One of the followback clusters is
much larger than the others and associated with the core pro-Trump cluster,
one is smaller and an off-shoot of the larger pro-Trump followback cluster,
and the smallest cluster is associated with the core pro-Biden cluster.

As with previous clusters, we can investigate the nodes which bridge one
cluster to another. In this case, however, both followback clusters have no
connections to non-followback clusters not aligned with their preferred
candidate. In all three cases, an important point of cross-cluster connection
are the accounts for the two presidential candidates themselves (@JoeBiden and
@realDonaldTrump), but unlike other groups, most nodes in the followback
clusters are connected to these nodes. This feature of dense cross-cluster
connectedness reflects a critical function of the engagement from followback
clusters: to retweet the followback community, but also to retweet the
influential nodes supporting their presidential candidate of choice.

Multiple followback clusters can exist supporting the same presidential
candidate, and different clusters may share unique user characteristics. By
inspecting the usernames and user-entered profile descriptions of the smaller
pro-Trump followback cluster, we found that 29/41 accounts provided some
indicator of Italian American identity, either explicitly identifying as such,
identifying as part of “Trump’s Italian Army,” or including Italian flag
emojis paired with US flag emojis. While the larger pro-Trump followback also
contained some self-identified Italian users, they were by no means the
majority as in the smaller cluster. This smaller cluster was linked to the
main pro-Trump followback cluster only through four bridging nodes, two of
which identified as part of Trump’s Italian army, and two which self-described
as duplicate accounts for the same user. This cluster otherwise only formed
internal connections and connections with @realDonaldTrump and @Llinwood, the
account of L. Lin Wood, a lawyer who advanced many false conspiracy theories
relating to the election and litigated on Trump’s behalf in some post-election
court cases.

All three followback clusters reflect the curious quality of coengagement
networks where users can play both the role of edge and node. Recall that
coengagement networks are graph projections, where each edge is comprised of
engagements from many unseen nodes present in a ordinary, unprojected version
of the graph. Almost 38% of the unprojected nodes that comprise the edges in
followback clusters are themselves represented as nodes in the coengagement
networks, compared to less than 1% of the accounts in the core pro-Trump, pro-
Biden, and pro-socialist clusters. This is to say that followback clusters
uniquely play the role of their own audience, relentlessly sharing their own
members’ content in addition to their chosen candidates. This feature sheds
light on how the smaller pro-Trump followback cluster can separate itself in
this graph, as its self-identified Italian American accounts specifically and
at the threshold we set retweet only each other. This analysis does not allow
us to speculate as to the level of explicit coordination or the motives behind
these accounts for acting in this way. While some of these clusters’ unusual
behavior could indicate automated or semi-automated activities, previous
reporting has indicated that at least some of these accounts are likely
operated and coordinated by otherwise ordinary users that simply aim to
support Trump’s candidacy on Twitter (Gallagher 2020).

### Synthesis: Cluster Contingency and Differential Moderation

By examining the full n/s parameter space, we can develop a sense for the
relative size and sharing characteristics of the clusters that we have
identified so far. We calculate the results of repeated network clusterings
using the Louvain algorithm on a range of possible n and s parameter
combinations, assigning cluster labels via high-degree landmark nodes
associated with each previously-identified cluster. In Figure 8, we illustrate
the boundaries at which these clusters are no longer salient. The pro-Trump,
pro-Biden, and pro-socialist clusters are all contained within a maximum
number of engaged users that steadily declines with restrictions on these
users’ number of retweets. By contrast, the followback clusters cannot be
detected unless users’ minimum retweets are elevated, but are never salient at
a size of greater than 110 users. We note that the point at which a cluster
fails to be detectable is not the point at which all nodes in these clusters
are removed, but rather the point at which these nodes become subsumed into
larger clusters. For example, the two highest degree pro-socialist nodes from
Case 2 are relatively-marginal nodes in Case 1, while many of the most popular
Trump followback accounts in Case 3 are found in the core pro-Trump cluster in
Cases 1 and 2. These facts reiterate that the clusters we identify here are
not found with respect to the actual ideologies or social ties between nodes,
but rather through an understanding of how nodes are perceived by their
engaged audience variously defined.

Figure 8: Existence map for clusters across filtering values n, s. For each
cluster identified in these three case studies, we determine the maximum
parameter values at which these clusters can be identified. Clusters are
labeled at each (n, s) parameter value if they contain specific high-degree
landmark nodes identified in the previous case studies. In the shaded regions,
individual clusters are salient to the clustering algorithm. Outside of the
shaded regions, these clusters cannot be identified either because their
constituent nodes are not present or because they have been subsumed into
other clusters. Figure 9: Visualization of suspensions across all case
studies, labeled (1-3). The original networks from all three case studies are
visualized in increasing order from left-to-right. Nodes that have been
suspended by Twitter as of January 6, 2021 are colored in black, otherwise
coloring remain the same as in the original case studies.

