# Anatomy of Elite and Mass Polarization in Social Networks

Ali Salloum  Ted Hsuan Yun Chen  Department of Environmental Science and
Policy, George Mason University  Mikko Kivelä

###### Abstract

Existing methods for quantifying polarization in social networks typically
report a single value describing the amount of polarization in a social
system. While this approach can be used to confirm the observation that many
societies have witnessed an increase in political polarization in recent
years, it misses the complexities that could be used to understand the reasons
behind this phenomenon. Notably, opposing groups can have unequal impact on
polarization, and the elites are often understood to be more divided than the
masses, making it critical to differentiate their roles in polarized systems.
We propose a method to characterize these distinct hierarchies in polarized
networks, enabling separate polarization measurements for these groups within
a single social system. Applied to polarized topics in the Finnish
Twittersphere surrounding the 2019 and 2023 parliamentary elections, our
analysis reveals valuable insights: 1) The impact of opposing groups on
observed polarization is rarely balanced, and 2) while the elite strongly
contributes to structural polarization and consistently display greater
alignment across various topics, the masses have also recently experienced a
surge in issue alignment, a special form of polarization. Our findings suggest
that the masses may not be as immune to an increasingly polarized environment
as previously thought.

##  1 Introduction

Distinct aspects of political polarization are reportedly higher today in many
societies [ILL+19, Sil18, KYA21, FZQ+23, BG20, KC24]. It is a serious and
consequential phenomenon that has garnered significant attention in recent
years [FGT+22, XGM+22, FGF+23, Kaw22]. It refers to the increasing divergence
of political attitudes, beliefs, and values between individuals and groups
[DEB96, FA08], often leading to heightened hostility [FH23, BC18], ideological
rigidity [WCY+22], and a breakdown in constructive dialogue [FH23]. As
polarization intensifies, it poses significant challenges to democratic
societies [MRS18], hindering effective governance [Jon01] and compromising
social cohesion [Mas15]. Hence the implications of political polarization are
far-reaching, affecting policy-making [WHD+21], public discourse [RM16], and
even personal relationships [WCP21, GS23]. Understanding the causes, dynamics,
and consequences of political polarization is crucial in order to address its
negative impacts.

Current methods for quantifying polarization in social networks assign a
single score to an entire social system (e.g., [GMGM18, GMJCK13, MBLB15,
LAD24, GAS+15]), simplifying the complexity of the underlying phenomenon.
While these methods are often intuitive and straightforward to implement, they
produce high polarization values for random networks [SCK22] and fail to
capture many crucial nuances related to the multifaceted nature of
polarization. They do not distinguish between the elite and the mass
polarization, nor do they reveal whether opposing groups contribute equally to
the observed division in a given network. These methods are also typically
designed to measure the decreasing communication between groups and/or
increasing cohesion within them, thus missing other aspects of polarization,
such as political alignment [Mas15, Tör22, CSG+21].

We argue that knowing whether both conceptually and structurally dominant
individuals are more polarized than the less dominant individuals is important
for understanding such polarized systems. Moreover, describing overall
polarization with a single score can create the false perception that all
groups are equally influential in the observed polarization online. Despite a
growing body of work reporting increasing polarization worldwide [AR21, CH23,
KYA21, FGT+22, KC24, BG20], we lack standardized methods to break down the
measured division into more interpretable components that provide deeper
insights into the phenomenon. In this study, we aim to extend the
informativity of common polarization measures. We begin by identifying the
assortative clusters [ZP20] representing the polarized groups. Then, we
further partition each identified group into a hierarchy using the well-known
core-periphery structure [BE00, RPFM17, GYW21] to separate the elite from the
mass. Finally, we study two distinct aspects of polarization: structural
polarization and issue alignment, at both the group and hierarchical levels.
For the former, this involves decomposing an existing polarization measure,
while for the latter, we measure the degree of alignment separately for the
identified hierarchical groups.

Some polarization studies focus solely on the political elite (e.g. [Nea20]),
such as elected officials, media figures or influential individuals, while
some are more limited to assess the division among the general mass (e.g.
[FA08, Lel16]). Researchers have also explored the interplay between these two
groups (elite and mass) in polarized settings (e.g. [BC18, BCRR23, Sky21,
RM16, FA08, LC02]), resulting in valuable insights into their differing levels
of political responsiveness [BCN+19] and ideological consistency [LC02].
Generally, the political elite are perceived as exhibiting greater ideological
coherence and predictability in their positions across various domains
compared to the mass. Some theories (e.g. [LC02, KK17]) suggest that the
public are actually immune to elite polarization, whereas political elites in
the opposing groups, are growing more ideologically distant from each other
while also becoming more internally homogeneous [RM16]. However, recent body
of work has found people’s views converging and becoming more interwined,
possibly indicating that certain ideological realignment is taking place among
the mass too [KM21]. To our knowledge, prior research on distinguishing the
contribution of the elite and the mass to distinct aspects of polarization in
online systems is rather limited. Our approach will help us evaluate whether
polarization among the mass has remained at the same level, independent of the
polarization trends observed among the elite.

Additionally, there exists work underscoring the importance of reconsidering
the idea of treating both sides of a political discussion as equal [JBD22,
GR14, SRZ19]. The language [KC24], rhetoric, and distribution of power can
vary significantly between different ideological groups. Equivalently,
polarization can be also asymmetrical [JBD22], which indicates that a group
can become more extreme while the other group staying at the same level. The
majority of current methodologies struggle to discern which group or groups
contribute more significantly to overall structural polarization or issue
alignment. Implementing a consistent partition strategy with an intuitive
characterization of identified structures allows for a more precise
description of disparities between these polarized groups and enables
comparisons across different topics represented by networks.

Generally, assuming that each individual or group contributes uniformly to
polarization can be suboptimal for several reasons. As the literature has
shown, we should expect that the nature of an individual’s divisiveness
depends on their role in society. We also know that the mass can take social
and political cues from political elites [VBS21, DL19, GEN+20, Sky21], and are
often influenced by them [All23, MS21, KT18, GEN+20, Ber07, MS18]. When elites
indicate a divergence in ideology or display hostility towards out-groups,
their followers interpret these signals and may adjust their behavior
accordingly [BC18, FH23, Ber07, MS18]. This phenomenon, commonly referred to
as elite signaling or elite influence, has been linked to political
polarization [BCRR23, Sky21]. While the impactful role of elite in political
and social systems is evident, one should not underestimate the impact of the
mass either. Previous research suggests that politically-engaged individuals
within the public may be the ones setting the political agendas for
discussions on online platforms [BCN+19]. Not only might the people be leading
the conversations, they are often needed for reaching wider audience in the
social networks [BWB+15]. While the connection between these unique
characteristics of the elite and mass groups and polarization is complex and,
for the most part, unknown, it is crucial to examine how they differ in their
contribution to the political polarization observed online.

In recent decades, three distinct aspects of polarization have received
significant attention in the community: 1) the bimodality of opinion
distributions, 2) the extent of issue and partisan alignment, and 3) the
affective dimension, which refers to the development of strong negative
attitudes and emotions towards members of opposing political or ideological
groups. While these three aspects of polarization can potentially strengthen
each other, it is important to recognize that they are different concepts,
both theoretically and in terms of how they are measured empirically [HDC23].
In this paper, we focus on the first two aspects.

