# Navigating Multidimensional Ideologies with Reddit’s
Political Compass: Economic Conflict and Social Affinity

Ernesto Colacrai  Politecnico di Torino, Turin, Italy
<EMAIL_ADDRESS>,  Federico Cinus  Sapienza University, Rome,
ItalyCENTAI, Turin, Italy<EMAIL_ADDRESS>,  Gianmarco De Francisci
Morales  CENTAI, Turin, Italy<EMAIL_ADDRESS>and  Michele Starnini  Universitat
Politecnica de Catalunya, Barcelona, SpainCENTAI, Turin, Italy
<EMAIL_ADDRESS>
(2024)

###### Abstract.

The prevalent perspective in quantitative research on opinion dynamics
flattens the landscape of the online political discourse into a traditional
left–right dichotomy. While this approach helps simplify the analysis and
modeling effort, it also neglects the intrinsic multidimensional richness of
ideologies. In this study, we analyze social interactions on Reddit, under the
lens of a multi-dimensional ideological framework: the political compass. We
examine over 8 million comments posted on the subreddits /r/PoliticalCompass
and /r/PoliticalCompassMemes during 2020–2022. By leveraging their self-
declarations, we disentangle the ideological dimensions of users into economic
(left–right) and social (libertarian–authoritarian) axes. In addition, we
characterize users by their demographic attributes (age, gender, and
affluence).

We find significant homophily for interactions along the social axis of the
political compass and demographic attributes. Compared to a null model,
interactions among individuals of similar ideology surpass expectations by 6%.
In contrast, we uncover a significant heterophily along the economic axis:
left/right interactions exceed expectations by 10%. Furthermore, heterophilic
interactions are characterized by a higher language toxicity than homophilic
interactions, which hints at a conflictual discourse between every opposite
ideology. Our results help reconcile apparent contradictions in recent
literature, which found a superposition of homophilic and heterophilic
interactions in online political discussions. By disentangling such
interactions into the economic and social axes we pave the way for a deeper
understanding of opinion dynamics on social media.

Homophily, Polarization, Socio-Demographic, Reddit

††copyright: acmcopyright††journalyear: 2024††doi:
XXXXXXX.XXXXXXX††conference: The Web Conference; May 13-17, 2024;
Singapore††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Applied computing
Sociology††ccs: Human-centered computing Social networks††ccs: Information
systems Social networks Figure 1. The political compass (Lester, 1996): the
horizontal axis delineates the economic ideologies, moving from left
(equality-focused) to right (liberty-focused), and represents views on
resource allocation. The vertical axis delineates the social ideologies, from
libertarian at the bottom to authoritarian at the top, and represents views on
personal freedom.

##  1\. Introduction and background

Understanding the dynamics of opinion formation in a population is a goal
shared by researchers from different disciplines, from social and computer
science to physics. Numerous studies have revealed the presence of opinion
polarization in political discussions (McCoy et al., 2018): The phenomenon
whereby two distinct groups tend to have opposite and potentially extreme
views on a specific controversial topic, spanning religion, race, political
stance, and more (Mejova et al., 2022; Garimella et al., 2018; Schmidt et al.,
2018; Weber et al., 2013; Layman et al., 2006).

When social interactions among individuals are taken into account, we observe
homophily: individuals prefer to interact with peers that hold similar
opinions (Conover et al., 2011; Garimella et al., 2018; Cota et al., 2019).
The combination of opinion polarization and homophilic interactions leads to
_echo chambers_ (Garrett, 2009; Cinelli et al., 2021), a situation where
existing beliefs can be reinforced by exposure to similar opinions. Echo
chambers, in turn, contribute to opinion polarization by reinforcing
ideological separation and strengthening the social identity of opposing
groups (Gilbert et al., 2009; Mok et al., 2023; Del Vicario et al., 2016).
These phenomena are easy to observe on platforms like Facebook or Twitter,
where people share their opinions in informal settings (Conover et al., 2011;
Garimella et al., 2016; An et al., 2014; Bessi et al., 2016; Garimella et al.,
2018).

Other studies, however, find contrasting results (Dubois and Blank, 2018;
Guess et al., 2018; Bail, 2021; Törnberg, 2022). For instance, researchers
have observed heterophilic interactions between Clinton and Trump supporters
on Reddit, a preference for cross-cutting political interactions that
contradicts the echo chamber narrative (De Francisci Morales et al., 2021).
Likewise, some scholars attribute opinion polarization to demographic and
socioeconomic factors rather than to social media (Duca and Saving, 2016;
Storper, 2018; Monti et al., 2023). Indeed, the profound disparities and
imbalances found in our society, by gender, race, age, and affluence, have
become more closely tied to party affiliation and ideological stances (Bradley
and Chauchard, 2022; Phillips, 2022; Stewart et al., 2020). This phenomenon,
known as partisan sorting (Mason and Wronski, 2018), may be one of the causes
of political polarization (Törnberg, 2022).

The answer to this apparent contradiction may lie in recognizing the intrinsic
multidimensional nature of opinion dynamics. Indeed, the process of opinion
formation builds upon views on multiple topics, as they might be discussed at
the same time. When considering multiple topics, an interesting phenomenon
frequently observed is issue alignment, i.e., the presence of correlations
between opinions on different topics, particularly along the left–right
dimension (Freire, 2008; Falck et al., 2020). For instance, individuals with
strong religious beliefs tend to oppose abortion legalization (Adamczyk,
2022), and various other non-trivial correlations manifest (Baldassarri and
Gelman, 2008; Benoit and Laver, 2012; DellaPosta et al., 2015). However, the
prevalent view in quantitative research on opinion dynamics has focused on the
simplest case of one-dimensional opinions concerning a single topic, both at
the level of the analysis (Barberá, 2015; Conover et al., 2011) and modeling
efforts (Deffuant et al., 2000; Hegselmann and Krause, 2002; Crawford et al.,
2013; Baumann et al., 2020). Only recently researchers have started taking
into account a comprehensive multidimensional modeling framework for opinion
dynamics (Baumann et al., 2021; Schweighofer et al., 2020; Chen et al., 2021).