We conclude these case studies by noting how the clusters we identify here
were subject to different levels of moderation in the wake of the US
presidential election (Figure 9). On January 6, 2021, a large group of
rioters, associated with a range of pro-Trump movements contesting the results
of the 2020 presidential election, entered the US Capitol building while its
representatives were in session. Following this event, Twitter suspended a
large number of accounts said to have encouraged this violent protest, as well
as accounts associated with the QAnon conspiracy movement (Romm and Dwoskin
2021). To measure the effect of these and other more recent suspensions on our
dataset, we identified all accounts that were suspended as of September 3,
2021. Since the election, most of the pro-Trump followback cluster in this
case study (71%) has been suspended by Twitter, compared to 32% of accounts in
the core pro-Trump clusters, 2% of the core pro-Biden cluster, 7% of the pro-
Biden followback cluster, and 2% of the pro-socialist cluster. In other words,
these suspensions have disproportionately affected the pro-Trump followback
users we identified in Case 3, which is to say groups of pro-Trump users with
frequent intracommunity retweet activity.

### Comparison: Directed Engagement Graph and Flows

We conclude our case studies with a brief comparison to a more standard
network form, which we refer to as the directed engagement graph. In directed
graphs, each node is an account, each edge is a directed retweeting
relationship, and edge weights signify the frequency of engagement. In this
present dataset, the unfiltered directed graph consists of 22M nodes and 299M
edges. This graph cannot be visualized effectively with our current software
and computing power, although a version with heavily filtered edge weights can
be observed in Figure 1.

We first assess whether coengagement graphs represent nodes typically
perceived to be important in the directed graph. We compare the weighted
(in)degrees of both graph, which in the case of the directed engagement graph
is simply equal to the number of times an account has been retweeted. We find
that of the top 1000 most retweeted accounts, 95%, 96% and 85% are represented
across Cases 1, 2, and 3 respectively. Furthermore, we find that the nodes we
choose to include in Cases 1, 2, and 3 account for 54%, 64%, and 51% of all
retweets in this dataset, despite only representing fewer than 0.1% of the
accounts in our data. There are some highly-retweeted nodes in the directed
graph that are not visible in each coengagement network. However, these
missing nodes are also mostly unrelated to the 2020 US presidential election.
For example, the most retweeted nodes missing from Cases 1, 2, and 3
respectively are a fan account for pop musician Justin Beiber, a fan account
for the Korean pop music group BTS, and the account for Billboard, a music
media outlet. These nodes are highly retweeted in these datasets because of
competitions in which fans “vote” for their favorite musicians. They may be
missing from our coengagement visualization simply because they are
essentially apart from the apparent main topic of the dataset, and thus have
fewer opportunities to “share” an audience with another node that is collected
under these terms. Some relevant highly-retweeted accounts, such as the
Twitter account for Donald Trump’s daughter Ivanka Trump, are also missing
from Case 3 likely because they did not publish enough election-related tweets
to reach the 25-retweet threshold.

We second assess whether structures found in the directed graph are
significantly different from the coengagement graph. To do this, we cluster
the directed graph using the Infomap algorithm (Rosvall, Axelsson, and
Bergstrom 2009), a different clustering method which views directed, weighted
edge interactions as flows of information between nodes. We find that the pro-
Biden, pro-Trump, socialist, and pro-Trump followback clusters are all still
found under this algorithm, with the pro-Biden followback cluster being
subsumed into the pro-Biden cluster. No other large cluster is found which
combines nodes found in these case studies into new mixtures. We do find,
however, that clusters apparent in these case studies, such as the pro-Biden
cluster, often are subdivided into smaller clusters in the directed network.

These smaller clusters reveal the tradeoffs between directed networks and
coengagement networks, as they most likely stem from Infomap’s tendency to
privilege connections between high-indegree nodes, i.e. retweets between
influential accounts. For example, a pro-Biden subcluster in the directed
graph is centered around official accounts and reporters for the media outlet
The New York Times. These popular accounts frequently retweet each other,
probably to promote each others’ work, which strengthens their association in
graph forms and algorithms which understand high-degree nodes as routes
through which users travel. By contrast, in coengagement graphs, the influence
of interactions between influential nodes on the resulting form and clusters
is dampened, due to its focus on large but not necessarily important shared
audiences. In social engagement data in which user engagements can be seen as
travelling from node to node, such as use clicks through profile pages,
directed retweet networks and clustering algorithms that consider them may be
more appropriate. In data such as this Twitter retweet dataset, where
engagements are enacted from relatively stable positions and circumstances,
coengagement graphs may be more appropriate.

## Discussion

We introduce coengagement networks and illustrate their value for a mixed
methods analysis of a dataset of Twitter posts related to the 2020 US
presidential election. We illustrate how the number of apparent clusters
perceived in these networks is contingent on the minimum proposed size and
activity level of their engaged audiences. When seen through the lens of
large, momentarily interested audiences, there appears to be two dominant pro-
Trump and pro-Biden communities. When including smaller engaged audiences, a
pro-socialist cluster emerges, and when focused on highly-active but even
smaller engaged audiences, unique followback clusters emerge with severely
different user behavior. These networks make clear that Twitter’s moderation
in the wake of the attack on the US Capitol Building disproportionately
affected these pro-Trump followback accounts, while also affecting a number of
pro-Trump accounts with more ordinary retweeting behavior. Taken together, the
insights from these networks depict an ecosystem of popular and activist
discourse communities in the presidential election with few but crucial points
of overlap, and the de facto removal of the majority of influential member is
one of these communities in the wake of the US Capitol attacks.