Researchers have traditionally examined the bimodality and alignment aspects
of polarization using survey-based data, estimating ideological shifts and
correlations between distinct issue positions from participants’ responses
(e.g. [KM21]). However, the research community has increasingly turned its
attention to social media platforms, as, in today’s world, these platforms
serve as crucial venues and spaces for daily discussions on various political
topics and debates. Additionally, these online platforms have become arenas
for political competition, with millions of interactions and users, leading to
valuable insights into the ideological distances among the users, their
limited communication, and elite influence on public opinion worldwide (e.g.
[Bri18, CRF+11, CMK+20, GEN+20]). Social media platforms enable researchers to
monitor conversations and interactions in almost real-time, allowing for
timely analysis of emerging trends and shifts in public opinion. Besides the
inexpensive access to large datasets, it is also reasonable to expect that
there are less response bias in users’ online behavior compared to traditional
survey experiments. These advantages may explain the growing interest in
analyzing polarization through the structural properties of networks.

Within social networks the bimodality of opinion distribution can be estimated
by measuring the limited communication or reachability between the opposing
groups in a given network [MBLB15], although some recent work have advocated
for handling these separately [HDC23]. The intuition is that sharper and
further peaks in an opinion distribution related to a topic X would correspond
to a more isolated communities in a network where the topic X is being
discussed. This type of isolation is often referred to as structural [SCK22]
or interactional [FZQ+23] polarization. Most of the methods designed to
capture this assesses the separation of interaction patterns among two
distinct groups [GMGM18] — although some work have been recently done for
handling multiple groups [NI22, MGLB23]. Typically, quantifying the structural
polarization of a political communication network involves a three-step
process [GMGM18, SCK22]. Firstly, the network is constructed based on the
collected data. Following this, community detection techniques are employed to
identify functional groups, often referred to as polarized groups, within the
network, which frequently represent opposing sides of a political discourse.
Finally, the quality of the division is evaluated using some structural
polarization score, such as Random Walk Controversy [GMGM18] or Adaptive EI-
index [CSG+21]. Again, the assumption these methods often have is that this
division between the two clusters corresponds to either the lack of
interactions between the clusters or the degree to which individuals are
trapped within their own communities. Moreover, many of these methods require
identifying the key figures in a network in order to quantify the extent of
separation [GMGM18, SCK22, MBLB15], in which typically researchers rely on the
number of endorsements a user has as a proxy for their importance or
“eliteness”. While some polarization studies utilize known node labels
[FGT+22, XGM+22] to identify e.g. politicians, this information is not always
available and might miss other crucial users without clear affiliations, such
as nonpolitical opinion leaders, which are shown to be more popular than the
political figures on social media [MJL22]. Although our method is not
specifically designed to address this potential limitation, there are
situations where inferring the key figures in a network solely from certain
structural properties may be preferable.

While these methods have produced valuable insights into societal division
trends, their approach to measuring polarization may be somewhat restrictive.
In addition to the inflexibility in choosing the number of groups, they are
primarily designed to quantify the extent of interactional patterns within a
single issue, and hence, not being capable of capturing the potential
correlation between individuals’ stances on different topics. Some political
and social polarization literature identifies issue alignment as more
deleterious aspect of polarization [Mas15, CSG+21, Tör22]. Issue alignment
refers to the degree of agreement or similarity between individuals or groups
on a set of issues. In issue alignment, individuals or groups may align
themselves based on shared values or ideologies leading to a collective stance
on range of issues, potentially deepening divisions in the society [Mas15,
Tör22].

Our contribution comprises both methodological and exploratory aspects. We
study five topics that have become drastically more polarized online in
Finland over the course of four years, spanning the period from 2019 to 2023,
which contains major events, such as a global pandemic and a war in Europe. In
this paper, we apply this framework to analyze polarization trends and
highlight their differences between two snapshots of the Finnish
Twittersphere. Specifically, we will examine the periods surrounding two
parliamentary elections, in 2019 and 2023, and demonstrate how polarization
dynamics vary across networks. We find that the polarized groups, as well as
the elite and mass within them, shape polarization trends in very different
ways.

To reiterate, our objective is to propose a method capable of measuring two
distinct types of polarization—structural polarization and issue
alignment—within both elite and public spheres, utilizing network
representations of social or political systems. Additionally, our method
unravels the imbalances in the impact that distinct polarized groups have on
the observed polarization—something that a single polarization value would not
reveal.

This paper is organized as follows: In Section 2, we outline the steps for
inferring hierarchical structures in a polarized network and introduce our
methods, which utilize these hierarchies to describe polarization. The main
findings are discussed in Section 3, where we study the trends and patterns in
hierarchical polarization on Twitter during the Finnish parliamentary
elections in 2019 and 2023. Finally, in Section 4, we conclude with a
discussion, list of identified limitations and ideas for future work.

##  2 Hierarchies in polarized networks

Figure 1: Observed network is depicted by the gray block, and it is divided
into two assortative groups (pink and turquoise blocks) only when there is
sufficient evidence of such groupings in the network. Once the polarized
groups are identified, each block is further partitioned into a core and
periphery independently, given significance of the hierarchy. Following the
hierarchy decomposition, there are four potential scenarios: both polarized
groups exhibit a hierarchical structure (1), one of the groups demonstrates a
hierarchical structure (2-3), or neither of the groups displays indications of
a hierarchical structure (4).

###  2.1 Finding the groups and hierarchies

Polarization is often assumed to significantly impact social network
structures, giving rise to two internally tightly-knit groups with sparse
connections between them [GMGM18, HDC23]. Therefore, the level of polarization
in a system can be quantified by examining the patterns of interactions within
that configuration [FGT+22, GMGM18]. To conduct our experiments, our initial
step is to identify the polarized groups. Following this, we need to
distinguish between the elite and the mass members within each polarized
group. This means that each individual in a network belongs to a specific
polarized group (for-against) and a hierarchical group (elite-mass). While
from now we will focus on two polarized groups for this study, this method can
be readily adapted to scenarios involving more than two opposing groups.

Decomposing a network involves breaking it into modular and structurally
distinct components. The conventional approach typically entails identifying
the polarized groups in a network using clustering algorithms [GMGM18].
However, prior research has highlighted that many clustering algorithms can
generate convincing group structures even in entirely random networks [LRR10,
GSPA04, ZM14]. This tendency results in non-controversial networks having
inflated polarization values, leading to a high number of false positives
[SCK22]. To alleviate this issue, we employ the stochastic block model (SBM)
to identify both polarized groups and hierarchical groups in a network,
exploiting the clustering method’s ability to compute the description length
for the given model and partition. Minimum description length is a principle
for model selection that favors models with the shortest code length needed to
describe the data, consistent with Occam’s razor, which suggests simpler
explanations are generally preferable [Grü07]. Thus, an observed community
structure that could be explained due to randomness, would be flagged with
SBM, as the description length for an SBM with no community structure would be
lower than that of a model attempting to overfit by dividing the network into
two communities [Yan16, Pei19, Pei14b, ZP20]. Note, however, that this does
not completely eliminate the need of normalizing the polarization scores for
instance in classification tasks [SCK22].

