21 cm Line Astronomy and Constraining New Physics

A Thesis Submitted
in Partial Fulfilment of the Requirements
for the Degree of
Doctor of Philosophy

by
Pravin Kumar Natwariya
Roll No. 17330022
Under the guidance of
Prof. Jitesh R. Bhatt
Theoretical Physics Division
Physical Research Laboratory, Ahmedabad.
to the

Discipline of Physics
Indian Institute of Technology Gandhinagar,
Gujarat 382355, India.

List of Acronyms

  * $\Lambda$CDM

$\Lambda$ Cold Dark Matter

  * BBN

Big-Bang Nucleosynthesis

  * WMAP

Wilkinson Microwave Anisotropy Probe

  * COBE

COsmic Background Explorer

  * QCD

Quantum ChromoDynamic

  * EoR

Epoch of Reionization

  * IGM

InterGalactic Medium

  * JWST

James Webb Space Telescope

  * EDGES

Experiment to Detect the Global Epoch of Reionization Signature

  * ISM

InterStellar Medium

  * CMB

Cosmic Microwave Background

  * SIDM

Self-Interacting Dark Matter

  * WDM

Warm Dark Matter

  * THESEUS

Transient High Energy Sky and Early Universe Surveyor

  * NuSTAR

Nuclear Spectroscopic Telescope Array

  * CMBR

Cosmic Microwave Background Radiation

  * LIGO

Laser Interferometer Gravitational-Wave Observatory

  * PBH

Primordial Black Hole

  * AGN

Active Galactic Nuclei

  * LUX

Large Underground Xenon

  * PandaX

Particle and astrophysical Xenon

  * CRESST

Cryogenic Rare Event Search with Superconducting Thermometers

  * PICO

PICASSO and COUPP

  * PICASSO

Project in CAnada to Search for Super-symmetric Objects

  * FIRAS

Far Infrared Absolute Spectrophotometer

  * COUPP

Chicagoland Observatory for Underground Particle Physics

  * AMEGO

All-sky Medium Energy Gamma-ray Observatory

  * PMFs

Primordial Magnetic Fields

  * MHD

MagnetoHydroDynamics

  * ARCADE

Absolute Radiometer for Cosmology, Astrophysics and Diffuse Emission

  * LWA

Long Wavelength Array

  * HESS

High Energy Stereoscopic System


# What’s in a Niche? Migration Patterns in Online Communities

Katherine Van Koevering,1 Meryl Ye,1 Jon Kleinberg,1
1Cornell University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Broad topics in online platforms represent a type of meso-scale between
individual user-defined communities and the whole platform; they typically
consist of related communities that address different facets of a shared
topic. Users often engage with the topic by moving among the communities
within a single category. We find that there are strong regularities in the
aggregate pattern of user migration, in that the communities comprising a
topic can be ordered in a partial order such that there is more migration in
the direction defined by the partial order than against it. Ordered along this
overall direction, we find that communities in aggregate become smaller, less
toxic, and more linguistically distinctive, suggesting a picture consistent
with specialization. We study directions defined not just in the movement of
users but also by the movement of URLs and by the direction of mentions from
one community to another; each of these produces a consistent direction, but
the directions all differ from each other.

We show how, collectively, these distinct trends help organize the structure
of large online topics and compare our findings across both Reddit and
Wikipedia and in simulations.

## Introduction

Social media is a remarkably dynamic landscape. Users, information, and even
platforms are constantly on the move. Users’ interests and interactions evolve
and as they do the users move from one corner of the internet to another.
Studying these user migrations has been popular (Kumar, Zafarani, and Liu
2011; Newell et al. 2016; Davies et al. 2021) especially in recent years,
where issues around conspiracies and ‘rabbitholes’ has sparked renewed
interest in understanding where and how users move into increasingly extreme
communities (Fernandez, Asif, and Alani 2018; Wang et al. 2023).

As a factor of these migrations, many online sites follow an organizational
scheme composed of numerous groups or communities, each structured around a
topic of shared interest. Many of the earliest forums and message boards on
the Internet had this design, and it is a central feature for several of the
most active sites today, such as Reddit. This type of structure based on
shared-interest communities is also a feature of how people pursue interests
in the offline world, with clubs and organizations bringing people together
(Glorieux et al. 2010; Gelber 1999).

Moreover, the internet is a big place. On large sites, it is rare for there to
be one singular community dedicated to any broad interest. Instead, we find a
collection of communities each with a different theme, but all related to this
interest (Tan and Lee 2015; TeBlunthuis et al. 2022; TeBlunthuis and Hill
2022). For instance, instead of one community discussing US politics, one
might find several communities at different points of the political spectrum,
or reflecting different levels of involvement.