Given these considerations, the main research question of this work is: _“How
do interactions on social media align with respect to a fine-grained
characterization of users in terms of their ideological coordinates and
demographic features?”_ In addition, we explore how conflictual interactions
across and within ideological groups are.

To fulfill this research goal, we disentangle the social interactions of
Reddit users along two ideological axes of the political compass (Lester,
1994) as depicted in Figure 1: one for economic values and resource allocation
(left–right), and one for social values and personal freedom
(libertarian–authoritarian). We take the multidimensional political ideology
of each user from self-declared tags associated with active accounts on the
/r/PoliticalCompass and /r/PoliticalCompassMemes subreddits during the period
2020–2022. This approach allows us to avoid inference techniques and achieve
greater accuracy in the identification of users’ ideology. We infer socio-
demographic features (age, gender, affluence, and partisanship) and relate
them to the users’ ideological orientation.

Our results show that libertarians are older, richer, and more left-wing than
authoritarians, while those on the left are younger, have a higher female
representation, and are generally less wealthy than right-wing users.

Then, we examine the interactions among users along the two axes and according
to their demographics. Comparing them to those obtained from a null model of
interactions, our empirical observations show marked homophily on the social
axis and for demographic characteristics. Conversely, the interactions are
more heterophilic than expected on the economic axis. Finally, by analyzing
the toxicity of the language associated with the interactions, we find that
ideological cross-group interactions present higher-than-expected toxicity.
Within-group interactions, on the contrary, show toxicity levels lower than
the one expected from the null model, hinting at a certain degree of social
affinity (Vela-McConnell, 1999).

Overall, this multidimensional approach allows us to reconcile the apparent
contradictions observed in the literature, particularly on the superposition
of heterophily and homophily in online political discussions.

The main contributions of this paper are as follows:

  * $\bullet$

We examine a large dataset of political interactions on social media where
users provide self-declared ideological positions on the political compass,
and characterize it both in terms of ideological and demographic composition
(Section 2);

  * $\bullet$

We study the political discussions occurring in this dataset by reconstructing
the social interaction network among users, and discover that interactions
along the social axis (libertarian–authoritarian) are more homophilic than
expected (and similarly for demographic features), while interactions along
the economic axis (left–right) are more heterophilic than expected (Section
3);

  * $\bullet$

We analyze the textual content of these interactions, and find that cross-
group interactions are characterized by a higher level of toxicity, which
hints at a conflictual discourse between axes’ poles (Section 4).

The main body of the paper describes the results of our analysis for
/r/PoliticalCompass. The ones for /r/PoliticalCompassMemes are qualitetively
and quantitatively similar and are shown in the Appendix. To allow for
reproducibility our code is available
online.111https://anonymous.4open.science/r/political-compass-D741

##  2\. Ideology and Demographics of Users

###  2.1. Ideological Features

Reddit’s Political Compass is a community that self-describes as “A subreddit
for posting and discussing test results as well as political self-tests and
political theory”. On Reddit, _flairs_ are tags that users add to their
usernames or posts for additional context or to signal their identity. In the
/r/PoliticalCompass (/r/PC) and /r/PoliticalCompassMemes (/r/PCM) subreddits,
these flairs act as self-declarations and specify the political ideologies of
users. Flairs are organized along the two axes shown in Figure 1: economical
(left, center, right) and social (libertarian, center, authoritarian).

Using the PushShift API,222https://github.com/pushshift/api we collect
comments and posts from /r/PC and /r/PCM. Henceforth, we analyze a dataset
covering the period from 2020 to 2022, which is the most active one, see
Figure A.1 in the Appendix. The dataset comprises $79\,368$ posts and
$952\,550$ comments for /r/PC and $383\,169$ posts and $22\,653\,346$ comments
for /r/PCM, respectively. These represent 95% and 96% of the entire content in
/r/PC, and 96% and 98% in /r/PCM.333Our decision to focus on the 2020-2022
time frame is justified by this high coverage. Furthermore, while the /r/PC
subreddit has been active since 2012, political ideologies have become a
prominent feature only since November 2019.

For our analysis, we select real active users identified with a unique
political ideology, thereby filtering out potential bots (Rollo et al., 2022),
deleted accounts, those not affiliated with any political ideology or
affiliated to multiple ideologies.444Within the /r/PC dataset for 2020-2022,
$22\,503$ users are aligned with a single ideology, whereas $2544$ have
multiple ideological affiliations. In /r/PCM, these numbers are $258\,428$ and
$30\,658$ respectively. We defer the analysis of users with multiple
ideological declarations to a further study. The breakdown of posts and
comments by these user types is detailed in Table A.1.

Economic Axis. The economic or distributive axis measures possible opinions of
how people should be endowed with resources. The _left_ (equality) pole is
defined as the view that assets should be redistributed by a cooperative
collective agency: the state in the socialist tradition, or a network of
communes in the libertarian or anarchist tradition. The _right_ (liberty) pole
is defined as the view that the economy should be left to the market system,
to voluntarily competing individuals and organizations. This is the classical
left–right conflict that dominated the Cold War (Lester, 1994, 1995, 1996;
Petrik, 2010).