There are two assumptions that are translated into our choices of SBM models.
According to our first assumption a polarized network consists of two
assortative groups, and thus, we apply the planted partition model variation
of the SBM [ZP20, Pei14b]. We sweep the number of blocks (groups) between 1 to
2, and select the configuration leading to the lowest description length.

Our second assumption is the existence of hierarchies in each polarized group.
Social groups tend to naturally self-organize into hierarchies, where members
exhibit varying levels of influence, expertise, or dominance [KXO15]. The
emergence of these hierarchies is considered both inevitable and often
advantageous for social cohesion [KXO15, MG08]. We utilize the well-
established concept of core-periphery structures (see e.g. [RPFM17]) in
approximating the hierarchical nature of these polarized groups. The core-
periphery structure is a common pattern observed in various systems, including
social networks [BE00, KM17, YZS+18], economics [CLWU13, Wan16], and many
other fields [CLWU13, KM17]. It consists of a densely interconnected core,
typically representing nodes with higher connectivity, influence, or resource
access, surrounded by a sparsely connected periphery representing nodes with
lesser influence. In our context, the core is equivalent to the elite, and the
periphery is equivalent to the mass (or public).

We anticipate that the elite are typically positioned within structurally
dominant areas of networks, forming cohesive structures [MJRW20]. On Twitter,
the elite often include influential figures such as politicians, journalists,
activists, and experts who are active on social media. In contrast, we assume
the surrounding periphery comprising the mass, consisting primarily of users
who endorse or amplify the core’s voice [BWB+15]. These assumptions are later
assesed for our data in Section 3.

For identifying the hierarchical structures that are embedded in the already
identified polarized groups, we follow the method proposed by Gallagher et al.
to infer the core-periphery structure [GYW21] within the SBM framework. We
treat each group independently, which enables us to compare their observed
hierarchies to the null model separately. We use the common Erdős–Rényi model
as the baseline, as done for instance in [KM18], because it assumes no
inherent hierarchy, even one that might be explained by the network’s degree
sequence. This approach has its pros and cons that we address separately in
the Appendix A.7. A visual depiction of the overall network decomposition
pipeline is shown in Figure 1. Note that alternative methods for identifying
core-periphery structures exist, such as the extended version of the original
method [BE00] for finding multiple core-periphery pairs [KM17] or the method
based on detecting core-periphery structures by surprise [dJCS19].

(a) Full network

(b) Cores

(c) Peripheries

Figure 2: The complete retweet network representing Climate discussion (2023)
in (a), subgraph containing the identified cores in (b), and subgraph
containing peripheries in (c). The complete observed graph is drawn with
ForceAtlas2 -layout algorithm [JVHB14]. For the others graphs, same node and
edge positions are applied with specific hierarchy groups filtered out.

###  2.2 Defining structural polarization and issue alignment for hierarchies

Prior work has shown that the most common quantitative methods for measuring
structural polarization in social networks perform similarly after they are
denoised [SCK22]. According to the same work, the best performing method in
detecting polarized networks was Adaptive EI-index (AEI), which is based on
the more popular EI-index (e.g. [EDVBP22, Yan16, DVB18, Bor23]) developed by
Krackhardt and Stern [KS88]. This method produces a score based on the
distribution of connections among nodes within and between groups, where a
greater density of links within groups relative to those between groups
indicates a more polarized system.

Let $G=(V,M)$ represent a network, where $V$ is the set of nodes and $M$ is
the set of links. We define a partition of $G$ into two groups, denoted as $A$
and $B$, such that: $V=V_{A}\cup V_{B}$ and $V_{A}\cap V_{B}=\emptyset$, where
$V_{A}$ and $V_{B}$ represent the sets of nodes belonging to groups $A$ and
$B$ respectively. The AEI is defined for such a system as

$P_{AEI}=\frac{i_{A}+i_{B}-2\times e_{AB}}{i_{A}+i_{B}+2\times e_{AB}}\,,$ (1)

where $i_{X}$ denotes the internal links density (observed links divided by
the possible number of links) within group $X$ and $e_{AB}$ denotes the
external links density between groups $A$ and $B$.

In the AEI-index, all links are treated homogeneously within each group, which
prevents it from distinguishing the activity driven by the elite versus the
mass. However, one should not view these links equal as the elite are often
seen as highly influential individuals on social platforms due to their large
following, prominent status in society and access to resources that can
amplify their messages and shape the public opinion. An elite individual can
also connect to other elite members enabling them to reach even a wider
audience and have a greater impact on conversations and trends within their
communities, making them “more organized”. The amplification and social
validation come mostly from the followers, the surrounding mass, who both take
cues from the elite and can be mobilized by them to take specific actions.
Therefore, we want to distinguish the link densities according to the
hierarchies in the network. We do this by decomposing the elements in
$P_{AEI}$ further

$i_{X}=\frac{I_{c_{X}}+I_{cp_{X}}+I_{p_{X}}}{\frac{1}{2}n_{X}(n_{X}-1)}\,,$
(2)

where $I_{c_{X}}$ represents the interaction between group $X$’s core nodes,
while $I_{cp_{X}}$ denotes the interaction between its core and periphery,
$I_{p_{X}}$ denotes the interaction among its periphery nodes, and $n_{X}$
denotes the number of nodes in group $X$. We can now subsitute the
corresponding terms of Eq. (2) into Eq. (1), which gives us

$P_{AEI}=(\underbrace{\frac{i_{c_{A}}}{\alpha}}_{\text{elite
cohesion}}+\underbrace{\frac{i_{cp_{A}}}{\alpha}}_{\text{mass
amplification}}+\underbrace{\frac{i_{p_{A}}}{\alpha}}_{\text{mass
cohesion}})+(\frac{i_{c_{B}}}{\alpha}+\frac{i_{cp_{B}}}{\alpha}+\frac{i_{p_{B}}}{\alpha})-\underbrace{\frac{2\times
e_{AB}}{\alpha}}_{\text{bridge}}\,,$ (3)

where $\alpha$ is the normalization factor equal to
$(i_{c_{A}}+i_{cp_{A}}+i_{p_{A}})+(i_{c_{B}}+i_{cp_{B}}+i_{p_{B}})+2\times
e_{AB}$.

We call this decomposition polarization spectrum as it allows us to further
study the direct effects of different groups and hierarchies on the
polarization score. By splitting the observed polarization score into
components, we can detect the differences between groups and their hierarchies
on the polarization measure and determine whether their contributions have
changed over time. When we say ’contribution’ or ’impact’ of a certain group
on observed polarization, we specifically refer to these components. This is
valuable information in the context of polarization, where we are often
interested in monitoring the evolution of the opposing groups and their power
dynamics.