We pursue the question of user migration in this context. In particular, does
a single user become active in multiple communities on the same topical
category? And if so, is there a directionality to their movement—do the
communities on the same topic form a kind of partial ordering or gradient
exhibiting more motion in the direction of the gradient than against it? We
cannot expect such properties to fully express the idiosyncratic decisions
that every user makes on a large site, but we can look for evidence of
aggregate effects, much as we might for directionality in the physical world:
a collection of twigs in a body of water, might either be moving uniformly in
all directions, or they might have a general drift one way, indicating a
current.

We study these questions across a range of social media topics, each of which
contains multiple communities. Within each of the topics we study, we find
strong evidence to support the existence of these directional migrations. We
focus our analysis on topics on Reddit, where the subreddits on a particular
topic form the related communities, and we consider analogues on Wikipedia as
well.

The consistent directionality of user migration then raises the question of
what factors are influencing the direction. There are two common narratives of
the internet that are often used informally to describe user migration:
radicalization, in which users move to communities with increasingly extreme
behavior (Fernandez, Asif, and Alani 2018), and specialization, in which users
move to communities with increasingly focused, niche interests (Waller and
Anderson 2021). In our analysis, we find that, overall, users tend to move to
smaller communities that linguistic measures evaluate as lower in toxicity but
higher in lexical distinctiveness (Fig 1). We find that this change in the
content of communities over users’ aggregate direction of movement provides an
interesting perspective on these two narratives in our context: the aggregate
decrease in toxicity as a user moves between communities works against the
narrative of radicalization, whereas the increase in linguistic
distinctiveness is consistent with specialization. We emphasize, of course,
that each individual user will have their own motivations in moving between
communities, but the contrasts stands out at the level of the overall user
population.

In addition to user migration, we also investigate information migration (Jin
et al. 2023) in a similar fashion using two measures: posting URLs and
comments with direct references to other communities. Specifically, we find
three distinct types of trends that emerge from this analysis. The first two
types of trends are (i) an individual (user) migration of the type described
above, defined by consistent patterns in how people move between communities
within a topical category; and (ii) an informational migration, defined by the
analogous movement of discrete pieces of information between communities,
encoded by following URLs as they are subsequently posted in various
communities. Both of these migrations are marked by temporal patterns of
engagement.

Figure 1: The user gradient for sports communities on Reddit, one of the
topics we study, demonstrates how three of our measures (size, toxicity, and
linguistic distinctiveness) are related to the aggregate progression of users
along a gradient - communities become smaller, less toxic, and more distinct.
Edges proceed downward. The size of nodes is a logarithmic scale of the size
of communities. The inner color of the nodes represents the toxicity of the
community, where darker nodes are more toxic. The outer color of the node
represents linguistic distinctiveness, where darker edges represent less
distinctive communities.

Our third type is (iii) a referential directionality, which looks at
references, or mentions, from one community to another, asking which
communities talk about others more. In this case, we ask whether it is
possible to define an overall direction based on the asymmetry in how much
different communities mention each other. We find that all three measures
produce strong trends, but are distinct in their orderings and influences.
Finally, we compare our findings to similar analysis of Wikipedia editors and
the migrations extant in that space, as well as simulated networks.

In the course of this analysis, we discover a number of interesting factors
that augment our understanding of online migration patterns. For instance,
users have a tendency to move wholly from one community to another, rather
than straddle many communities simultaneously within a category. However, when
users talk about other communities within a category, the sentiment trends
negative.

### Additional Related Work

Many prior works have considered categories of topically related communities.
One strand of research has studied spinoff communities (Hessel, Tan, and Lee
2016; Zhu et al. 2014), focusing on the creation (and eventual death) of
highly related shared-interest communities. Additional work has turned the
focus to what makes these communities different and how they evolve alongside
each other (Rajadesingan, Resnick, and Budak 2020; Zhang et al. 2021; McThenia
2021). This includes instances of how communities interact and respond to each
other outside a user perspective. (TeBlunthuis and Hill 2022; TeBlunthuis et
al. 2022).

Understanding the movement of users and information through social networks
has been a richly studied topic, although much analysis in this literature is
not directly relevant to us here. Instead, we specifically note several papers
that focus on user life cycles (Yang et al. 2010; Tan and Lee 2015; Rowe
2013), and more recent work on user embeddings (Waller and Anderson 2021). All
of these works attempt to understand or explain why users participate in
certain online communities, and how these groups evolve over time (and how
users might choose to join, participate, or leave). Some of this work,
including Tan et al. (Tan and Lee 2015) Zhang et al (Zhang et al. 2021), and
Rowe (Rowe 2013) focus on instances of users participating in shared interest
groups and highly related communities, principally from a user perspective.
Finally, a recent paper established URL information flow using YouTube links
on Reddit as an interesting and complex problem (Jin et al. 2023).