Social Axis. The other axis—cross-cutting the first one—is concerned with
values of fraternity, understood as axiological principles driving
institutionalization, community, forms and actors of democracy, and the
quality of the process of collective outcomes. This dimension measures
possible political opinions either in a communitarian or procedural sense,
considering the appropriate amount of personal freedom and participation:
_libertarianism_ is defined as the idea that personal freedom as well as
voluntary and equal participation should be maximized. This would entail the
full realization of liberty in a democratic sense. Parts of that view are
ideas like autonomous, direct democratic institutions beyond the state and
market, the transformation of gender roles, and self-determination over
traditional and religious orders. On the opposite end of the axis,
_authoritarianism_ is defined as the belief that authority and religious or
secular traditions should be complied with. Equal participation and a free
choice of personal behavior are rejected as being against human nature, or
against necessary hierarchies for a stable society (Lester, 1994, 1995, 1996;
Petrik, 2010).

Collected data is representative of all classes. Figure 2 shows a
comprehensive overview of user engagement and content distribution across
various political ideologies for /r/PC (Figure B.3 for /r/PCM). This includes
a breakdown of the number of users and comments classified by political
ideology. The goal is to provide a perspective on the ideological composition
and activity within the community. The plot shows that the collected data is
representative of all classes, in terms both of users and activity.
Additionally, Figure A.1 in the Appendix displays the temporal distribution of
posts and comments. It shows a marked increase starting from 2020, especially
considering the ones authored by users with an ideological flair (self-
declaration).

Ethics statement. This work follows the guidelines and the ethical
considerations by Eysenbach and Till (2001); Moreno et al. (2013); Ramírez-
Cifuentes et al. (2020). All the results provide aggregated estimates and do
not include any information that focuses on single authors. The users in our
study were fully aware of the public nature and free accessibility of the
content they posted since the subreddits are of public domain, are not
password-protected, and have thousands of active subscribers. Reddit’s
pseudonymous accounts make the retrieval of the true identity of users
unlikely, therefore our research did not require informed consent. Since this
research does not involve interventions, no approval was required by our
Institutional Review Board.

Figure 2. Composition of /r/PC in terms of the number of users and comments
classified by political ideology. Figure 3. For each axis (social and
economic) displayed in the rows, and for each user characteristic (age,
gender, affluence, partisanship) displayed in the columns, the plot shows the
average quantile score and its 95% confidence intervals for /r/PC.
Libertarians tend to be older, while right-leaning users are predominantly
male. Left-leaning users are more likely to be female. Additionally, a
correlation between partisanship and the left–right economic axis emerges,
which further validates our data collection methodology.

###  2.2. Demographic Features

Inferring socio-demographics. To infer the socio-demographic characteristics
of Reddit users—including age (old/young), gender (male/female), affluence
(poor/rich), and partisanship (left/right)—we employ the unsupervised
representation learning approach by Waller and Anderson (2021). This method
assigns to each subreddit a z-score for each of these four
characteristics.555The study by Waller and Anderson (2021) uses the 10k most
popular subreddits from 2004-2018. Since our dataset begins in 2020, we focus
on the overlap between our dataset and theirs, with a coverage of 57% in /r/PC
and 68% in /r/PCM. As a result, socio-demographic characteristics could not be
determined for 4% of users in /r/PC and 5% in /r/PCM.

To determine the z-score for each user $u$ based on a characteristic $c$ from
the set $\\{\text{age},\text{gender},\text{affluence},\text{partisanship}\\}$,
$z_{u}^{(c)}$, we compute a weighted mean of the z-scores $z_{s}^{(c)}$ of all
subreddits $s\in S$. The weight is determined by the number of comments
$N_{u,s}$ that user $u$ posted in subreddit $s$. The weighted average is thus:

(1) $z_{u}^{(c)}=\frac{\sum_{s\in S}N_{u,s}z_{s}^{(c)}}{\sum_{s\in
S}N_{u,s}}.$

Subsequently, we normalize the z-scores of users by using quantiles for each
characteristic. These normalized scores will henceforth be referred to as
quantile scores ($Q$). Quantile scores in the top 25% are classified as
“high”, those in the bottom 25% as “low”, and quantile scores in between
belong to the reference class. For instance, within the age characteristic, a
high score would correspond to an “old” user, while a low score represents a
“young” one.

We stress that a score indicating an old-leaning user does not imply that such
a user is necessarily old, as Reddit is known to be participated by more young
than old users (Barthel et al., 2016). Rather, it indicates a user who
frequents subreddits more likely to be participated by older users. A similar
argument holds for the other categories: demographic attributes are always to
be considered relative to the Reddit user base, and never on an absolute
basis.

Political declarations align with inferred ideologies. Figure 3 illustrates
the distribution of demographic characteristics among different political
ideologies for /r/PC. In each plot, the y-axis represents the average quantile
score of a demographic characteristic for users with the corresponding
ideology on the x-axis. Notably, the distribution of inferred ideologies
(left/right) closely mirrors the declared ideologies from the Political
Compass along the economic axis, lending validation to our analytical
approach. Exactly the same holds for /r/PCM, as depicted in Figure B.4.

Authoritarian and left-wing users are younger. Figure 3 reveals noticeable
variations in the characteristics of Reddit users based on their diverse
socio-demographic attributes and political ideologies on /r/PC. Libertarians
are typically older, wealthier, and more left-leaning compared to
authoritarians. On the contrary, left-wing users tend to be younger and have a
higher female representation but are generally less rich compared to right-
wing users. Such results hold also for /r/PCM, as shown in Figure B.4.