Introducing hierarchical groups also enables us to operationalize the
polarization processes. We call $i_{c_{X}}/\alpha$ the elite cohesion of group
$X$. The higher the value, the more organized is the core. Similarly for
$i_{p_{X}}/\alpha$, which we call mass cohesion of group $X$. Structurally,
this value quantifies the centralization of leadership in a given group. Lower
values indicate more centralized leadership, as most links would be directed
towards amplifying the core rather than connecting with peers in the
periphery. Finally, we name $i_{cp_{X}}/\alpha$ as mass amplification,
capturing the intuition that surrounding periphery act as an amplifying body
for its group’s core voices.

We expect the hierarchy of users playing a significant role in terms of
polarization. To better understand the impact on score, we also measure the
marginal polarization for both core and periphery nodes. We define marginal
polarization as the change in structural polarization when a node of a
specific type is added to the network, which enables us to assess how
introducing an average elite user versus an average non-elite user affects the
network’s overall polarization.

A core node in group $A$ has $k_{c_{A}}$ links to other core nodes in the same
group, it receives $k_{cp_{A}}$ links from the periphery and forms $k_{out}$
external links to group $B$. For a periphery node belonging to group $A$, it
can have $k_{p_{A}}$ links to other periphery nodes. It amplifies $k_{cp_{A}}$
core nodes and connects to $k_{out}$ links outside its own group. By
substituting the average of observed values into these “decomposed” degrees,
we can approximate the marginal polarization for a mean core node as

$\Delta_{core_{A}}P_{AEI}\approx 2\times\frac{\frac{\langle
k_{c_{A}}\rangle+\langle k_{cp_{A}}\rangle}{(n_{A})^{2}}-\frac{\langle
k_{out_{B}}\rangle}{n_{A}n_{B}}}{\alpha}\,,$ (4)

where $n_{X}$ denotes the number of nodes in group $X$. The formula is
equivalent for the situation where an average periphery node is added to the
polarized system $(\Delta_{periphery_{A}}P_{AEI}$), but you need to substitute
the $k$ values with those reflecting the link distribution of peripheral
nodes. Interestingly, the formula suggests that having an equal number of
links in and out is not enough for having zero net impact on polarization when
group $A$ is smaller than group $B$. See A.6 for full derivation of this
marginal polarization formula.

In addition to structural polarization, we aim to quantify the degree of issue
alignment separately for the elite and mass. Issue alignment contributes to
political polarization by creating a situation where individuals or groups
align themselves strictly along specific issues. When people strongly identify
with a particular issue or group and use it as a defining factor for their
political stance towards other topics, it can lead to several detrimental
consequences [CSG+21]. We measure alignment using the normalized mutual
information (NMI) method [CSG+21] for cores and peripheries separately. In
essence, it quantifies the extent to which knowing the group of an user in one
topic informs us about the users’s group in another topic.

Let $G_{1}=(V_{1},M_{1})$ and $G_{2}=(V_{2},M_{2})$ denote networks
representing issues 1 and 2, respectively. For estimating each user’s stance
on a specfic issue, we use their group membership in the given network. Since
this assessment requires user’s presence in both networks, we add the stances
for each node in $V_{1}\cap V_{2}$ in stance vectors, $s_{1}$ and $s_{2}$,
corresponding to the respective issues, i.e. these stance vectors effectively
encode each user’s group membership in the networks. Finally, we compute the
issue alignment separately for the distinct hierarchies; NMI($s_{1}^{c}$,
$s_{2}^{c}$) for the cores (elite alignment) and NMI($s_{1}^{p}$, $s_{2}^{p}$)
for the peripheries (mass alignment). This measure is constrained between 0
and 1, where 0 indicates no alignment, and 1 indicates complete alignment
between the issues in question.

##  3 Hierarchical polarization in Finnish Parliamentary Elections 2019 and
2023

Topic | $|N|_{19}$ | $|E|_{19}$ | $|N|_{23}$ | $|E|_{23}$ | $AEI_{19}^{*}$ | $AEI_{23}^{*}$
---|---|---|---|---|---|---
Climate politics | 16 639 | 50 605 | 11 264 | 34 509 | 0.45 | 0.53
Immigration politics | 6 739 | 20 923 | 9 519 | 36 889 | 0.50 | 0.61
Social policies | 13 926 | 39 778 | 11 395 | 34 173 | 0.36 | 0.49
Economic policy | 5 880 | 10 962 | 10 031 | 37 111 | 0.28 | 0.57
Education policy | 9 843 | 20 314 | 12 814 | 35 619 | 0.29 | 0.47
Table 1: Retweet networks were constructed for both election years,
encompassing five distinct networks each. These networks were constructed
based on Twitter data obtained over a span of 12 weeks leading up to the
respective election day, which were 14.4. for 2019 and 2.4. for 2023. $|N|$
represents the count of unique nodes (users), and $|E|$ denotes the count of
unique edges (retweets) in network after the preprocessing. $AEI^{*}$ denotes
the normalized structural polarization computed according to [SCK22], where
AEI stands for Adaptive EI-index. Last column of the table shows how
polarization has increased significantly for all topics studied here.

###  3.1 Data

We use data collected from Twitter (Public API v1 and v2) during the Finnish
parliamentary elections in 2019 and 2023. To ensure consistency when comparing
these two snapshots, we consider a 12-week period leading up to the election
day, resulting in the following periods: from January 21, 2019, to April 14,
2019 for the first elections, and from January 9, 2023, to April 2, 2023 for
the second elections. For each year, five networks are constructed from sets
of keywords related to larger topics, such as immigration and climate change
(see Appendix in [CSG+21] for more details). Before constructing the networks,
retweets that contain keywords belonging to two or more topics are removed
from the dataset, as they would produce a systematic error, i.e. inflate the
extent of alignment between networks.

The largest connected component is extracted for each network, as done in
[GMGM18, CSG+21, SCK22], with the subsequent removal of self-loops and
parallel edges, as performed in [SCK22, CSG+21]. Finally, a network can be
viewed as a graph where each user contributing to the topic is assigned to one
node. In this graph, an edge between two nodes represents a retweet, which can
be deemed as an agreement or a shared point of view on a selected issue
between the corresponding users [GMGM18, CSG+21, SCK22]. Basic information
about each network can be found in Table 1, and a summary of the network
partition outputs in Table 2.

We tested the assumptions stated in the previous section independently: To
verify the presence of political figures within cores, we assessed the
probability of a politician being found in the core compared to the periphery.
We found that politicians who ran for elections are at least 3-4 times more
likely to be in the core than in the periphery (see A.1 for more details). To
view the connections between the outer periphery and the core nodes as mass
amplification, we confirmed this assumption by checking that the majority of
retweets flow towards the core nodes (see A.2 for more details). Thus, we find
our assumptions reasonable.