## Data Description

The data used in this research is collected from Reddit, a popular social
media platform, using Pushshift, a nearly-comprehensive archive of Reddit
(which until recently was publicly available). This discussion-based platform
has a large number of user-created individual forums known as subreddits. Each
subreddit has an explicitly stated topic, usually related to its name that is
chosen by the founder. This clear definition of communities and topics makes
Reddit ideal for studying community interaction. Due to this, Reddit is also
an extraordinarily well-studied platform (Medvedev, Lambiotte, and Delvenne
2019; Proferes et al. 2021; Jamnik and Lane 2017), with extensive research
into Reddit behavior (Amaya et al. 2021), user demographics (Tigunova et al.
2020), user motivations (Bogers and Wernersen 2014; Moore and Chuang 2017)
communities (Monti et al. 2023; Soliman, Hafer, and Lemmerich 2019),
information sharing (De Francisci Morales, Monti, and Starnini 2021; Zimmer et
al. 2018), and more (Baumgartner et al. 2020). As such, we refer to these
papers, and many others on similar subjects, for detailed analysis of Reddit,
Reddit demographics, and description of Reddit as a data source and social
media platform.

Reddit also holds a unique role on the internet. Unlike other large social
media platforms such as Facebook, Twitter, Instagram, TikTok or YouTube,
Reddit represents a socially mediated form of expertise. Reddit users do not
generally engage in ”following” particular users the way one might on most
other social media, nor is there a ”friend” mechanic - this centers activity
on Reddit around the information presented rather than the users themselves.
It is primarily a source for finding information and it is common for users to
interact with other invested, niche, and expert users and engage with posts or
comments months or even years old. It is also community focused in a way that
other platforms are not, with its structuring around themed communities giving
distinct boundaries for topics, cultures, and users. Together, this presents
Reddit as a prime location to identify and study these themed community
groups, where the communities are highly distinct and the themes influential.

The selection of our communities is an important step in our analysis that
requires detailed explanation111Code and data will be made available upon
publication. To begin, we selected thirteen categories of topics that
represent a wide array of users and have many different yet related
communities on Reddit. Among these is politics, which we chose for comparative
purposes. The rest were selected according to three principles: (i)
representation of multiple levels of specificity (e.g., the category ’sports’
is considerably more broad than the category ’Indoor Plants’), (ii)
representation of various demographics (i.e., choose categories that have a
good chance of representing diverse demographics, particularly in terms of
age, race, and gender), (iii) representation of various degrees of
politicization (political radicalization is a well-studied phenomenon that
should not strongly influence all of our categories). See the appendix for
more detail on community choice.

## User Migration

category | # communities | # posts/comments | # users
---|---|---|---
Computer Science (CS) | 19 | 54,304,503 | 1,800,653
Camping | 13 | 13,690,588 | 620,178
Crafting | 16 | 24,658,755 | 1,383,781
Early Retirement | 14 | 56,866,540 | 5,938,592
Fitness | 20 | 5,4256,622 | 3,483,201
Indoor Plants | 16 | 13,302,290 | 1,221,205
Makeup | 19 | 58,141,329 | 1,701,287
PC Building | 14 | 82,819,446 | 3,426,274
Politics | 13 | 308,943,275 | 91,915,842
Sports | 16 | 86,770,040 | 8,610,698
Survival | 14 | 17,298,659 | 605,557
Veganism | 11 | 15,591,562 | 977,653
Table 1: Categories, number of associated communities, and the total number of
posts/comments and users collected for each.

Within a category, users must choose which communities to participate in and
when, but what choices they make and what factors influence those choices is
unclear. Understanding user flow across the Internet has been a long-standing
goal of Web research over many years (Hessel, Tan, and Lee 2016; Huberman et
al. 1998; Rowe 2013; Tan and Lee 2015; Yang et al. 2010), and while a number
of influences have been proposed, we are far from truly understanding the
elements that influence our online lives.

User behavior on Reddit is a well-studied phenomenon and we refer to the
myriad of prior work for more detailed descriptions (Amaya et al. 2021;
Tigunova et al. 2020; Bogers and Wernersen 2014; Choi et al. 2015; Newell et
al. 2016; Thukral et al. 2018; Glenski, Pennycuff, and Weninger 2017). It is
sufficient to say that users follow previously established behavioral
patterns—exponentially distributed amounts of activity, bursty activity, and
exponentially distributed number of communities.

### Defining User Migration

Figure 2: Histogram of the asymmetry scores for all pairs of communities.
Adjusting $k$ has little effect. Note that the average asymmetry of user
directions is far more than would be expected with random movement (with a
p-value of nearly 0), and random movement would produce an exponential
distribution rather than a more uniform distribution.

Given that users are migrating between communities, this raises the question
of whether all users are moving in the same way or if they move differently.
As a first step, we need a way to formalize a user’s association with a given
community; in particular, it is useful for the analysis to require a threshold
of involvement that may be more than a single post, but which is permissive
enough to include a wide range of activities in the community. In particular,
we say that a user has entered a community when they first exceed our
definitional threshold for involvement. We have explored a wide range of
definitions for this process of a user entering a community, and they give
markedly similar results. As such, we compare the following two possible
definitions:

######  Definition 1 (Thresholding)

A user must post/comment in a community at least $k$ times, and enters on the
$k$th post/comment

######  Definition 2 (Expanding)

A user must post/comment in a new community at least $k$ times without
posting/commenting in any other new communities, and enters on first
post/comment in this sequence.