The purpose of our case study is not necessarily to draw definitive
conclusions or cause-and-effect relationships, but to use visualization to
expose those features of this discourse ecosystem that deserve further study.
We describe a mostly binary structure of pro-Trump and pro-Biden engagement in
English-language US election discourse, and provide a typification of the
points of crossover between these two clusters of accounts. We identify a
detectable and yet marginal tide of third-party US political discourse in the
pro-socialist cluster whose growth and comparative influence on non-election
discourses may yet have further importance. We identify the phenomenon of
satellite audiences, where high-degree nodes, and particularly the account
@realDonaldTrump, serve as singular points of reference for many communities
with only marginal connection to English-language election discourse. And we
characterize followback communities, which use unique posting strategies and
engage in unusually partisan rhetoric to support specific candidates in our
election dataset.

The social network visualization approach described here makes transparent
many features which contribute to this understanding of Twitter discourse
around this election, while reducing visual clutter and artifacts typical to
equivalent large datasets. However, we stress that the sum total narrative
could not be found from any of these visualizations taken alone, and indeed
some visualizations have contradictory features which could, when viewed in
isolation, generate misleading insights as to the nature of these communities.
For example, we have shown that the pro-Trump followback cluster is from the
perspective of the size of their engaged audience a somewhat marginal
phenomenon, ceasing to be organizationally coherent at the level of more than
a hundred users. Portraying them with equal prominence to more typical
political communities may mislead viewers as to their relative impact on the
overall election conversation compared to, for example, the core pro-Trump
cluster. Yet small groups of coordinated users can nonetheless have an outsize
impact in online communities (Center for Countering Digital Hate 2021), and
our analysis of Twitter suspensions since the presidential election show that
this followback community was specifically targeted for moderation at a much
greater rate than other communities. Depending on the visualization goals of
the researcher — proportional impact, or behavioral diversity — forefronting
this community may or may not be informative to their chosen audiences.
Portraying multiple visualizations that provide both interpretations, and
analyzing the discrepancies between them, provides a level of explanatory
power that no single visualization is likely to otherwise achieve.

Overall, what we describe in this paper is an interpretative, mixed-methods
workflow, in which visual artifacts derived from quantitative network
transformations are combined with a deep qualitative understanding of the US
Twitter election context, to the benefit of both. Given the contingency of
network visualizations on their initial parameters, deep understanding of the
context of a given dataset is necessary to interpret and triangulate their
different incarnations. However, given the size of contemporary social
interaction datasets, quantitative methods such as the coengagement network
are necessary to reduce the complexity of online conversations to artifacts
which are tractable to qualitative researchers. We advocate for the continued
use of coengagement networks in mixed-methods research, as a tool that both
stimulates and benefits from deep contextual understanding of increasingly
unwieldy social data.

### Ethical Considerations, Limitations, and Software Sharing

As network visualizations continue to be central in social media analysis, we
believe it is necessary to briefly examine the ethical considerations on
whether network visualizations such as these should be used in every
circumstance. We have chosen here to visualize users participating in a high-
prominence topic, consisting of mostly public-facing accounts such as
politicians and media outlets. The same methods applied to communities with a
higher expectation of privacy, or who face higher risks from exposure, may be
unethical surveillance if researchers have not derived consent from members of
these communities. There are also ethical implications to naming accounts
visualized as nodes in networks. Some users, due to gender, race, or other
factors, are at higher risk of harassment if identified as influential in a
given community, while other users who explicitly seek attention in online
communities may use their identification in networks as a propaganda tool in
hateful campaigns. In our reporting of this work, we have declined to name
some accounts for both reasons. We also stress here how the data collection
procedure, and subsequent description of that procedure, affects which
communities appear to be participating in a phenomenon. Several politically-
active Twitter communities in the US that have been previously described in
research, such as Black Twitter (Clark 2019) and non-English language
communities (Fang 2021; Soto-Vásquez et al. 2020), are not explicitly visible
in our analysis, likely due to their different posting volumes and the choice
of terms and topics on which we chose to center our data collection.
Researchers working with such visualizations whose research bears on policy
and public perception must explain such limitations in the communication of
their work.

We have described the basic form of an approach for visualizing engagements in
social network data, and there are many ways in which this method can be
modified to more saliently capture engagement dynamics. For example, the
current formulation of this method places emphasis on users who frequently
share content, which is not necessarily undesirable given the external impact
of this behavior. However, different formulations of the network projection
scheme, such as those that weight users’ engagements relative to their average
level of engagement, may be a better reflection of real discourse communities
that exist at lower sharing volumes. We also observe that many of these
coengagement networks create densely-connected subgraphs in which most nodes
are connected to most other nodes, making internode relationships difficult to
visually identify. Accordingly, these networks may be complementary with other
techniques to improve visualizations of social networks, such as the edge
sparsification procedures for densely-connected networks proposed by Nocaj et
al. (2015).