Table 1 demonstrates the drastic increase in structural polarization for all
networks from 2019 to 2023. Over these four years, normalized AEI more than
doubled for the economy network. The climate network showed the smallest
percentage increase in polarization, approximately 18 %. For the immigration
network, AEI increased from 0.50 to 0.61, reaching the highest polarization
value observed among our networks. This extent of development in polarization
over such a short period made us investigate its structure in more detail.

| Assortative | Hierarchy
---|---|---
Topic | $G_{19}$ | $G_{23}$ | $A_{19}$ | $B_{19}$ | $A_{23}$ | $B_{23}$
Climate politics | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Immigration politics | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Social policies | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Economic policy | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Education policy | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Table 2: Each network studied here has an assortative structure and each
identified polarized group has some type of hierarchy that could not be
explained with Erdős–Rényi model. We confirm assortativity if the two-group
model yields a lower description length than the single-group model.
Similarly, hierarchy within each group is confirmed if the CP-model results in
a lower description length compared to the null model.

###  3.2 Groups have unequal impact on the overall structural polarization

Figure 3: Polarization spectrum for structural contributions of different
groups and their hierarchies to AEI-score. The figure illustrates the
predominant influence of elite cohesion ($i_{{c}_{A}}$ & $i_{{c}_{B}}$), mass
amplification ($i_{{cp}_{A}}$ & $i_{{cp}_{B}}$), and mass cohesion
($i_{{p}_{A}}$ & $i_{{p}_{B}}$) on the overall score. The green part of
spectrum represents the impact of the bridge between the opposing entities
($2\times e_{AB}$). The contributions of different hierarchical members not
only vary within individual networks but also across the distinct networks.
The part of the spectrum that corresponds to the internal structures is
shifted to the left by an amount equal to the cross-interactions. This enables
us to read the unnormalized AEI score for each network directly from the
figure.

We encapsulated the contributions of different components into a single plot
(see Figure 3), where separate spectrums are composed for each year separately
for all topics. Here, groups labeled A (in red) refer to left-leaning groups,
while groups labeled B (in blue) refer to right-leaning groups, respectively
(see A.3 for labeling details). This allows us to track the changes in the
dynamics of left-right groups and their embedded hierarchies in the polarized
milieu online.

Results show many interesting findings. First, the influence of polarized
groups on the observed structural polarization varies widely. In fact, in no
network was the influence of both groups close to balanced, even when the
groups were of similar sizes. In 2019, left-leaning groups contributed the
most to the structural polarization in the climate, social security, and
economic policy networks, whereas right-leaning groups were more dominant in
the immigration and education networks. However, in 2023, right-leaning groups
accounted for the majority of the observed polarization in 4 out of 5
networks. The left-leaning group remained the more contributing group in the
economic policy network, although its share became slightly smaller.

The polarization spectrums also reveal the influence and evolution of distinct
hierarchical groups. For instance, in the immigration network, the most
polarized topic in both years, the elite cohesion of the right-leaning group
had a relatively large impact on polarization. The same elite were also
significantly more organized than those in the left-leaning group, although
the elite cohesion of the latter increased in 2023. The opposite phenomenon
was observed for economic policy network, where the left-leaning group’s elite
cohesion was higher, while right-leaning group elite evolved into more
organized structure in 2023. Note, how in all networks, at least one of the
elite groups had become more cohesive during the four years; Right-leaning
elite became more cohesive in four networks and left-leaning elite in three
networks, indicating a stronger role of elite in certain topics. The most
drastic swap in terms of contributions of a hierarchical groups was witnessed
in the mass amplification patterns in climate network, where right-leaning
public outrun the previously dominating left-leaning public. Although in the
absolute terms, the publics’ interaction patterns tend to explain most of the
observed polarization in all networks, the relative contribution (when
compared to the sizes of these hierarchical groups) of the elite members is,
on average, greater, which we will see later in Figures 4 & 5B.

Communication between polarized groups has mostly dropped or remained at the
same level across all topics. This type of communication, labeled as bridge,
is the only interaction that could structurally decrease polarization. Its
share of the spectrum has shrunk in all networks except the immigration
network.

To summarize the changes in power dynamics between the two groups, we
visualize the polarization-increasing components of AEI alongside the size
distributions of groups in a more compact format (see Figure 4). For each
network and year, we plot a thicker rectangle to illustrate the fraction each
group (including their hierarchical groups) contributed to the observed AEI. A
thinner bar shows the relative sizes of each group. The lower rectangles
correspond to the data from 2019, while the upper rectangles represent the
results from 2023. These rectangles, which depict the distribution of
contributions, sum up to one. In a balanced scenario, the red and blue shaded
areas would be symmetrical along the vertical line.

In the context of immigration, the overall impact of the right-leaning group
exceeded that of the left-leaning group in both years, despite the right-
leaning group being smaller in size. A drastic change was observed in the
climate network, where in 2019, the left-leaning group was more dominant.
However, after four years, the right-leaning group managed to organize its
elite and public components more effectively, resulting in a greater impact on
polarization, despite left-leaning group being larger in 2023. The plot for
the economy network also illustrates how the size of a group does not
necessarily correlate with its impact on the polarization score. In both
years, the right-leaning groups were significantly larger, yet the left-
leaning groups accounted for a larger portion of the observed polarization.

Three main findings from the evolution graphs (Fig. 4) stand out: first, the
increase in structural polarization does not follow uniform patterns, second,
a larger group size does not always lead to a higher contribution to the
observed polarization, and third, the elite play relatively higher role than
the mass. The networks shown are the most polarized ones, yet the underlying
contributions of the groups are very different. We see, for instance, how
right-leaning group compensated its smaller size with more organized core and
greater mass amplification in both years, leading to significantly higher
impact on polarization when compared to the impact of left-leaning group. The
greater relative contribution of the elite members is generally evident across
all networks when reflecting it to the size of their groups shown in the
thinner bars. We also confirm this by computing the marginal polarization
separately for the elite and mass individuals in Figure 5.

Figure 4: Evolution of power structures in most polarized networks:
Contributions of different structures to overall polarization is shown in the
thicker bars for the most polarized networks. Mass amplification and cohesion
are combined here into one color. Thinner bars illustrate the size
distributions of both elite and mass groups within the networks. The upper set
of bars corresponds to the results for 2023, while the lower set represents
those for 2019. Each bar sums up to 1. These illustrations demonstrate
visually how the increases in structural polarization does not follow uniform
patterns, larger group sizes do not always lead to higher contributions to
observed polarization, and the elite play a relatively higher role than the
masses. The similar plots for the remaining topics can be found in Appendix
A.5.

###  3.3 Elite is significantly more aligned than the mass

Figure 5: A: Elites are consistenly more aligned than masses across all topic
pairs. Elites became more aligned in 2023, together with a smaller increase in
the alignment of mass opinion on various issues. To capture the uncertainty
around the observed values, we bootstrapped 500 pairs of networks for each
topic pair. Each bootstrap sample represents a perturbed version of the
original network, where the sizes of cores and peripheries are subject to
random fluctuations. B: Elite tend to have higher marginal polarization as
well compared to the mass. In both years, adding a new elite member to the
economy network had the greatest impact. A weighted average of both groups’
marginal values is applied to obtain a single value representing the
hierarchy’s mean effect on AEI.

Elite members are consistently more aligned than the public in all topics
studied here, as shown in Figure 5A. Higher alignment suggests that the elite
regularly maintain more predictable stances on various political themes, as
typically expected. In 2019, among both elite and mass, topic pairs
immigration & education and immigration & climate were the most aligned.
Conversely, the lowest alignment was found between the topics immigration &
social and education & social.