The most obvious definition for entering a community may be the thresholding
definition. However, this definition produces arbitrary behavior when, for
instance, a user is alternating between two new communities. Thus using this
definition produces a noisier picture. The expanding definition, on the other
hand, is much more stringent. It reduces the noise and focuses our analysis on
users who make more definitive moves to a new community. We choose the
expanding definition for the rest of our analysis, but the thresholding
definition yields nearly identical, albeit noisier, results.

We also impose a minimum number of posts/comments in order for a user to be
viewed as participating in a communities, rather than a transient
contribution; for our purposes, we require at least $k=20$ posts/comments,
although raising and lowering $k$ has minimal effect on results. One more
novel aspect of user behavior is how users change their behavioral patterns
after entering a new community - users tend to either remain primarily tied to
their original community or switch completely, with only a small percentage of
users participating in both relatively equally. This emphasizes the temporal
aspect of user participation in a community and lends credence to our
expanding definition.

### Gradients of User Migration

We now consider pairs of communities: for a given pair of communities $(A,B)$,
we say that the orientation of $(A,B)$ is directed from $A$ to $B$ if a
majority of users associated with both enter $A$ before $B$; the orientation
is directed from $B$ to $A$ if a majority enter $B$ before $A$; and the
orientation is void if the number of pairs who enter both is below some
specified threshold. We define the asymmetry of the orientation of $(A,B)$ to
be the fraction of users that move in the majority direction. For instance, we
say that the orientation of $(nba,fantasybball)$ is from $nba$ to
$fantasybball$ since most users who are part of both communities enter $nba$
first.

######  Definition 3 (Orientation & Asymmetry)

Let $A$ and $B$ be two communities s.t. some $n\geq k$ number of users have,
by our definition, entered both communities. Let the number of users who
entered $A$ before $B$ be $a$ and the number of users who entered $B$ before
$A$ be $b$ s.t. $a+b=n$. Let $a>b$.

Then, we say the orientation is $(A,B)$ and the asymmetry is $\frac{a}{n}$.

For the communities within a category, we can form a directed graph whose
nodes are the communities, and where there is a directed edge from $A$ to $B$
if the orientation of $(A,B)$ is from $A$ to $B$. We say that a set of
orientations on pairs of communities forms a gradient if this directed graph
is acyclic.

######  Definition 4 (Gradient)

For each pair of communities $A$, $B$, if there is an orientation $(A,B)$ then
let there be two nodes $A$ and $B$ with a directed edge between them from $A$
to $B$. This graph, when acyclic, represents a partial ordering on all
communities. We refer to this partial ordering as the gradient.

Given random movement, as describe in the next section, we would not expect
any patterns in this network. However, surprisingly, across all 13 of the
categories we study, the graphs of strong edges are acyclic. This means for
all our categories, the set of orientations on the pairs of communities forms
a gradient, indicating a consistent direction of user movement through the
category. This is extraordinarily unlikely to happen at random, as described
below, and indicates there are likely strong influences on the direction of
movement between communities. We will spend much of the rest of this paper
exploring these influences.

Figure 3: Small graphs representing the user gradient for politics and CS,
where all edges move downwards. Note that not all communities in each category
appear in these graphs. Those that do not, do not have strong enough user
overlap with any other community ($<20$ users by our definition). For CS, we
remove all edges of less than 0.75 asymmetry to remain acyclic, our only
thresholding for the use gradient.

In these gradients, a large enough fraction of community pairs have edges,
such that there are relatively few distinct topological orderings of the
graphs: they are often essentially linear (Fig 3). Additionally, the fact that
every community present in the graph is very connected within it, despite not
all communities being represented, suggests we have found a natural boundary
of the category.

### Null Models

We compare our small acyclic graphs with two null models - randomizing edge
directions and randomizing edge placement. In the first, we randomly re-assign
the edge direction in a graph. We do this many times, and ask how many result
in an acylcic graph. For the second, we create a new graph with the same
number of edges and nodes, where the edges are placed uniformly at random.
Again, we repeat this process and count the number of acyclic graphs. For most
of our categories, we find significantly few acylic graphs (less than 1%) Tab
2. The exceptions are the few categories with very few edges. This suggests
that it is extremely unlikely for the acyclic graphs we see here to have
happened at random. As a comparison, less than $4.1*10^{-4}$% of all possible
directed graphs of 10 nodes are acyclic.

Category | # communities | #Edges |  Fraction Randomly Acyclic
---|---|---|---
Computer Science (CS) | 13 | 28 |  $<0.01$**
Camping | 7 | 11 | 0.21
Crafting | 10 | 28 |  $<.01$**
Early Retirement | 10 | 12 | 0.39
Fitness | 10 | 23 | 0.01*
Indoor Plants | 9 | 24 |  $<0.01$**
Makeup | 6 | 10 | 0.19
PC Building | 11 | 43 |  $<0.01$**
Politics | 11 | 34 |  $<0.01$**
Sports | 11 | 48 |  $<0.01$**
Survival | 4 | 6 | 0.34
Veganism | 6 | 8 | 0.34

Table 2: The size of the acyclic graphs, and the fraction of random graphs
that have acyclicity under the edge-swap model.