In the hopes that others may replicate our methods of both visualization and
analysis on new datasets, we make the code available for generating these
graphs either from structured data received from the Twitter API, or in
general JSON and CSV-based formats.111https://github.com/uwcip-
research/Coengagement-Networks This code uses the visualization capabilities
of the open-source network visualization library Gephi, and its implementation
of the ForceAtlas2 algorithm for network visualization (Bastian, Heymann, and
Jacomy 2009; Jacomy et al. 2014). We have packaged this code in publicly-
available Docker containers, a relatively portable and stable code format
which can be run on many machines with relatively few installation
requirements. Additionally, we have made available node and link data for all
visualizations displayed in this paper, as well as a list of Twitter ID
numbers for tweets and users corresponding to data used to generate these
graphs. We hope by making the code for generating these graphs open-source,
other researchers both qualitative and quantitative will both explore the
potential and limitations of this method, as well as contribute modifications
to this scheme as appropriate.

## Acknowledgments and Funding

We would like to acknowledge Clément Levallois for his advice on the open-
source implementation of this project, and Paul Lockaby for ample support in
data collection. We received funding from the University of Washington’s The
Center for an Informed Public, The John S and James L Knight Foundation
(G-2019-58788), Craig Newmark Philanthropies, and the Omidyar Network.

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# Slowly rotating quantum dust cores and black holes

R. Casadioab and L. Tabarronia

aDipartimento di Fisica e Astronomia, Università di Bologna
via Irnerio 46, 40126 Bologna, Italy

bI.N.F.N., Sezione di Bologna, I.S. FLAG
viale B. Pichat 6/2, 40127 Bologna, Italy E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the effect of rotation on the spectrum of bound states for dust cores
that source (quantum) black holes found in Eur. Phys. J. C 82 (2022) 10. The
dust ball is assumed to spin rigidly with sufficiently slow angular velocity
that perturbation theory can be applied. Like the total mass, the total
angular momentum is shown to be quantised in Planck units, hence so is the
horizon area. For sufficiently small fraction of mass in the outermost layer,
the model admits ground states which can spin fast enough within the
perturbative regime so as to describe regular rotating objects rather than
black holes.

##  1 Introduction

Many proposals for resolving the classical spacetime singularities [1] in
approaches to quantum gravity have been put forward (for a necessarily limited
selection, see Refs. [2, 3, 4, 5, 6, 7]). Quite remarkably, a general feature
obtained in semiclassical models of the gravitational collapse is the
appearance of a bounce at a minimum radius [8, 9, 10], which suggests that
understanding the dynamics of matter in the description of black hole
formation [11] (and subsequent evolution [12]) is crucial. However, the non-
linearity of Einstein’s equations makes it impossible to study realistic
classical models analytically, and this furthermore renders their quantum
description intractable in general.

One can still hope to make some progress by studying those (over)simplified
models obtained by forcing a strong symmetry and unphysical equations of state
for the collapsing matter which are employed to solve the Einstein equations
at the classical level. One such example is given by the Oppenheimer and
Snyder model of a ball of dust collapsing solely under its own weight [13]. A
discrete spectrum of bound states for this prototype of matter core was found
in Ref. [14] (see also Ref. [15] for more results 111For thin shells, see
Refs. [16, 17, 18, 19, 20].) for which we will here estimate the effect of
rotation. The key idea in Ref. [14] was to quantise the radial geodesic
equation for dust located at the areal radius of the ball in the Schwarzschild
spacetime as an effective quantum mechanical description of the outermost
layer. In order to deal with rotation, we shall here include the term in that
geodesic equation containing the angular momentum for a rigidly rotating ball
and compute its perturbative effects on the spectrum for sufficiently slow
angular velocity.

##  2 Core quantum spectrum

Let us start by considering the collapse of a perfectly isotropic ball of dust
with total ADM mass [21] $M$ and areal radius $R=R(\tau)$. Here $\tau$ is the
proper time of the dust particles following radial geodesics $r=r(\tau)$ in
the Schwarzschild spacetime metric 222We shall always use units with $c=1$ and
often write the Planck constant $\hbar=\ell_{\rm p}\,m_{\rm p}$ and the Newton
constant $G_{\rm N}=\ell_{\rm p}/m_{\rm p}$, where $\ell_{\rm p}$ and $m_{\rm
p}$ are the Planck length and mass, respectively.

$\displaystyle\mbox{${\rm d}$}s^{2}=-\left(1-\frac{2\,G_{\rm
N}\,M_{0}}{r}\right)\mbox{${\rm d}$}t^{2}+\left(1-\frac{2\,G_{\rm
N}\,M_{0}}{r}\right)^{-1}\mbox{${\rm d}$}r^{2}+r^{2}\left(\mbox{${\rm
d}$}\theta^{2}+\sin^{2}\theta\,\mbox{${\rm d}$}\phi^{2}\right)\ ,$ (2.1)

where $M_{0}=M_{0}(r)$ is the (constant) fraction of ADM mass inside the
sphere of radius $r=r(\tau)$.