After four years the gap between elite and mass alignment widened – not only
did the elite show a much higher level of issue alignment (the mean level
doubled), but the disparity between the mass and the elite also increased in
2023. The overall level of mass alignment did not stay at the same level as in
2019 either. Instead, the mass also experienced a notable increase in
alignment across all topics, as the mean line, corresponding to the average of
alignment over all the topics, rose from 0.13 to 0.44. For the elite groups,
eight out of ten pairs of topics had an NMI value over 0.8, which can be
considered very high. Results show that knowing elite member’s stance on
climate politics would reveal the most information on their views about
immigration policies compared to the other topics included in this study.
Interestingly, climate & economy experienced the smallest increase in
alignment.

To evaluate deeper the relationship between the elite and mass alignment, we
employ bayesian linear regression. As our study contains five distinct topics,
the total number of pairs are ten for each year, resulting in a total of
twenty samples for the entire period. We fit a linear regression using elite
alignment value as an independent variable and mass alignment as dependent
variable (see Figure 6). The bayesian framework enables the quantification of
uncertainty in model parameters, which we find crucial due to the small sample
size. While we acknowledge that our approach may oversimplify the relationship
between the elite and mass, we believe that our analysis sheds light on the
behavior of these distinct hierarchical groups and highlights how the
interdependence between the variables has intensified over time. To be more
specific, here we consider the slope of the regression to correspond to the
strength of the relationship between the elite and mass alignment. It’s also
worth reminding that the linear dependence between the variables we are about
to report doesn’t necessarily imply causality. However, the slope of linear
regression is tightly connected to the pearson correlation coefficient, as

$\text{slope}=r\times\frac{\text{standard deviation of }y}{\text{standard
deviation of }x}\,,$ (5)

where slope is the slope estimate of the fitted regression line and $r$ is the
pearson correlation coefficient.

The posterior distribution for the slope $\beta$ is centered at 0.607 (95% CI
[0.60, 0.61]), with the model’s intercept being -0.08. Hence, the expected
slope, and consequently, the correlation, is positive. This suggests that as
elite alignment increases, mass alignment also increases. It’s evident from
the regression results that the elite tend to be more politically aligned than
the general public, which aligns with common expectations. However,
intriguingly, the model also offers an estimate for the “baseline alignment”,
i.e. the degree of alignment among the elite even when the mass shows no
alignment whatsoever. According to our model, the point estimate for the
baseline alignment is approximately 0.13.

Figure 6: Each data point represents the observed alignment between pair of
topics separately for the elite (x-axis) and public (y-axis). To assess and
compare the strength of the relationship between these variables, bayesian
linear models were fitted. The subplot illustrates the posterior distribution
of the slope parameter. The linear correlation is evidently positive between
the variables, and for most topics in 2023, the mass exhibits higher issue
alignment than the expected posterior mean.

##  4 Conclusion

###  4.1 Discussion

This study investigated whether polarized groups and their respective
hierarchies have equal influence on polarization observed in online systems.
We proposed a method to identify these structures within a single social
system and analyzed their different impacts on two aspects of polarization:
structural polarization and issue alignment-based polarization. We illustrated
the use of the method by applying it to Finnish polarized networks.

Our findings revealed that polarized groups had an unequal impact on overall
structural polarization across all the networks studied, indicating lack of
balanced systems where both opposing groups would contribute equally to the
polarization score. Specifically, we found that left-leaning groups played a
more dominant role in three out of five topics (climate, social, and economy)
in 2019. However, by 2023, the right-leaning group became the dominant force
in both climate and social politics, mostly driven by a more organized elite
and greater mass amplification. Finland elected a left-leaning government in
2019 and right-leaning government in 2023 [GS23].

The hierarchy of the user in a polarized networks matters. Not only did we
find that the elite consistently aligned more on distinct political topics
than the mass within our networks, but we also observed that the elite had a
relatively higher impact on structural polarization, as evidenced by their
respective marginal values. Additionally, we discovered that all networks,
which experienced an overall increase in polarization during the study period,
had at least one elite group that increased its cohesion. This reminds us that
a network’s increased polarization does not necessarily depend on greater
public amplification. Rather, it suffices for the elite to become more
organized, as exemplified for the right-leaning group in the education
network.

Elite are significantly more aligned than the public online, and the alignment
intensified over the four years across all the topics. The absolute difference
in the average level of elite alignment more than doubled over the same time
span. We asked whether the public online is somehow immune to the increasingly
polarized environment. Although this study cannot respond, whether the drastic
increase in the mass alignment is due to the elite being more polarized, our
results clearly suggests that certain ideological realignment has taken place
among the public too. The relative increase in the issue alignment was more
pronaunced for the mass (+228 %) than for the elite (+121 %). We also observed
that in 2023, a significant number of observations clustered above the mean
for various topics, suggesting a stronger association between elite and mass
alignment. Hence, there is evidence that the mass has at least not remained
isolated from the trend towards increased alignment.

Again, while our analysis does not definitively pinpoint whether the elite
drove this increase in mass alignment, it’s clear that socially influential
figures are becoming more politically aligned. This trend could potentially
influence the public, aligning with past research highlighting how they often
take cues from elite figures. Furthermore, our model suggested that
theoretically, the alignment among the elite would persist even if the mass
showed no indication of aligning on issues. This could suggest that
maintaining predictable stances on a range of political topics serves as a
distinguishing feature (or strategy) that sets the elite apart from the
general public.

Our contribution does not address the concept of an “ideal polarized network”.
Would such a network be non-hierarchical, or would it exhibit balanced
contributions from both opposing parties? It is important to acknowledge that
some degree of disagreement is not only inevitable but also essential in a
healthy democracy. Ideally, disagreement fosters debate, encourages diverse
perspectives, and promotes critical thinking, which can be deemed as
fundamental components of democratic processes. How these potentially
desirable dynamics would be reflected in the network structure is an open
question. However, we believe it is crucial to recognize the prevalent
imbalances we have in today’s polarized networks and to take them into account
in the polarization-related literature. Our study also reveals a concerning
trend: a rapid decline in communication patterns and a sharp rise in issue
alignment, both which have happened within a relatively short timeframe.

Social media is sometimes seen as a decentralized platform for content
creation [TLT+18, LPBF19] that is not monopolized by a few influential
entities. However, our observations could challenge this perception. The elite
groups consisted of very small number of individuals, and these were literally
in the core of the polarized networks, making them structurally dominant. Yet
they had greater marginal impact on polarization and were significantly more
politically aligned. This raises questions about the true decentralization of
social media platforms and whether they genuinely represent the diversity of
political voices in our societies.

###  4.2 Limitations

While our study provides valuable insights into some structural mechanisms of
polarization, it is not without limitations.