### Extremists and Specialists

# users | Decreases **
---|---
total activity | Decreases*
activity per user | Increases
toxicity | Decreases**
distinctiveness | Increases**
Table 3: Binomial tests comparing source and target communities for all edges
in user movement graphs. * is p-value $<.05$, ** is p-value $<.01$. We measure
toxicity with the Google Persepctive API.

Given the existence of these gradients, it is natural to ask what is informing
their ordering. The mechanisms behind this pattern of user migration (and,
indeed, informational and referential directionality as well) is likely
complex, but we address some aspects of the mechanism here. One simple test
for relevance is to ask for correlation of various measures with the edges of
our graphs - asking if there is a significant difference in measures between
communities $A$ and $B$ where we have a trend of movement from $A$ to $B$. By
this analysis, we find three measures with any significant relevance: size,
toxicity, and distinctiveness. These three measures are statistically
independent and relevant to two competing narratives of the Internet:
radicalisation and specialization.

#### Size

Size can be measured in a variety of ways - number of distinct users, number
of distinct posts/comments, number of distinct URL’s, etc. Unsurprisingly,
these measures are all highly correlated with each other. If we use number of
users as our proxy, we see strong correlation with our gradient. That is,
generally, users tend to move from large to small communities.

#### Toxicity

Toxicity is another measure. While one might expect users to be drawn into a
‘rabbithole’ of increasingly toxicity, we actually find the opposite. There is
a significant correlation with decreasing toxicity and user gradients. Users
tend to move from more toxic to less toxic communities.

#### Distinctiveness

Finally, we investigate distinctiveness. That is, we ask how different the
language used in a subreddit is from the language used in other subreddits. We
test this using tfidf and a custom term frequency measure, but both produce
similar results. Users generally move to communities with more distinct
language. In practice, this distinctiveness tends to represent increased usage
of technical terminology within the category.

These three measures together bring to mind three competing narratives of the
internet. The first narrative, radicalization, harkens to research in
political radicalization and suggests that over time, internet denizens tend
to move to more radical and extreme spaces (Wang et al. 2023). A second
narrative, specialization, references human behavior offline in that users
become more specialized in niche or advanced topics within a category (Waller
and Anderson 2019; McAuley and Leskovec 2013).

Both behaviors exist and both could plausibly be influential in these gradient
orderings. If we were to posit that these user gradients mirror the
radicalization seen in political spheres, the extreme communities narrative,
one might expect increased engagement or increased toxicity. Instead, we find
that the amount of engagement per user does not change and toxicity decreases.
If we were to instead suggest that that the narrative of specialization
informs the ordering, we might expect size to decrease, but distinctiveness to
increase, with agnosticism on toxicity and engagement - which we do see.

## Informational and Referential Movement

The definitions for movements and gradients defined above do not only apply to
users. We can also apply them to information and references. In this case, we
use URLs as an example of information movement. If a URL is first posted in
one community and then later in another, we can view this similarly to a user
entering these communities. This gives us a rough proxy for information
movement. On the other hand, references are quite different. We say a
community mentions another community if the text ”r/$community$” exists in any
post or comment. Unlike users and URLs, there is no real concept of movement,
but we retain the directed pairs structure.

While user migration has analogues in prior research, albeit with a variety of
definitions, URL movement and mentions are more novel under any formalism. As
such, it is useful to develop a better understanding of the underlying nature
of these measures. We begin by evaluating the size of each measure. The sizes
are closely correlated—larger subreddits also use more URLs and more mentions.
Note that with the differing scales, however, mentions are by far the most
common of the measures.

### URL & Mention Behavior

URLs and mentions have an additional structure in that they are references
outside of the community. URLs are a representation of retrieving additional
information and bringing that to an ongoing conversation. A flow of URLs is
akin to a flow of information. Thus, it is inherently interesting to
understand where this information is coming from (Tab 4). In fact, a large
percentage of these URLs reference Reddit (and we include these as ‘mentions’
in our later analysis). Additionally, not only do URLs link to other Reddit
communities, they may also link to another post or comment in the same
community. We see wide variation in these.

Since URLs are pointing at specific conversations on Reddit, this implies that
some categories may have a stronger ‘memory’ in that they review previous
material more often. It also implies that some categories share information
among themselves more than other categories do.

URLs, much like users, can be posted in a community multiple times. The vast
majority of URLs, however, are only posted once. In fact, a large percentage
of URLs that appear in many communities (particularly cross-categories) are
related to bot accounts or Reddit settings (e.x., links to spam reporting,
summarizer bots, auto-moderators, etc.). We filter most of these URLs out. It
is impossible to be entirely without bot accounts, but the few remaining
likely do not significantly impact our data.