In particular, we consider the outermost (thin) layer of (average) radius
$r=R(\tau)$ and mass $\mu=\epsilon\,M$, where $0<\epsilon<1$ is the fraction
of mass in that layer. 333See Ref. [15] for more details about the role of
$\epsilon$. The evolution of $R$ is governed by the mass-shell condition for
the massive layer of four-velocity $u^{\alpha}=(\dot{t},\dot{R},0,0)$, that is

$\displaystyle\frac{E_{\mu}^{2}}{\mu^{2}}-\dot{R}^{2}+\frac{2\,G_{\rm
N}\,M_{0}}{R}-\left(1-\frac{2\,G_{\rm
N}\,M_{0}}{R}\right)\frac{L_{\mu}^{2}}{R^{2}\,\mu^{2}}=1\ ,$ (2.2)

where dots denote derivatives with respect to $\tau$, $M_{0}=(1-\epsilon)\,M$,
and $E_{\mu}$ and $L_{\mu}$ are the conserved momenta conjugated to
$t=t(\tau)$ and $\phi=\phi(\tau)$, respectively.

###  2.1 Zero angular momentum and mass quantisation

The case of purely radial motion was analysed in Refs. [14, 15], which we
briefly review here. For $L_{\mu}=0$, Eq. (2.2) reads

$\displaystyle
H_{0}\equiv\frac{P^{2}}{2\,\epsilon\,M}-\frac{\epsilon\left(1-\epsilon\right)G_{\rm
N}\,M^{2}}{R}=\frac{\epsilon\,M}{2}\left(\frac{E_{\mu}^{2}}{\epsilon^{2}\,M^{2}}-1\right)\equiv\mathcal{E}\
,$ (2.3)

where $P=\epsilon\,M\,\dot{R}$ is the momentum conjugated to $R=R(\tau)$. The
canonical quantisation prescription with $\hat{P}=-i\,\hbar\,\partial_{R}$
allows us to write Eq. (2.3) as the time-independent Schrödinger equation

$\displaystyle\hat{H}_{0}\,\Psi_{\bar{n}}=\left[-\frac{\hbar^{2}}{2\,\epsilon\,M}\left(\frac{\mbox{${\rm
d}$}^{2}}{\mbox{${\rm d}$}R^{2}}+\frac{2}{R}\,\frac{\mbox{${\rm
d}$}}{\mbox{${\rm d}$}R}\right)-\frac{\epsilon\left(1-\epsilon\right)G_{\rm
N}\,M^{2}}{R}\right]\Psi_{\bar{n}}=\mathcal{E}_{\bar{n}}\,\Psi_{\bar{n}}\ ,$
(2.4)

which is formally the same as the one for the $s$-states of the hydrogen atom.
The solutions are thus given by the Hamiltonian eigenfunctions

$\displaystyle\Psi_{\bar{n}}(R)=\sqrt{\frac{\epsilon^{6}\left(1-\epsilon\right)^{3}M^{9}}{\pi\,\ell_{\rm
p}^{3}\,m_{\rm
p}^{9}\,\bar{n}^{5}}}\,\exp\\!\left(-\frac{\epsilon^{2}\,(1-\epsilon)\,M^{3}\,R}{\bar{n}\,m_{\rm
p}^{3}\,\ell_{\rm
p}}\right)L_{\bar{n}-1}^{1}\\!\\!\left(\frac{2\,\epsilon^{2}\,(1-\epsilon)\,M^{3}\,R}{\bar{n}\,m_{\rm
p}^{3}\,\ell_{\rm p}}\right)\ ,$ (2.5)

where the normalisation is defined in the scalar product

$\displaystyle\mbox{$\langle\,\bar{n}\mid\bar{n}^{\prime}\,\rangle$}=4\,\pi\,\int_{0}^{\infty}R^{2}\,\Psi_{\bar{n}}^{*}(R)\,\Psi_{\bar{n}^{\prime}}^{\phantom{}}(R)\,\mbox{${\rm
d}$}R=\delta_{\bar{n}\bar{n}^{\prime}}\ ,$ (2.6)

the functions $L_{{\bar{n}}-1}^{1}$ are Laguerre polynomials and the non-
negative integer quantum number ${\bar{n}}$ corresponds to the eigenvalues

$\displaystyle\mathcal{E}_{\bar{n}}=-\frac{\epsilon^{3}\,(1-\epsilon)^{2}\,M^{5}}{2\,\bar{n}^{2}\,m_{\rm
p}^{4}}\ .$ (2.7)

The expectation value of the areal radius on these states is given by

$\displaystyle\bar{R}_{\bar{n}}\equiv\bra{\bar{n}}\hat{R}\ket{\bar{n}}=\frac{3\,\ell_{\rm
p}\,\bar{n}^{2}\,m_{\rm p}^{3}}{2\,\epsilon^{2}\,(1-\epsilon)\,M^{3}}\ ,$
(2.8)

so that the quantum picture is the same that one would have in Newtonian
physics. In particular, the ground state $\bar{n}=1$ has a width
$\bar{R}_{1}\sim\ell_{\rm p}\,(m_{\rm p}/M)^{3}$ and energy
$\mathcal{E}_{1}\sim-M\,(M/m_{\rm p})^{4}$, which makes it practically
indistinguishable from a point-like singularity if $M\gg m_{\rm p}$.