Many influential figures use Twitter, and it is popular social media platform
among the public too. However, while the core-periphery concept has been
extensively employed to model hierarchies in networks, and our manual check
indicated that a majority of political candidates fell within the identified
cores, it’s essential to recognize that these findings may not necessarily
extend to the behavior of the “actual” political elite (which is also
relatively vague concept) in society. The same applies to the mass, which
represented the non-elite nodes in the network. However, it’s worth noting
that, as highlighted in previous studies, Twitter users are often perceived as
more politically engaged than an average citizen. This limitation is
interesting, as it can imply that the mass (and the elite) outside Twitter
might be even less aligned than the elite on social media, or the other way
around. The function of the public within a polarized network was simplified
to act as amplifiers of the messages produced by the cores. While this
simplification is not uncommon in similar studies, we believe that the
temporal interactions and dynamics between a network’s cores and peripheries
are more complex.

While each method presented here could be easily extended to characterize
polarization in the case of multiple groups, we have chosen to concentrate on
the typical settings involving two groups to facilitate comparisons between
the networks.

Considering network ties as directed could enhance the accuracy of network
decomposition, as the direction of endorsements is not random. Assigning
greater importance to nodes with an in-degree larger than their out-degree may
be in some cases desirable. Likewise, when an elite endorses someone who does
not reciprocate, it might suggest a special user, such as an organizational
account. In this work, we interpretted the edge between two nodes more as a
mutual agreement than directed endorsement.

###  4.3 Future work

We mainly focused on studying structrual differences between the polarized
groups and embedded hierarchies in polarized networks. Building upon the
insights gained from this study, future research could investigate deeper the
content and attitudes within cores, and how they reflect to the structural
properties of the network. Political discussions are often perceived as a
clash between the status quo and change. Do these respective groups exhibit
similar patterns across all networks? Can the structural properties alone of a
polarized group offer insights into the positions held by its members? How are
these connected to the forces driving polarization online?

Social media platforms have experienced a surge in polarization in recent
years, with conflicting findings on whether it is the public or the elite
driving this phenomenon. We do not claim that the methods presented here
provide a direct answer to this question. However, by conceptualizing the
components linked to structural polarization, this study facilitates the
examination of temporal changes. Particularly when there are data available
from a longer period, it becomes possible to detect persistent structures and
identify strengthening groups, whether in terms of mass amplification or elite
cohesion. These factors could contribute to a more nuanced understanding of
the possible under- or overrepresentation of specific groups. In fact, the
introduced polarization spectrums enable us to compare how different social
media platforms exhibit disparities in power structures and dynamics among
competing political groups. Furthermore, treating elite and mass polarization
as distinct processes could assist researchers in detecting suspicious
activities that contribute to polarization in online networks—such as abrupt
spikes in mass amplification that deviate from their expected trajectory.

The significance of hierarchies and their impacts on online systems should not
be underestimated. Our findings enable the deeper exploration into these
matters, enriching the comprehension of mechanisms that drive polarization.

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##  Appendix A Appendix

###  A.1 Political actors in cores versus peripheries

To verify that the presence of political elite is higher in the cores than in
peripheries, we computed the following for each network: How much more likely
it was for a randomly selected node from the cores to belong to the list of
political candidates compared to a node sampled from the peripheries, i.e.

$\gamma=\frac{\text{Probability of core node being a
candidate}}{\text{Probability of periphery node being a candidate}}$

Topic | $\gamma$
---|---
CLIMATE | 3.3
IMMIGRATION | 2.8
SOCIAL SECURITY | 3.1
ECONOMIC POLICY | 4.0
EDUCATION | 3.6
Table 3: Values of $\gamma$ demonstrate the higher chance of finding a
politician from the cores versus peripheries.

###  A.2 Public Amplification

Figure 7: To view the core-periphery interactions as public amplification, we
confirm that the majority of connections are directed towards the core. In all
networks examined, most of the links between the core and periphery consist of
retweets originating from the outer periphery. This proportion remains
relatively consistent across different polarized groups. Therefore, it’s
reasonable to characterize this dynamic as public amplification.

###  A.3 Political parties presence in inferred polarized groups

Figure 8: Majority of the candidates from left-leaning parties are grouped
together in the inferred polarized groups, as are those from right-leaning
parties.

###  A.4 Characterics of polarized groups

| Group A | Group B |
---|---|---|---
Topic | C | P | E Coh. | Ampl. | P Coh. | C | P | E Coh. | Ampl. | P Coh. | B
CLI ’19 | 2.2 | 42.4 | 16.6 | 40.1 | 8.6 | 2.8 | 53.0 | 4.0 | 16.8 | 5.2 | 8.7
CLI ’23 | 3.6 | 65.8 | 6.3 | 12.9 | 4.7 | 1.7 | 29.0 | 11.9 | 52.0 | 9.1 | 3.1
IMM ’19 | 3.7 | 71.8 | 2.2 | 7.6 | 2.2 | 2.5 | 22.1 | 29.9 | 50.9 | 5.2 | 1.9
IMM ’23 | 3.7 | 51.8 | 9.9 | 17.6 | 4.0 | 3.6 | 41.0 | 20.5 | 39.1 | 5.7 | 3.1
SOC ’19 | 1.5 | 39.0 | 9.9 | 37.1 | 9.8 | 4.4 | 55.1 | 7.2 | 18.2 | 6.7 | 11.3
SOC ’23 | 4.2 | 57.3 | 8.4 | 15.6 | 6.7 | 2.7 | 35.8 | 13.7 | 42.0 | 8.7 | 4.9
ECO ’19 | 1.1 | 16.2 | 8.2 | 54.2 | 17.8 | 4.7 | 78.0 | 2.2 | 10.2 | 4.3 | 3.1
ECO ’23 | 2.4 | 28.9 | 23.8 | 36.0 | 8.5 | 4.1 | 64.7 | 7.3 | 17.2 | 4.1 | 3.2
EDU ’19 | 4.9 | 81.0 | 2.9 | 9.1 | 3.9 | 1.2 | 13.0 | 10.9 | 49.1 | 19.8 | 4.4
EDU ’23 | 4.0 | 60.4 | 5.6 | 14.2 | 6.0 | 2.4 | 33.2 | 16.8 | 45.1 | 8.3 | 4.0
Table 4: All values in the table represent percentages. ’C’ indicates the size
of the corresponding core, while ’P’ indicates the size of the corresponding
periphery, divided by the total number of nodes in the network. For the
polarization spectrum components, ’E/P Coh.’ (Cohesion) column reflects either
elite or public cohesion, while the ’Ampl.’ (Amplification) column shows the
public amplification values. Bridge denotes the component representing the
overall inter-communication between groups A and B. See more details in the
caption for Figure 3.

###  A.5 Power dynamics

Figure 9: Evolution of power structures for social and education networks. See
caption of Figure 4 for more details.