Table 4: Proportion of URLs that reference somewhere on Reddit and the community in which the URL is posted. The self category is percentage of URLs that link to Reddit, not total URLs. category | % reddit | % self
---|---|---
makeup | 0.21 | 0.70
politics | 0.36 | 0.78
veganism | 0.37 | 0.51
pc building | 0.38 | 0.85
crafting | 0.39 | 0.33
sports | 0.40 | 0.86
cs | 0.41 | 0.84
survival | 0.44 | 0.65
camping | 0.56 | 0.55
fitness | 0.70 | 0.82
indoor plants | 0.76 | 0.14
early retirement | 0.83 | 0.93

Unlike URLs, mentions represent talking about a community more generally than
just a specific instance. In this analysis, we include Reddit-directed URLs in
our mentions analysis as instances of talking about a community. It is also
worth noting that mentions align closely with size, and the amount a community
is mentioned is therefore generally closely aligned with the amount it
mentions other communities and thus also its size.

Finally, it is worth noting that a majority of mentions are in a negative
context. When we applied sentiment analysis (Loper and Bird 2002) to a sample
of mention comments for every pairs of subreddits, we found mostly negative
sentiment, with nearly every pair of communities having a majority negative
sentiment between them. One notable exception is r/hiking as the only
community where the sentiment on posts about other communities averages
positive. No community averages positive sentiment on posts about them.

### Gradients of Informational and Referential Migration

We now define gradients along our informational axis (URL movement) and
referential axis (mentions) in a way similar to how we define gradients for
user movement. However, we do not have the same concept of entering a
community that we did for users. For URLs, it is somewhat rare to be posted in
the same community twice—and posting many times in a community could be
considered spamming rather than genuine information diffusion. This makes our
thresholding for entering a bad fit for URLs. Mentions, on the other hand,
lack the concept of “movement”. One community might be mentioned many times in
another, but there is no obvious analog for the ordering of communities that a
user movement has.

Figure 4: Histograms of the asymmetry scores for all pairs of communities by
URLs. Figure 5: Histograms of the asymmetry scores for all pairs of
communities by mentions.

Despite these differences, we can still define the orientation between a pair
of communities $(A,B)$ for each of these modalities in a natural way: for
information, whether a majority of URLs common to $A$ and $B$ are first posted
to $A$ before $B$; and for mentions, whether there are more mentions from $A$
to $B$ than from $B$ to $A$. With these definitions in place, we can again
define directed graphs on the communities within a category — one for
orientations based on URLs (informational), and one for orientations based on
mentions (referential). Interestingly, we again find that these graphs are
acyclic, establishing the existence of informational and referential gradients
in analog to our user gradients.

We can immediately see that the informational and referential measures are
quite different by examining their asymmetry scores (Fig 5). URLs exhibit a
bell curve shape. Mentions, on the other hand have a more uniform
distribution. As with our other histograms, both measures have significantly
larger asymmetry scores than expected at random. For both mentions and URLs,
we restrict pair-wise analysis to pairs with at least 20 URLs/mentions between
them, similar to users.

measure | URL | Mention | Expanding
---|---|---|---
# users | Decreases* | Decreases | Decreases**
total activity | Decreases* | Decreases | Decreases*
activity per user | Increases | Increases | Increases
toxicity | Decreases* | Decreases | Decreases**
distinctiveness | Decreases* | Increases | Increases*
# URLs | Decreases* | Decreases* | Decreases*
URLs per user | Increases | Decreases | Increases
URL per post | Increases | Decreases* | Decreases
Self-mentions | Decreases** | Increases* | Decreases**
# times mentioned | Decreases* | Increases** | Decreases**
# times mentions others | Decreases** | Decreases* | Decreases**
self-mentions per post | Decreases | Decreases | Increases
mentions others per post | Increases | Increases** | Increases*

Table 5: Binomial tests comparing source and target communities for all edges
* is p-value $\leq.05$, ** is p-value $\leq.01$

Given that these new axes display a similar underlying directionality to our
user gradient, we examine the alignment of these new gradients. Generally,
user, URL, and mention gradients do not strongly agree on directionality
between pairs of communities - or even which pairs of communities have edges.
Examining possible influences, however, reveals similar influences for users
and URLs, but quite different influences for mentions (Tab 5).

Just as for users, URL orders seem to be strongly influenced by community
size—URLs move from large subreddits to smaller ones. Given our measures of
size are closely linked, users also move to communities with fewer URLs.
However, this does not mean that those subreddits are more active per user, as
we see no significant change in activity per user.

Additionally, mentions are significantly influential for all of our axes.
While users and URLs move to communities with fewer mentions (reflecting the
size tendency), mentions direct to communities that are more self-referential.
If we look at mentions relative to community size, mentions and users tend to
direct to communities than mention others more, although this is not true for
URLs. Users and URLs, additionally, move to communities than mention
themselves less. However, mentions also seem to be influenced by URLS—mentions
direct more strongly towards communities that have lower numbers of URLs per
post.