In fact, the only general relativistic feature that the model retains is given
by $\mathcal{E}=\mathcal{E}(E_{\mu})$ in Eq. (2.3). By then assuming that
$E_{\mu}$ is well-defined for the allowed quantum states, we obtain

$\displaystyle
0\leq\frac{E_{\mu}^{2}}{\epsilon^{2}\,M^{2}}=1-\frac{\epsilon^{2}\,(1-\epsilon)^{2}}{\bar{n}^{2}}\left(\frac{M}{m_{\rm
p}}\right)^{4}\ ,$ (2.9)

which yields the lower bound

$\displaystyle\bar{n}\geq
N_{M}\equiv\epsilon\,(1-\epsilon)\left(\frac{M}{m_{\rm p}}\right)^{2}\ .$
(2.10)

The fundamental state of the outermost layer hence corresponds to
$\bar{n}=N_{M}\gg 1$, with

$\displaystyle\mathcal{E}_{N_{M}}=-\frac{\epsilon\,M}{2}\ .$ (2.11)

This result clearly resembles Bekenstein’s famous area law quantisation [22],
since the allowed values of the mass and gravitational area are quantised in
Planck units according to 444We recall that $R_{\rm H}=2\,G_{\rm N}\,M$ is the
classical Schwarzschild (or gravitational) radius of the ball.

$\displaystyle{\mathcal{A}}_{M}=16\,\pi\,G_{\rm
N}^{2}\,M^{2}=\frac{16\,\ell_{\rm p}^{2}\,N_{M}}{\epsilon\,(1-\epsilon)}\ ,$
(2.12)

where the entire spectrum is given by $\bar{n}=N_{M}+n$, with $n$ a non-
negative integer.

We can now see that the singularity is precluded, as one expects from
semiclassical models [8, 9], since

$\displaystyle\bar{R}_{N_{M}}=\frac{3}{2}\,(1-\epsilon)\,G_{\rm
N}\,M=\frac{3}{4}\,(1-\epsilon)\,R_{\rm H}\ ,$ (2.13)

and $\bar{R}_{N_{M}}\sim R_{\rm H}\gg\ell_{\rm p}$ unless the value of
$\epsilon$ is extremely close to $1$. Furthermore
$\bar{R}_{N_{M}}<(3/4)\,R_{\rm H}$ for any value of $0<\epsilon<1$, so that a
non-spinning dust ball in the ground state can be the matter core of a quantum
black hole for any mass $M$. We also notice that the relative uncertainty in
the areal radius is given by

$\displaystyle\frac{\Delta
R_{\bar{n}}}{\bar{R}_{\bar{n}}}\equiv\frac{\sqrt{\bra{\bar{n}}\hat{R}^{2}\ket{\bar{n}}-\bar{R}_{n}^{2}}}{\bar{R}_{\bar{n}}}=\frac{\sqrt{\bar{n}^{2}+2}}{3\,\bar{n}}\
,$ (2.14)

which asymptotes to a minimum of $1/3$ for $\bar{n}\to\infty$. In the
following, for simplicity, we will mostly consider the ground state
$\bar{n}=N_{M}$ with $M\gg m_{\rm p}$. 555For instance, for $M=M_{\odot}\simeq
10^{30}\,$kg, one finds $N_{M}\sim 10^{76}$.

###  2.2 Rigidly rotating ball

Next, we consider the case of a dust ball which rotates rigidly with angular
velocity $\omega$. A proper general relativistic treatment would require
replacing the outer Schwarzschild geometry with the Kerr metric, but we will
here be satisfied with a perturbative approach for the quantum system
described previously. In particular, the outermost layer would have classical
angular momentum

$\displaystyle L_{\epsilon}=\frac{2}{3}\,\epsilon\,M\,R^{2}\,\omega\ ,$ (2.15)

which we assume is small enough and replace for $L_{\mu}$ in Eq. (2.2). This
yields the total Hamiltonian $H=H_{0}+V_{L}$, where $H_{0}$ is given in Eq.
(2.3) and

$\displaystyle V_{L}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\left(1-\frac{2\,G_{\rm
N}\,(1-\epsilon)\,M)}{R}\right)\frac{L_{\epsilon}^{2}}{2\,\epsilon\,M\,R^{2}}$
(2.16) $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{2}{9}\,\epsilon\,M\left[R^{2}-2\,G_{\rm
N}\,(1-\epsilon)\,M\,R\right]\omega^{2}\ .$

In perturbation theory, this term will result in a correction to the
eigenvalues (2.7) given by

$\displaystyle\Delta\mathcal{E}_{\bar{n}}\equiv\bra{\bar{n}}\,\hat{V}_{L}\,\ket{\bar{n}}\simeq\frac{2}{9}\,\epsilon\,M\,\bar{R}_{\bar{n}}\left[\frac{10}{9}\,\bar{R}_{\bar{n}}-2\,G_{\rm
N}\,(1-\epsilon)\,M\right]\omega^{2}\ ,$ (2.17)

where we used Eq. (2.14) for $\bar{n}\gg 1$ and $\bar{R}_{\bar{n}}$ is given
in Eq. (2.8).