##  3\. Homophily and Heterophily
in social interactions

This section showcases the results regarding interactions among users grouped
by political ideology and demographics. We start by describing the
reconstruction of the interaction network, and the levels of homophily and
heterophily among the distinct groups. Table 1 provides summary statistics for
these networks.

###  3.1. Reconstructing the Interaction Network

The interaction network is represented as a directed weighted graph
$G=(V,E,w)$, where users are nodes $V$ and the edges $E$ correspond to
interactions between them. An edge $(u,v)\in E$ indicates that user $u$
(source) replied to user $v$ (target) in a thread on Reddit. Each edge carries
a weight $w_{uv}$ which reflects the number of interactions between $u$ and
$v$.

Table 1. Network statistics: number of users $|V|$, edges $|E|$, average degree $\langle d\rangle$, and total number of interactions $W$. Subreddit | $|V|$ | $|E|$ | $\langle d\rangle$ | $W$
---|---|---|---|---
/r/PC | $18\,135$ | $173\,672$ | $9.58$ | $261\,078$
/r/PCM | $215\,111$ | $6\,197\,901$ | $28.81$ | $8\,065\,395$

Interaction Probabilities. For a user with ideology $X_{s}$ from {libertarian,
center, authoritarian} ($\\{B,C,A\\}$ for simplicity) on the social axis ($s$)
and $X_{e}$ from {left, center, right} ($\\{L,C,R\\}$) on the economic axis
($e$), the probability of observing them is given by $P(u=X)=N_{X}/|V|,$ where
$|V|$ is the total number of users, and $N_{X}$ stands for those users
identified with the ideology $X$. In particular, for /r/PC, the probability to
find a node labelled as $X_{s}\in\\{B,C,A\\}$ (or $X_{e}\in\\{L,C,R\\}$), is
respectively $P(B)\simeq 0.48$, $P(C)\simeq 0.30$, $P(A)\simeq 0.23$ (or
$P(L)\simeq 0.31$, $P(C)\simeq 0.38$, $P(R)\simeq 0.31$).

Joint probabilities. When considering a specific ideological axis, either
social or economic, the joint interaction probability between a user of
ideology $X$ and another of ideology $Y$ is $P_{X\to Y}=\frac{W_{X\to Y}}{W}$,
where $W$ is the total number of interactions in the network, and

$W_{X\to Y}=\sum_{u,v\in V:u=X\wedge v=Y}w_{u,v}\vspace{-1mm}$

is the total weight of directed edges from $X$ to $Y$. Empirically, in /r/PC
we find:

$\displaystyle P_{X_{s}\to Y_{s}}$
$\displaystyle\simeq\>\>\>\>\bordermatrix{&B&C&A\cr B&0.21&0.11&0.13\cr
C&0.11&0.08&0.08\cr A&0.12&0.07&0.10}\>,$

$\displaystyle P_{X_{e}\to Y_{e}}$
$\displaystyle\simeq\>\>\>\>\bordermatrix{&L&C&R\cr L&0.13&0.12&0.12\cr
C&0.12&0.12&0.09\cr R&0.12&0.09&0.08}.$

Conditional probabilities. Let $W_{X\to}=\sum_{Y}W_{X\to Y}$ be the number of
interactions generated from $X$, and $W_{\to Y}=\sum_{X}W_{X\to Y}$ those
received by $Y$. To take into account the volume of interactions from the
different groups (e.g., $P_{A\to}\simeq 0.29<P(A)\simeq 0.23$, $P_{B\to}\simeq
0.45<P(B)\simeq 0.48$), we consider the conditional probability of getting an
interaction $X\to Y$ given a source with ideology $X$

(2) $\displaystyle P_{X\to Y\mid X}=\frac{P_{X\to Y}}{P_{X\to}}=\frac{W_{X\to
Y}}{W_{X\to}}.$

Empirically, for /r/PC we have

(3) $\displaystyle P_{X_{s}\to Y_{s}\mid X_{s}}$
$\displaystyle\simeq\>\>\bordermatrix{&B&C&A\cr B&0.47&0.25&0.28\cr
C&0.42&0.28&0.29\cr A&0.41&0.25&0.34}\>,$

(4) $\displaystyle P_{X_{e}\to Y_{e}\mid X_{e}}$
$\displaystyle\simeq\>\>\bordermatrix{&L&C&R\cr L&0.36&0.32&0.32\cr
C&0.37&0.35&0.27\cr R&0.42&0.29&0.29}.$

If people interact with each other irrespective of their ideology, we would
expect everyone to have an equal chance of interacting with each group,
depending only on the group size. However, we observe that this is not the
case. Instead, Equations 3 and 4 show important deviations from this
expectation, in addition to differences between the two ideological axes.

On the social axis, the interactions tend to be more homophilic than
heterophilic (the on-diagonal entry has a larger weight in each column). This
fact is especially evident between libertarians and authoritarians:
authoritarians engage more with authoritarians (34%) compared to how much
libertarians do (28%), and a similar pattern is observed in the opposite
direction for libertarians (47% vs. 41%). Centrists also show reduced
interactions with both groups and favor instead their own group.

Surprisingly, on the economic axis the trend inverts. There is a noticeable
heterophily between left and right users (the off-diagonal entries have a
larger weight in the respective columns). Left users receive more interaction
from right ones (42%) than from within their group (36%), and the opposite
holds true for right users (29% from the right vs. 32% from the left).

###  3.2. Null model of Social Network

To determine whether the patterns observed in the previous section are
statistically sound, we need to compare the empirical interaction network with
a null model that ignores political ideologies. We use a configuration model,
which is a directed, weighted random network (RN) that preserves the in and
out-degree sequences but rewires the connections among them. This way, we
ensure that the political ideology of users does not affect their
interactions.