Overall, this suggests that users move from larger communities to smaller
ones, but also to communities that talk about others more. URLs similarly
direct to smaller communities, but is not as influenced by mentions. Mentions,
on the other hand, are not influenced by community size, but do direct to
communities with fewer URLs that reference themselves and others more.
However, none of the simple variables we check fully explain the gradient,
suggesting a more complex model is necessary to explain the strong
directionality.

Figure 6: URL gradient graph for CS (threshold at 0.55) and mention gradient
graph for CS. The thresholding for mentions to be acyclic is higher - 0.6 for
CS. For all categories, thresholding is at or below 0.6 except for fitness at
0.8 and politics and makeup at 0.7. However, asymmetry of mentions is
generally much higher than users or URLs, meaning higher thresholds does not
strongly influence the gradient.

For a visual demonstration of these gradients, we again create our small
directed graphs (Fig 6). Again, we note that these graphs are remarkably
linear, although perhaps not quite as strongly so as for the user gradient.
They also include different communities. While URL and user graphs seem to
have similar communities across all of our categories, mentions can be very
different. Additionally, URL and user gradient seem to have similar
influences, while mentions has different influences. One possible nuance to
this, is that sharing users or information may be seen as a measure of
connectedness of communities. That is, if two communities share users or
information, this may imply they are more highly connected. Mentions, however,
do not necessarily fit in this framework. Mentioning another community does
not imply connectedness in the same way, particularly given the generally
negative sentiment.

Most of our analysis thus far has focused on the pairs of communities $(A,B)$
where the orientation is directed either from $A$ to $B$ or from $B$ to $A$.
However, as noted above, there is a third option for the orientation: it could
be void. That is, there is not enough data to make an informed decision about
directionality. This is particularly interesting when not all orientations are
void for a particular pair of communities. For instance, Python is mentioned
by many communities in the CS category, but very few users in Python post
frequently enough to meet our criteria for entering, thus it does not appear
in our user gradients. In fact, there is not much correlation between how many
instances of our measures exist for each gradient, with correlation less than
0.15 across the board and with mentions and URLs particularly uncorrelated at
merely 0.01.

However if we look at only zero vs non-zero indicators, then the measures seem
to be slightly more aligned, with all three gradients generally agreeing on
whether an edge is void. In particular, Users and URLs agree on 64% of edges,
users and mentions agree on 60%, and URLs and mentions agree on 80%. This is
especially surprising, since if we look at only the size measures for each
gradient (for a community, number of users, number of URLs, number of
mentions) the sizes are much more highly correlated, since they are all
correlated with the number of posts and comments for a community. However, if
we compare our total sizes of each community with the gradient instances, we
see a strong correlation with our user gradient ($\sim 0.5$), a moderate
correlation with our mention gradient ($\sim 0.3$), but almost no correlation
with URL gradient ($\sim 0.04$). Overall, this suggests that even if the
direction of an edge has limited correlation for our three measures, the
existence of an edge is highly correlated.

## Experiments and External Data

### Wikipedia

Our framework has been structured to be general enough to apply to many social
media platforms. However, in order to perform analysis on Reddit, we have had
to make some Reddit-specific decisions in applying the framework there – where
communities are well-defined and users typically have many interactions with a
community. As an experiment, we create a similar user graph from Wikipedia
(Fig 7). To do this, we adapt the low-level definitions to the Wikipedia
context. If we consider a page on Wikipedia as a community, and any edit to
the page equivalent to a comment in a community (Geiger and Halfaker 2013),
then users on Wikipedia interact far less than Reddit users and there are far
fewer of them.

We first start with a seed page - say World War II. We then find all users who
have edited this page (or the associated talk page), and then find all pages
those editors have touched. From there, we only include pages with at least
100 editors - enough to reasonably expect overlap. Then we restrict to editors
that have touched a page at least three times since 2010, and pages have
connections if they have at least 30 editors in common (where at least 60% of
edits overlap in time frame). From here, we can pull a directed graph of
editor movement of 62 nodes. If we threshold those edges to 0.55 we see only a
few cycles and at 0.60 we have an acyclic graph of several hundred edges. This
example suggests that our definitions are applicable across other social
media, and the acyclicity suggests some of the findings might be similarly
applicable. The same definitions applied to the seed page ’American Civil War’
gave similar acyclicity results, with 42 nodes producing an acyclic graph
thresholded also at 0.6.

Figure 7: Graph of Wikipedia data seeded from World War II. Edges flow from
left to right.