For the ground state $\bar{n}=N_{M}$, we thus find

$\displaystyle\Delta\mathcal{E}_{N_{M}}$ $\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle\frac{m_{\rm
p}^{4}-\epsilon^{2}\,(1-\epsilon)^{2}\,M^{4}}{9\,\epsilon\,M}\,G_{\rm
N}^{2}\,\omega^{2}$ (2.18) $\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle-\frac{1}{9}\,\epsilon\,M\,(1-\epsilon)^{2}\,G_{\rm
N}^{2}\,M^{2}\,\omega^{2}\ ,$

in which we assumed that $\epsilon\,(1-\epsilon)\,M^{2}\gg m_{\rm p}^{2}$. We
then have

$\displaystyle\bra{N_{M}}\,\hat{H}\,\ket{N_{M}}=\mathcal{E}_{N_{M}}+\Delta\mathcal{E}_{N_{M}}\simeq\mathcal{E}_{N_{M}}\left[1+\frac{2}{9}\,(1-\epsilon)^{2}\,G_{\rm
N}^{2}\,M^{2}\,\omega^{2}\right]\ .$ (2.19)

This result is acceptable as long as
$|\Delta\mathcal{E}_{N_{M}}|\ll|\mathcal{E}_{N_{M}}|$, or

$\displaystyle|\omega|\ll\frac{3}{\sqrt{2}\,(1-\epsilon)\,G_{\rm
N}\,M}\equiv\omega_{\rm max}\ .$ (2.20)

We further notice that $\omega_{\rm max}$ corresponds to a total angular
momentum per unit mass given by

$\displaystyle a_{\rm max}\equiv\frac{L_{\rm
c}}{M}=\frac{2}{5}\,\bra{N_{M}}\,\hat{R}^{2}\,\ket{N_{M}}\,\omega_{\rm
max}\simeq 2\,(1-\epsilon)\,G_{\rm N}\,M\ .$ (2.21)

In the same approximation, we can compute the correction to the ground state
quantum number to leading order in $\omega$, which is given by

$\displaystyle N_{ML}\simeq N_{M}\left[1+\frac{1}{9}\,(1-\epsilon)^{2}\,G_{\rm
N}^{2}\,M^{2}\,\omega^{2}\right]\ .$ (2.22)

As expected, the angular momentum acts as an effective potential barrier and
increases the minimum allowed quantum number from $N_{M}$ to $N_{ML}$,
corresponding to a larger radius

$\displaystyle\bar{R}_{N_{ML}}\simeq\bar{R}_{N_{M}}\left[1+\frac{2}{9}\,(1-\epsilon)^{2}\,G_{\rm
N}^{2}\,M^{2}\,\omega^{2}\right]\ .$ (2.23)

In particular, for $\omega=\omega_{\rm max}$, one obtains

$\displaystyle N_{\rm
max}=\frac{3}{2}\,\epsilon\,(1-\epsilon)\left(\frac{M}{m_{\rm
p}}\right)^{2}=\frac{3}{2}\,N_{M}\ ,$ (2.24)

corresponding to a radius $\bar{R}_{\rm
max}=2\,\bar{R}_{N_{M}}=3\,(1-\epsilon)\,G_{\rm N}\,M$.

###  2.3 Outer geometry

Figure 1: Top left panel: ratio $\omega_{\rm H}/\omega_{\rm max}$ for
$0<\epsilon<1$ (shaded area corresponds to $\omega_{\rm H}<\omega_{\rm max}$).
Top right to bottom left panels: core radius (solid line), outer horizon
radius (dashed line) and inner horizon radius (dotted line) for
$0<\omega<\omega_{\rm max}(\epsilon)$ with $\epsilon=0.8$, $0.5$ and $0.1$
(all quantities in units of $G_{\rm N}\,M$).

We can now study the geometry outside the ball in the ground state by assuming
that it is indeed given by the Kerr metric. We recall that the outer horizon
for a Kerr metric is located at

$\displaystyle R_{+}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle G_{\rm
N}\,M+\sqrt{G_{\rm N}^{2}\,M^{2}-a^{2}}$ (2.25)
$\displaystyle\\!\\!\simeq\\!\\!$ $\displaystyle 2\,G_{\rm
N}\,M\left[1-\frac{1}{4}\,(1-\epsilon)^{4}\,G_{\rm
N}^{2}\,M^{2}\,\omega^{2}\right]\ ,$

whereas the inner Cauchy horizon would be at

$\displaystyle R_{-}$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle G_{\rm
N}\,M-\sqrt{G_{\rm N}^{2}\,M^{2}-a^{2}}$ (2.26)
$\displaystyle\\!\\!\simeq\\!\\!$
$\displaystyle\frac{1}{2}\,(1-\epsilon)^{4}\,G_{\rm
N}^{3}\,M^{3}\,\omega^{2}\ll\bar{R}_{N_{M}}\ .$

Since $R_{-}$ is located well inside the dust core, the relevant geometry will
differ from the Kerr vacuum there and the inner horizon cannot be realised (at
least) within the perturbative approximation. On the other hand, $R_{+}$ is
larger than the core for