Conditional probabilities in the null model. The conditional probability of
getting an interaction $X\to Y$, given the source’s ideology $X$ in the null
model is independent of the class $X$ of the source. Instead, it depends on
the in-degree of the target’s class $Y$

(5) $\displaystyle P^{RN}_{X\to Y\mid X}$ $\displaystyle=\frac{P^{RN}_{X\to
Y}}{P_{X\to}}=\frac{W_{\to Y}}{W}.$

For the /r/PC dataset, the conditional probabilities for the social and
economic axes are

(6) $\displaystyle P^{RN}_{X_{s}\to Y_{s}\mid X_{s}}$
$\displaystyle\simeq\>\>\bordermatrix{&B&C&A\cr B&0.44&0.26&0.30\cr
C&0.44&0.26&0.30\cr A&0.44&0.26&0.30},$

(7) $\displaystyle P^{RN}_{X_{e}\to Y_{e}\mid X_{e}}$
$\displaystyle\simeq\>\>\bordermatrix{&L&C&R\cr L&0.38&0.32&0.30\cr
C&0.38&0.32&0.30\cr R&0.38&0.32&0.30}.$

Figure 4. Odds ratios between empirical and random conditional probabilities
of interaction for /r/PC, with respect to social (left) and economic (right)
axes. The interactions show a homophilic pattern on the social axis (higher
values in the main diagonal) and a heterophilic one on the economic axis
(higher values in the anti-diagonal).

###  3.3. Effect of Ideologies

Homophily within Social Ideologies. Figure 4 illustrates that political
ideologies on the social axis (libertarian–authoritarian) for /r/PC exhibit
significant homophily. Interactions within the same ideology (on-diagonal) are
up to $12\%$ more likely than those predicted by the null model. The
interaction probabilities both in receiving and sending a comment from/to a
differing ideology are approximately symmetric. This pattern holds for /r/PCM
too, as shown in Figure B.5, where interactions between authoritarian users
are up to $18\%$ more likely than expected.

Heterophily across Economic Ideologies. Conversely, the right side of Figure 4
shows a pronounced heterophily across ideologies on the economic axes. Left-
and right-leaning users are approximately $10\%$ more likely to interact with
each other than what the null model predicts. This phenomenon of increased
interaction between left and right is even more prominent in the /r/PCM
subreddit, where is observed $15\%$ more likely than predicted, see Figure
B.5. Furthermore, within-group interactions for left and right are
considerably lower than expected, while the economic center group shows more
within-group interactions.

###  3.4. Effect of Demographics

Next, we test how social interactions are associated with demographic factors.
To this aim, we employ the logistic regression model by Monti et al. (2023)
which outputs the probability of interaction between demographic groups and
validates the statistical significance of the results.

Logistic regression model. We train the model for a link prediction task over
the directed interaction graph $G=(V,E)$ discussed in Section 3.1. For a given
node pair $u,v\in V$ the target variable is:

(8) $y_{u,v}=\begin{cases}1,&\text{if }(u,v)\in E\\\
0,&\text{otherwise.}\end{cases}$

We assume that the likelihood of observing an interaction $u\to v$ might be
influenced by the combined feature values of both $u$ and $v$. Thus, the
independent variable can be represented as:

(9) $\mathbf{X}_{u,v}\in\\{0,1\\}^{d}\subseteq F\times F$

where $F$ is the set of all feature values. Specifically, for demographic
features, we take $F$ to be $F:=\\{$young, old, male, female, poor, rich$\\}$,
and $d=|F|^{2}$. In this context, the logistic regression model is trained on
a series of pairs $(y_{u,v},\mathbf{X}_{u,v})$, producing as output
coefficients a feature-feature matrix $M$. This estimated matrix provides
insights into which demographic groups have a higher probability of
interacting. More explicitly, each matrix entry $a_{ij}$ denotes the log-odds
ratio of a node with feature value $i$ to interact with a node with feature
value $j$, compared to the probability of random interactions given by the
null model.

To define the non-existing edges for the training (where $y_{u,v}=0$), we
employ the configuration model in Section 3.2. By shuffling the existing
interaction network edges, we balance positive and negative edge examples. We
choose edges based on the activity of the source node $u$ and the
attractiveness of the target node $v$ (De Francisci Morales et al., 2021). If
such a pair already exists, it is omitted. This method ensures that the
estimated matrix $M$ correctly represents how the demographic features make
the observed edges deviate from this null model.

Homophily is evident in demographic features. Figure 5 reveals pronounced
homophily among users based on their demographic features: pairs on the
diagonal (i.e., within-class interactions) occur more frequently than
expected. Especially the interaction among ‘old’ users is 18% more likely than
expected, and interaction among ‘young’ and ‘female’ users is 10% more likely.
Figure B.6 shows the same pronounced homophily for /r/PCM, especially
interactions among ‘young’, ‘old’, and ‘poor’, which are respectively $15\%$,
$12\%$ and $8\%$ more likely than expected. Conversely, off-diagonal elements
show a lower-than-expected interaction probability between users with
different demographic features. In Figure B.6 for /r/PCM this effect is
particularly notable for the age feature, where ‘old’ and ‘young’ interactions
are infrequent (odds ratios below $0.90$).

Figure 5. Odds ratio (exponentiated logistic regression coefficients) for each
ordered pair of interacting features on /r/PC. The source user is in the rows
and the target user in the columns. Only coefficients significant at the
$\alpha=5\%$ level are shown. The results show homophily in the demographic
attributes with higher values in the main diagonal.

###  3.5. Modeling the Combined Effects of Ideological and Demographic
Features

Next, we analyze the interplay between ideological and demographic features
and their joint effect on user interactions. Additionally, we recognize the
potential influence of a user’s popularity on Reddit (confounding), which
might skew interactions.