###  A.6 Marginal polarization

How much does the overall interactional polarization increase if you added an
“average” core (or periphery) node to group A? Assume that an average core
node has $k_{c_{A}}$ links to other core nodes in the same group, it receives
$k_{cp_{A}}$ links from the periphery and forms $k_{out}$ links to the other
group B. The marginal polarization could be written then as

$\displaystyle\Delta P_{AEI}(\text{+ corenode to A})$
$\displaystyle=\frac{\frac{(I_{c_{A}}+k_{c_{A}}+I_{cp_{A}}+k_{cp_{A}}+I_{p_{A}})}{\binom{n_{A}+1}{2}}+\frac{(I_{c_{B}}+I_{cp_{B}}+I_{p_{B}})}{\binom{n_{B}}{2}}-2\times\frac{E+k_{out_{B}}}{(n_{A}+1)n_{B}}}{\alpha^{\prime}}$
$\displaystyle\quad-\frac{\frac{(I_{c_{A}}+I_{cp_{A}}+I_{p_{A}})}{\binom{n_{A}}{2}}+\frac{(I_{c_{B}}+I_{cp_{B}}+I_{p_{B}})}{\binom{n_{B}}{2}}-2\times\frac{E}{n_{A}n_{B}}}{\alpha}$
(6)

where $I_{\bullet}$ is the observed number of links within the corresponding
structure, and $E$ denotes the observed number of links between groups $A$ and
$B$. Each term is adjusted for density, as described in the main text. We
approximate $\alpha^{\prime}=\alpha$. Hence we get for average core node the
following marginal:

$\displaystyle\Delta P_{AEI}(\text{+ corenode to A})$
$\displaystyle=\frac{\frac{(I_{c_{A}}+k_{c_{A}}+I_{cp_{A}}+k_{cp_{A}}+I_{p_{A}})}{\frac{1}{2}n_{A}(n_{A}+1))}-2\times\frac{E+k_{out_{B}}}{(n_{A}+1)n_{B}}-\frac{(I_{c_{A}}+I_{cp_{A}}+I_{p_{A}})}{\frac{1}{2}n_{A}(n_{A}-1))}+2\times\frac{E}{n_{A}n_{B}}}{\alpha}$
(7)
$\displaystyle\approx\frac{\frac{(I_{c_{A}}+k_{c_{A}}+I_{cp_{A}}+k_{cp_{A}}+I_{p_{A}})}{\frac{1}{2}n_{A}(n_{A}))}-2\times\frac{E+k_{out_{B}}}{n_{A}n_{B}}-\frac{(I_{c_{A}}+I_{cp_{A}}+I_{p_{A}})}{\frac{1}{2}n_{A}(n_{A}))}+2\times\frac{E}{n_{A}n_{B}}}{\alpha}$
(8)
$\displaystyle=\frac{\frac{(k_{c_{A}}+k_{cp_{A}})}{\frac{1}{2}n_{A}(n_{A}))}-2\times\frac{k_{out_{B}}}{n_{A}n_{B}}}{\alpha}$
(9)
$\displaystyle=2\times\frac{\frac{(k_{c_{A}}+k_{cp_{A}})}{(n_{A})^{2}}-\frac{k_{out_{B}}}{n_{A}n_{B}}}{\alpha}$
(10)

And for the others:

$\displaystyle\Delta P_{AEI}(\text{+ core node to B})$ $\displaystyle\approx
2\times\frac{\frac{(k_{c_{B}}+k_{cp_{B}})}{(n_{B})^{2}}-\frac{k_{out_{A}}}{n_{B}n_{A}}}{\alpha}$
(11) $\displaystyle\Delta P_{AEI}(\text{+ periphery node to A})$
$\displaystyle\approx
2\times\frac{\frac{(k_{p_{A}}+k_{cp_{A}})}{(n_{A})^{2}}-\frac{k_{out_{B}}}{n_{A}n_{B}}}{\alpha}$
(12) $\displaystyle\Delta P_{AEI}(\text{+ periphery node to B})$
$\displaystyle\approx
2\times\frac{\frac{(k_{p_{B}}+k_{cp_{B}})}{(n_{B})^{2}}-\frac{k_{out_{A}}}{n_{B}n_{A}}}{\alpha}$
(13)

###  A.7 Statistical testing

Our analysis requires the network to be decomposed into $k$ polarized groups
(assortativity assumption) after which each polarized group is decomposed
further into separate hierarchical groups (hierarchy assumption). Therefore,
in case of $k=2$, the overall number of different groups would be four. From
now, we assume $k=2$, and label the identified polarized groups as $A$ and
$B$. For the hierarchical decomposition, we label the cores of $A$ as $c_{A}$
and its periphery as $p_{A}$ (similary for B).

We know that most existing methods for determining the modular structure of
networks face issues. They often overlook the statistical evidence supporting
the identified patterns, making it difficult to distinguish genuine structures
from noise. In our case, we want to be sure about two things: 1) Is there
enough evidence that the network can be decomposed into two groups? and 2) If
so, is there enough evidence that these groups can be partitioned further into
separate hierarchies?

Each network undergoes the following partitioning pipeline: 1) Network is
partitioned into two blocks with the non-uniform planted partition model. 2)
To find the embedded hierarchies, we consider the groups independently, and
run the core-periphery algorithm provided here. We apply the traditional hub-
and-spoke description of core-periphery structure as it has been shown to
explain online amplification better than the layered one. We run this two-step
decomposition for 100 times and save each obtained partition for the
statistical significance testing.

We want to be explicit about how we measure significance and what we do with
that information. First, if it turns out that the network has no significant
assortative groups, then we conclude that there is no evidence for the network
being at least structurally polarized. In our experiment, all networks were
better described with two groups in terms of description length. We then shift
our attention to the inferred hierarchies within each group.

Fist, defining a hierarchy is harder than defining the absence of it. Perhaps
the most common baseline model in network science, Erdős–Rényi model,
generates random graphs, where each edge $(i,j)$ is included in the graph with
probability $p$ independently from every other edge. Hence, the possible
heterogeneity that emerge in such graph is purely due to random fluctuations,
making it decent candidate for null model.

A stricter null model would be the configuration model, another common option
for testing the statistical evidence of an observed structure. Essentially, it
is a mathematical model used to generate random networks with a specified
degree sequence. Interestingly, previous research has shown how the degree
sequence of a network can fully explain the core-periphery structure in
certain situations [KM18]. We selected ER-model over the configuration model
for three reasons:

  1. 1.

We use the core-periphery structure to identify the structurally dominant
users, and it is acceptable for this significance to stem from a user’s higher
degree.

  2. 2.

Some work (e.g. [YS23]) has discussed the possible shortcomings of using
configuration model in detecting core-periphery structures, claiming it likely
leading to low power for the statistical test. They illustrate this with an
ideal core-periphery structure, for which the same original network is the
only one that maintains the fixed degree sequence. Therefore, no rewiring is
possible, resulting in a hypothesis test yielding a p-value of 1. [KM18]
claims that an additional block, such as a community or another core-
periphery, is required to conclude the significance of the observed core-
periphery. However, as far as we know, there isn’t a generative model capable
of inferring multiple core-periphery structures simultaneously, and such model
is typically required to measure the description length.

  3. 3.

Erdős–Rényi model is an intuitive and simple model to understand, and it does
not produce self-loops or parallel edges.

Ideally, we would apply directly a constrained version of nested stochastic
block model [Pei14a] because of the hierarchical nature of our overall
decomposition approach. Unfortunately, it is out of scope of this work to
develop such method, and hence we employ the method described in the main text
for model selection.