$\displaystyle|\omega|<\omega_{\rm
H}=\sqrt{\frac{3+9\,\epsilon}{(1-\epsilon)^{3}\,(5-3\,\epsilon)}}\,\frac{1}{G_{\rm
N}\,M}\ ,$ (2.27)

which is approximately equal to $\omega_{\rm max}$ in Eq. (2.20) for
$\epsilon\simeq 0.5$ (see Fig. 1). The inner matter core approaches the outer
horizon radius $R_{+}$ for $\omega\simeq\omega_{\rm H}$, with $\omega_{\rm
H}<\omega_{\rm max}$ for $0<\epsilon\lesssim 0.5$ and $\omega_{\rm
H}>\omega_{\rm max}$ for $0.5\lesssim\epsilon<1$. In general, it would seem
that one can consider small enough values of $\omega$ so that the outer
horizon is realised. Moreover, for sufficiently small $\epsilon$, one can also
describe cores that spin fast enough, so as not to form a black hole, within
the perturbative regime. We will come back to this point in the following.

###  2.4 Angular momentum quantisation

In the above analysis, we considered the angular velocity $\omega$ as a (free)
perturbative parameter, but Eq. (2.22) implies the necessary quantisation of
the angular momentum. In fact, both $N_{M}$ and $N_{ML}$ must be (positive)
integers in order to correspond to allowed states in the spectrum (2.5). This
implies that $N_{ML}=N_{M}+n$, with $n$ a (non-negative) integer (much smaller
than $N_{M}$ for the perturbative result to hold), and we can write

$\displaystyle\omega^{2}\simeq\omega_{n}^{2}=\frac{9\,\epsilon\,n}{(1-\epsilon)\,N_{M}^{2}\,\ell_{\rm
p}^{2}}\ ,$ (2.28)

corresponding to a quantised angular momentum

$\displaystyle L_{n}=\frac{3}{\epsilon}\,\ell_{\rm p}\,m_{\rm
p}\,\sqrt{N_{M}\,n}\ ,$ (2.29)

or

$\displaystyle a_{n}\equiv\frac{L_{n}}{M}=3\,\ell_{\rm
p}\,\sqrt{\frac{1-\epsilon}{\epsilon}\,n}\ ,$ (2.30)

which is plotted in the left panel of Fig. 2 for a first few values of $n$ and
the same values of $\epsilon$ used in Fig. 1. Note that the condition (2.20)
of small angular velocity then simply yields $n\ll N_{M}/2\sim N_{M}$.

Figure 2: Left panel: angular momentum-to-mass ratio (2.30) (in units of
$\ell_{\rm p}$). Right panel: ratio $n_{\rm c}/N_{M}$ in Eq. (3.1) (shaded
region marks the perturbative regime).

We can further notice that the quantisation of the mass and angular momentum
lead to a quantisation of the outer horizon radius (2.25) and area in Planck
units, namely

$\displaystyle{\mathcal{A}}_{+}=4\,\pi\,R_{+}^{2}\simeq{\mathcal{A}}_{M}\left[1-\frac{9}{2}\,(1-\epsilon)^{2}\,\frac{n}{N_{M}}\right]\
,$ (2.31)

where ${\mathcal{A}}_{M}$ is the horizon area (2.12) for a Schwarzschild black
hole of mass $M$. We then see that increasing the mass (that is $N_{M}$) makes
the horizon larger, whereas increasing the angular momentum (that is $n$)
makes it smaller, consistently with the classical theory.

##  3 Final remarks

We have considered the perturbation induced by rotation on the quantum
spectrum of dust balls found in Ref. [14]. Quite expectedly, we have found
that adding angular momentum increases the size of the ground state. Moreover,
no inner Cauchy horizon can appear within the perturbative regime, and the
event horizon area (2.31) is now quantised in terms of two quantum numbers,
namely $\bar{n}=N_{M}$ corresponding to the total mass $M$ and $n$ for the
quantised total angular momentum of the system.

Increasing $N_{M}$ makes both core and horizon larger, whereas increasing $n$
makes the core larger but the event horizon smaller. There is therefore a
critical angular velocity $\omega_{n_{\rm c}}\simeq\omega_{\rm H}$, that is

$\displaystyle\frac{n_{\rm
c}}{N_{M}}\simeq\frac{1+3\,\epsilon}{3\,(1-\epsilon)\,(5-3\,\epsilon)}\ ,$
(3.1)

above which the core remains larger than the (would-be outer) horizon and the
system is not a black hole. From the right panel of Fig. 2, we see that this
can occur within the perturbative regime if $0<\epsilon\lesssim 0.5$,
consistently with the top left panel of Fig. 1. An example is provided by the
bottom right panel of Fig. 1, where $\epsilon=0.1$ and the transition from a
rotating black hole to a horizonless compact object occurs for $G_{\rm
N}\,M\,\omega\simeq 1$. Of course, beside the qualitative behaviour, the
precise values obtained from such a simple model should not be trusted to bear
a phenomenological relevance.

## Acknowledgments

R.C. is partially supported by the INFN grant FLAG and his work has also been
carried out in the framework of activities of the National Group of
Mathematical Physics (GNFM, INdAM).

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