To ensure the significance of our findings, we utilize the logistic regression
approach introduced in Section 3.4. Aware of the risk of multicollinearity,
where independent variables are highly correlated and can distort results, we
opt for a selected combination of ideological and demographic values rather
than examining all pairwise combinations. As indicated by Equation 9, the
independent dummy variables for our analysis are denoted as
$\mathbf{X}_{u,v}\in\\{0,1\\}^{29}$. The 29 selected features include:

i. For ideological features, we consider the following set of pairwise
features associated with each source-target edge ($u\to v$). Given
$X_{u},X_{v}\in\\{\mathrm{left},\mathrm{right}\\}$, we define

  * •

Economic Homophily $=1$ if $X_{u}=X_{v}$

  * •

Economic Heterophily $=1$ if $X_{u}\neq X_{v}$

Given $X_{u},X_{v}\in\\{\mathrm{libertarian},\mathrm{authoritarian}\\}$, we
define

  * •

Social Homophily $=1$ if $X_{u}=X_{v}$

  * •

Social Heterophily $=1$ if $X_{u}\neq X_{v}$

In both cases, users labeled as ‘center’ do not contribute to the features.
The directionality of the interaction is taken into account in a separate
feature denoted by an arrow. This design choice allows to separate
homophilic/heterophilic effects and their possible asymmetry. Therefore, we
consider 6 possible ideologic features: 2 economic features, 2 social
features, and 2 features for asymmetry.

ii. For demographic features, we consider all the pairwise combinations of
values of each feature (age: young-old, gender: male-female, affluence: poor-
rich) as those employed to analyze the effect of demographics. Therefore, we
have $6\times(6+1)/2=21$ possible demographic features.

iii. For the confounding effects, we define the popularity of a user on a
subreddit as the average score666A comment’s score is the difference between
its received upvotes and downvotes. of its comments on that subreddit. Figure
A.2 shows the distribution of popularity of users for both /r/PC and /r/PCM.
Then we quantile-normalize such scores to define 4 classes of popularity, from
low to high score, and we consider the top and bottom quartiles as two
distinct binary features. By taking into account the target’s popularity, we
define

  * •

Target is popular $=1$ if the target is in the top quartile

  * •

Target is not popular $=1$ if it is in the bottom quartile

We thus add 2 confounding features for popularity.

Figure 6 reports the results of the logistic regression model.

Heterophily in economic ideologies. Having opposite economic ideologies
increments the odds of an interaction by 10% in /r/PC (Figure 6) and higher
than 16% in /r/PCM (Figure B.7). This effect is mirrored by a decrement of the
odds by 5% if both users have the same economic ideology for /r/PC, and by 12%
for /r/PCM. Results for directional heterophily are not statistically
significant both in /r/PC and /r/PCM.

Homophily in social ideologies. Conversely, pairs of users with the same
social ideology are more likely to interact, with an increment of the odds of
8% for /r/PC (Figure 6) and slightly higher than 5% for /r/PCM (Figure B.7).
Similarly, heterophilic interactions across different social ideologies are
less likely by 7% for /r/PC and slightly higher than 5% for /r/PCM. Results
for directional heterophily are not statistically significant both in /r/PC
and /r/PCM.

Homophily in demographics. Age homophily significantly predicts interactions,
with a 13% increase in odds for old-old pairs and a 6% increase for young-
young pairs in /r/PC (Figure 6). In /r/PCM (Figure B.7), these odds increase
by 13% and 14%, respectively. Demographic homophily in terms of poor-poor,
female-female, and male-male interactions shows higher likelihood, with odds
increasing by 8%, 7%, and 5% for /r/PC and 8%, 4%, and 2% for /r/PCM. Results
for rich-rich interactions are not statistically significant for /r/PC.
However, in /r/PCM, they cause an odds increment of 4%.

No heterophily in demographics. Generally, heterophily in demographics reduces
the odds of interactions. In particular, poor-rich and young-old interactions
in /r/PC decrease odds both by 4% (Figure 6). In /r/PCM (Figure B.7), these
reductions are 5% and 11%.

Furthermore, cross-feature interactions generate different results. Old-poor
and old-female interactions in /r/PC decrease odds by 4% and 10% (Figure 6).
In /r/PCM (Figure B.7), these reductions are both of 4%. Old-rich interactions
cause a 3% increase in odds for both /r/PC and /r/PCM. Female-rich and young-
rich interactions yield opposite results, with odds increasing in /r/PC and
decreasing in /r/PCM. Old-male and young-male interactions lead to a decrease
in odds for /r/PC but are not statistically significant for /r/PCM. Young-
female interactions are not statistically significant for /r/PC but result in
an odds increment for /r/PCM.

Figure 6. Coefficients and 99% confidence intervals for logistic regression
features on /r/PC. The features are displayed in the rows and they represent
pairs of classes in ideological characteristics (blue), demographic
characteristics (orange), or popularity (green). A coefficient greater than 0
means a positive impact on the likelihood of interaction. Statistical
significant results are highlighted with full markers.

##  4\. Toxicity in Social Interaction

We found a higher rate of heterophilic interactions on the economic axis of
the political compass. Here we show that heterophilic interactions are in
general characterized by higher _toxicity_ , which hints at conflictual
interactions between axes poles. Toxicity is defined as “a rude,
disrespectful, or unreasonable comment that is likely to make you leave a
discussion”.777https://developers.perspectiveapi.com/s/about-the-api-
attributes-and-languages

Estimating language toxicity. We use Google’s Perspective
API888https://perspectiveapi.com to determine the toxicity scores of a sample
of 100k comments from both /r/PC and /r/PCM, excluding comments containing
only emojis or links. A toxicity score $\tau$ ranges between 0 (lowest) and 1
(highest), and indicates the likelihood that an individual perceives the text
as toxic. For each pair of interacting ideologies ($X,Y$),
$\overline{\tau}_{\scaleto{X\rightarrow Y}{4pt}}$ denotes the average toxicity
of the comments exchanged.