### Simulations

We also attempt two simulations to compare our results. We create a graph of
$n$ nodes, with $m$ users. These users visit $k$ random nodes in some order,
where $k$ is determined by an exponential distribution to,
$f(x,\lambda)=\lambda exp(-\lambda x)$, to simulate the exponentially
distributed number of communities entered by the users in our data. First, we
assume there is some canonical ordering of nodes in a graph. We then assume
some $p$ percentage of users follow this ordering exactly. Given this, we find
that, for larger graphs, 25% or more of users must follow this ordering in
order to have reliably acyclic graphs. This could suggest a strong tendency
towards ordering is required for repeated acyclicity. Note, however, that this
is not a perfect representation of our data. Importantly, the histogram of the
asymmetry of edges given this model is a bell curve, but it has significantly
lower mean and standard deviation than our data.

We attempt a second simulation with a similar set-up, but instead using the
Mallows model to give a partial ordering of nodes (Collas and Irurozki 2021).
Although acyclicity is reached with relatively low $\phi$, we again see a
narrow, low bell curve for asymmetry values. These two results suggest that,
while total ordering seems to be present to account for the acyclicity, the
high levels of asymmetry suggest even higher agreement with this ordering than
mere acyclicity.

Figure 8: Number of acylic graphs from 100 simulations with some $P$
percentage of users following a prescribed ordering. Solid lines depict a path
length with $\lambda=1$, dotted is $\lambda=2$.

## Conclusion

In this work, we have defined three types of gradients of movement within
online communities—at the individual (user), informational, and referential
levels. We developed methods to identify and analyze these gradients, and
studied their properties in a broad set of topical categories on Reddit. We
find that each of these three gradients defines an acyclic ordering to the
communities, and interestingly, these orderings are not consistent with each
other, with referential gradients tending to be particularly different.

User migration is influenced by four metrics: decreasing size of the
community, decreasing toxicity, increasing linguistic distinctiveness, and
decreasing self-mentions/increasing other-mentions. Combined, these measures
paint a picture of users moving to smaller, friendlier, and more niche
communities, echoing a narrative of online specialization while seeming to
dispute a narrative of radicalization. This type of progression is compatible
with a number of psychological mechanisms, particularly the so-called hedonic
treadmill of Brickman and Campbell, where people become satiated on their
current form of an activity and seek out increasingly complex versions of it
(Brickman 1971; Brickman, Coates, and Janoff-Bulman 1978; Diener, Lucas, and
Scollon 2006). However, this raises new questions about how users find and
choose communities, the interplay of all of these measures, and the dynamics
of particular categories of Reddit.

Beyond these questions of mechanism, there are a number of potential further
applications that could be explored using gradients. Knowledge that gradients
exist could help inform recommendations of new communities for users to
explore on large platforms, and in cases where gradients potentially represent
an intensification of behavior the platform views as undesirable, they could
provide guidance on how to steer users away from these directions. Given that
gradients represent likely patterns of behavior in the user community,
significant deviations from a gradient could also be a potential signal in the
detection of bots or spam accounts.

Finally, given that we found these gradients in every location we looked -
across a range of instances in both Reddit and Wikipedia - we note that they
seem to be relatively pervasive across social media platforms. As such, future
work focusing on meta-communities should take into account that these
gradients are likely to be a baseline for any further analysis.

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##  Appendix A Additional Data Description

Given our seed topics, we used Reddit’s search function to identify a set of
starter communities for each topic. This gave us the largest communities for
any given topic. Following this, we expanded our set by utilizing the ”Related
Communities” sidebar feature containing other communities identified by the
community moderator as having similar topics. Not all related communities were
large enough for analysis and some did not align with the topic closely. We
only included communities of sufficient size and specificity. Finally, we
completed our community selection by identifying mentions as described in
Section Gradients of Informational and Referential Migration. For all
communities that we identified for each category, we retrieved all communities
mentioned by posts or comments in that community. We then included new
communities that were mentioned frequently and had significant user overlap
with at least one other community in the category. As before, we excluded
communities that were not topical, such as large generic subreddits that have
high user overlap with many communities. While there was no explicit cut-off,
we aimed for about 15 to 20 communities per category. Categories with fewer
communities than this did not have enough links to large, topical communities
for a larger selection. In practice, this size for each category allows enough
data for analysis but also ensures that all links between communities are of
sufficient strength to be worth analyzing. Our final selection presents a wide
array of highly related communities across the thirteen categories, which
gives us a broad view of community interaction ideal for this analysis
(Laumann, Marsden, and Prensky 1989).

While there are many other categories we could have analyzed and many other
choices we could have made for communities to include, these selections
represent a diverse selection of topics and demographics. This enhances the
applicability of our results, demonstrating that our results are applicable to
many different topics and communities (Tab 1). It also suggests that our
methods and results may be potentially applicable outside of Reddit.

Note also that some communities are older than others. For most communities,
we collected data from the time of creation until Fall 2022. Two communities
were banned by Reddit before then (chapotrapouse and The_Donald, both
political subreddits) and rnba was too large for this so we used a partial
dataset containing a large fraction of years but excluding early data. For any
comparative analysis, we restricted our data to only times when all
communities in a category were active. For any additional analysis, the entire
time frame was used.