We then compare this empirical measure against the one obtained in a null
model of interactions that preserves the degree sequences and the toxicity
distribution. We sample 100 randomized interaction networks from the null
model for each subreddit via an MCMC edge-swapping algorithm (Fosdick et al.,
2017). Randomization is achieved by executing $Q\cdot|E|$ edge endpoint swaps,
with $|E|$ being the edge count of the network. Although there is no definite
proof for Markov Chain convergence, we adhere to a conservative value of
$Q=\log|E|$, as recommended by Uzzi et al. (2013).

Heterophilic interactions show higher toxicity. Figure 7 depicts the ratio of
the average empirical toxicity to the average toxicity in the null model for
/r/PC. On both axes, heterophilic comments present a higher toxicity than
expected, while homophilic comments present a lower one (on-diagonal values
are lower than 1, while off-diagonal ones are higher than 1). On the economic
axis, heterophilic comments (i.e., between left- and right-leaning users) are
4-to-7% more toxic than expected, and statistically significant in both
directions (p-values of $0.0004$ and $<10^{-6}$). Conversely, interactions
between users of the same ideology demonstrate lower-than-expected toxicity
with an effect of 8-to-12% (p-values $<10^{-6}$).

Regarding the social axis, homophilic comments between users of identical
political ideologies are 5-to-9% less toxic than expected (p-values
$<10^{-6}$). Instead, comments from authoritarians towards libertarians are
significantly more toxic (10%, p-value $<10^{-6}$), with a similar albeit
smaller effect in the opposite direction.

People who interact with others who have different political views (Figure 4)
are more likely to engage in toxic behavior, while people who interact with
others who have the same political views are less likely to engage in toxic
behavior.

The same considerations apply to /r/PCM, but the results are less evident
and/or not statistically significant, as shown in Figure B.8. This suggests
that a more humorous discussion involving memes, as those in /r/PCM, may be
less conflictual than a serious discussion, such as those in /r/PC.

Figure 7. Ratio between the average toxicity in empirical data
($\overline{\tau}_{\mathrm{emp}}$) and random null model
($\overline{\tau}_{\mathrm{null}}$) for /r/PC. Values above 1 indicate higher
toxicity than expected. Non-significant results at the $\alpha=5\%$ level are
omitted (via a t-test). The left heatmap shows interactions between ideologies
on the social axis, and the right heatmap focuses on the economic axis.
Heterophilic interactions (lib-auth, left-right) exhibit higher toxicity than
expected.

##  5\. Discussion

In our quantitative analysis of interaction patterns on Reddit’s communities
/r/PoliticalCompass and /r/PoliticalCompassMemes from 2020 to 2022, we
uncovered dynamics that go beyond the conventional left–right political
dichotomy. Users with similar social ideologies and demographics interacted
more frequently and typically used non-toxic language, which denotes a high
level of homophily on the social axis of the political compass. Instead,
conversations reflecting “social” heterophily, particularly between
authoritarian and libertarian ideologies, occurred 6% less frequently than
expected. In contrast, heterophilic and conflictual interactions were
pronounced on the economic axis, with users of opposing economic ideologies
displaying higher signs of toxicity and engaging with each other 10% more than
expected.

In light of our results, some of the apparently puzzling differences in the
recent literature may be reconciled. On the one hand, social media platforms
such as Twitter and Facebook, which emphasize social connections, reportedly
exhibit a high level of homophily (De Choudhury, 2011). The presence of
political echo chambers (Cinelli et al., 2021; Garimella et al., 2018; An et
al., 2014) is therefore to be expected if this social homophily is a driver of
the interactions, as suggested by our results. On the other hand, platforms
such as Reddit, where status homophily is less dominant and interactions are
centered on shared topical interests, tend to experience more conflictual
interactions across ideologies (Törnberg, 2022). This fact is particularly
true for Reddit’s political spaces, which skew towards the U.S., a nation
where the political spectrum largely corresponds to a singular economic left-
right dimension (Klar, 2014). As a result, echo chambers may be rarer on
platforms like Reddit (Cinelli et al., 2021; De Francisci Morales et al.,
2021; Monti et al., 2023). Building on our findings, it is evident that the
nature of interactions, in terms of toxicity along with the specific
ideological axis, significantly influences the manifestation of homophily and
heterophily.

Our findings also highlight the inclination of users to group based on
demographics and similar social ideologies, hinting at a certain degree of
social affinity (Vela-McConnell, 1999). Belonging to such digital echo
chambers provides a sense of community but also poses risks. Being
predominantly exposed to only one type of ideology can fuel misinformation
(Törnberg, 2018; Del Vicario et al., 2015). It can strengthen existing biases
and further spread incorrect beliefs. If these online tendencies continue,
they might intensify real-world divisions with consequences in voting patterns
and everyday interactions (Rhodes, 2022).

Limitations and future work. Our study is not exempt from limitations.
Firstly, our dataset lacks geographical granularity; it focuses on English-
speaking users predominantly from a U.S.-centric platform, Reddit. Future
studies should aim to infer and incorporate geographical locations of users to
control for regional effects and limit sampling bias. Extracting self-
declarations from plain text or inferring them via machine learning models
could extend the dataset to additional subreddits and social networks, thus
improving result generalizability. Secondly, our reliance on self-declarations
means that we excluded users who change their political orientation in time.
An interesting direction for future research is analyzing such opinion
changes, under the lens of the political discussions that occurred within the
subreddit and outside of it.

Moreover, user declarations may not strictly align with the true ideology of
users nor with the political content in the observed subreddits. While
declared ideologies along the economic axis are validated by comparison to
left/right ideologies inferred by following Waller and Anderson (2021), we
could not validate self-declarations along the social axis. Future work could
be devoted to building a more comprehensive embedding of Reddit users that
takes into account their political stance in a multidimensional way, e.g., by
adding the libertarian–authoritarian axis.

Likewise, the textual content of the interactions is a rich resource that can
yield further insights and has not yet been fully utilized.

Lastly, our findings pave the way for multidimensional modeling and
intervention studies. For instance, a social compass model has been recently
proposed to explore depolarization dynamics in multidimensional topics
represented in a polar space (Ojer et al., 2023a, b). Similarly interesting
directions for future work are the exploration of algorithms that offer
diverse content by taking into account the multidimensional nature of targeted
users (Garimella et al., 2017a, b).

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##  Appendix A Dataset Description

This Appendix provides additional information regarding the data sets used in
this work, from /r/PC and /r/PCM. Table A.1 reports the fractions of deleted
users and users with unique or non-unique political flairs in both subreddits.
Figure A.1 shows the number of posts and comments in the years from 2018 until
2022 in both subreddits. Figure A.2 shows the probability density function of
activity and popularity of users in both subreddits.

Table A.1. Fraction of posts and comments by user category in /r/PC and
/r/PCM. The rows represent the following user types, in order: deleted users,
bots (as listed in (Rollo et al., 2022)), users lacking a unique political
declaration, and users with a unique political declaration. The last row
indicates the fraction of analyzed data.

Subreddit | /r/PC | /r/PCM
---|---|---
User Type | Posts | Comments | Posts | Comments
Deleted |  $34.72$% |  $9.49$% |  $30.25$% |  $10.82$%
Bots |  $0.25$% |  $0.6$% |  $0.73$% |  $2.57$%
Non-unique |  |  |  |
political flairs |  $28.41$% |  $30.83$% |  $21.98$% |  $22.28$%
Unique |  |  |  |
political flairs |  $36.62$% |  $59.08$% |  $47.04$% |  $64.33$%

Figure A.1. Number of posts and comments over time in both subreddits. Figure
A.2. Activity ($a$) and popularity (average score $s$) distributions for both
subreddits.

##  Appendix B Additional Figures for /r/PoliticalCompassMemes

This Appendix shows additional figures for /r/PCM, analogous to the ones
presented in the main text. In all cases, results are qualitetively and
quantitatively similar to /r/PC.

Figure B.3 shows the number of users and comments grouped by political
ideology for /r/PCM, equivalent to Fig. 2 in the main text for /r/PC. Figure
B.4 shows the mean and confidence intervals of quantiles for each socio-
demographic feature with respect to ideologies for /r/PCM. Figure B.5 shows
the odds ratios between empirical and random conditional probabilities of
interaction for /r/PCM, equivalent to Figure 4 in the main text for /r/PC.
Figure B.6 shows the odds ratios for each ordered pair of demographic features
for /r/PCM, equivalent to Figure 5 in the main text for /r/PC. Figure B.7
shows the the coefficients and confidence intervals for logistic regression
features for /r/PC, equivalent to Figure 6 in the main text for /r/PC. Figure
B.8 shows the ratio between average toxicity in empirical data and random null
model for /r/PC, equivalent to Figure 7 in the main text for /r/PC.

Figure B.3. Composition of /r/PCM in terms of the number of users and comments
classified by political ideology. Figure B.4. For each axis (social and
economic) displayed in the rows, and for each user characteristic (age,
gender, affluence, partisanship) displayed in the columns, the plot shows the
average quantile score and its 95% confidence intervals for /r/PCM.
Libertarians tend to be older, while right-leaning users are predominantly
male. Left-leaning users are more likely to be female. Additionally, a
correlation between partisanship and the left–right economic axis emerges,
which further validates our data collection methodology. Figure B.5. Odds
ratios between empirical and random conditional probabilities of interaction
for /r/PCM, with respect to social (left) and economic (right) axes. The
interactions show a homophilic pattern on the social axis (higher values in
the main diagonal) and a heterophilic one on the economic axis (higher values
in the anti-diagonal). Figure B.6. Odds ratio (exponentiated logistic
regression coefficients) for each ordered pair of interacting features on
/r/PCM. The source user is in the rows and the target user in the columns.
Only coefficients significant at the $\alpha=5\%$ level are shown. The results
show homophily in the demographic attributes with higher values in the main
diagonal.  Figure B.7. Coefficients and 99% confidence intervals for logistic
regression features on /r/PCM. The features are displayed in the rows and they
represent pairs of classes in ideological characteristics (blue), demographic
characteristics (orange), or popularity (green). A coefficient greater than 0
means a positive impact on the likelihood of interaction. Statistical
significant results are highlighted with full markers. Figure B.8. Ratio
between the average toxicity in empirical data
($\overline{\tau}_{\mathrm{emp}}$) and random null model
($\overline{\tau}_{\mathrm{null}}$) for /r/PCM. Values above 1 indicate higher
toxicity than expected. Non-significant results at the $\alpha=5\%$ level are
omitted (via a t-test). The left heatmap shows interactions between ideologies
on the social axis, and the right heatmap focuses on the economic axis.
Heterophilic interactions (Lib-Auth, Left-Right) exhibit higher toxicity.