# Emergence of Polarization in Coevolving Networks

Jiazhen Liu  Department of Physics, University of Miami, Coral Gables, Florida
33142, USA  Shengda Huang  Department of Physics, University of Miami, Coral
Gables, Florida 33142, USA  Nathaniel.M.Aden  Department of Physics,
University of Miami, Coral Gables, Florida 33142, USA  Neil.F.Johnson  Physics
Department, George Washington University, Washington D.C. 20052, USA  Chaoming
Song<EMAIL_ADDRESS>Department of Physics, University of Miami, Coral
Gables, Florida 33142, USA

###### Abstract

Polarization is a ubiquitous phenomenon in social systems. Empirical studies
document substantial evidence for opinion polarization across social media,
showing a typical bipolarized pattern devising individuals into two groups
with opposite opinions. While coevolving network models have been proposed to
understand polarization, existing works cannot generate a stable bipolarized
structure. Moreover, a quantitative and comprehensive theoretical framework
capturing generic mechanisms governing polarization remains unaddressed. In
this paper, we discover a universal scaling law for opinion distributions,
characterized by a set of scaling exponents. These exponents classify social
systems into bipolarized and depolarized phases. We find two generic
mechanisms governing the polarization dynamics and propose a coevolving
framework that counts for opinion dynamics and network evolution
simultaneously. Under a few generic assumptions on social interactions, we
find a stable bipolarized community structure emerges naturally from the
coevolving dynamics. Our theory analytically predicts two-phase transitions
across three different polarization phases in line with the empirical
observations for the Facebook and blogosphere datasets. Our theory not only
accounts for the empirically observed scaling laws but also allows us to
predict scaling exponents quantitatively.

Recently published discourse around opinion polarization, a process by which
the opposition of opinions increases with time, has received much attention
Bramson  _et al._ (2016); DiMaggio  _et al._ (1996); Matakos  _et al._ (2017);
Manrique and Johnson (2018). Empirical studies observe a typical bipolarized
pattern where individuals are divided into two groups with radically opposite
opinions. For instance, the political division between liberal and
conservative parties reflects the heterogeneity in opinions regarding
political orientations. These different attitudes are observed on social media
Bakshy  _et al._ (2015); Cinelli  _et al._ (2021); Del Vicario  _et al._
(2016a); Garimella  _et al._ (2018a); Del Vicario  _et al._ (2016b); Borge-
Holthoefer  _et al._ (2015), finding that the most frequently shared political
opinions are aligned with a large proportion of the liberal or conservative
population, i.e., the opinion bipolarization. On the other hand, empirical
data also show a depolarization phase for some systems, where opinion
distribution peaks around a neutral state with a significant variance in non-
political fields Bakshy  _et al._ (2015).

Empirical studies show that the echo chamber effect underlies the opinion
polarization in social networks Cinelli  _et al._ (2021); Del Vicario  _et
al._ (2016a); Garimella  _et al._ (2018b); Cota  _et al._ (2019), suggesting
that like-minded users tend to interact Bakshy  _et al._ (2015); Cinelli  _et
al._ (2021); Lazer  _et al._ (2009). Indeed, the coexistence of polarization
of opinions and network structures implies that the interplay between opinion
and network dynamics plays an essential role in polarization. While growing
coevolving models are proposed to understand mutual interactions between
opinion and network structures, these studies did not focus specifically on
opinion polarization Holme and Newman (2006); Iniguez  _et al._ (2009);
Vazquez  _et al._ (2007); Flache and Macy (2011).

More recently, a reinforced coevolving model has been proposed to explain
network polarization Baumann  _et al._ (2020, 2021). The reinforcement model
(RM) generates either a stable mono-polarized phase where everyone leans
towards one-sided opinions or a global consensus phase where everyone has the
same neutral opinion. While a metastable bipolarized pattern has been observed
numerically, it only appears temporarily and degenerates rapidly to a mono-
polarized pattern. Moreover, its occurrence heavily relies on the initial
state.

While the partial success of the RM model hints that polarization patterns
might originate from the coevolving dynamics, the discrepancy between the
existing models and the stable bipolarized pattern observed empirically has a
deep origin. Indeed, the existing polarization modeling framework is a
variation of synchronization models, where individuals synchronize to a single
state when the system falls into an ordered phase. To capture a bipolarized
structure, however, one requires a different framework. Moreover, coevolving
network models are infamously difficult to solve due to the complicated
interplay between opinion and network structure. Hence, existing models mostly
rely on numerical simulations and qualitative approximations, and a
quantitative model that accounts for the empirically observed bipolarized
pattern is missing.

In this Letter, we report a universal scaling law characterized by a set of
scaling exponents for empirical opinion distributions. These exponents
classify social systems into polarization and depolarization phases. To
explain this finding, we propose two ubiquitous mechanisms for the
polarization dynamics: 1) Opinion homogenization. Individuals’ opinions are
influenced by their neighbors in a social network and tend to converge to
similar views Starnini  _et al._ (2016); Sasahara  _et al._ (2021); Clifford
and Sudbury (1973); Holley and Liggett (1975); Black (1948), and 2) Homophily
clustering. Social connections evolve with time, where individuals tend to
connect to those with similar beliefs. Consequently, similar individuals group
together to form clusters Cinelli  _et al._ (2021); Del Vicario  _et al._
(2016a); Garimella  _et al._ (2018b, a). These two mechanisms lead to the
entanglement of network evolution with opinion dynamics. Therefore, we propose
a generic coevolving framework capturing the interplay between opinion
dynamics and network evolution. We find the exact solution of the proposed
modeling framework and predict analytically the universal scaling laws
observed in the empirical data. We calculate the phase diagrams analytically
for the proposed model, predicting three stable new phases: (i) polarization,
(ii) partial polarization, and (iii) depolarization, where both polarization
and partial polarization phases show a stable bipolarized pattern. To our best
knowledge, we offer the first coevolving network model that predicts a stable
bipolarized pattern analytically, validated by numerical simulation and
empirical measurements.

Experimental observations. We use two datasets to uncover the polarization.
The first dataset consists of the 500 most shared online domains on Facebook
collected Bakshy  _et al._ (2015). These domains are classified as either hard
content (FB-HC) or soft content (FB-SC). The second dataset consists of 1,490
blogs and 19,090 references in Blogosphere Adamic and Glance (2005). We
compute the mean score of political leanings $s$ for each domain or blog,
where $s$ ranges from $-1$ (liberal) to $1$ (conservative) (see SM for
details). Figure 1a depicts the Blogosphere network, suggesting that this
network is polarized into two opposite communities, a phenomenon known as the
echo chamber Cinelli  _et al._ (2021); Del Vicario  _et al._ (2016a);
Garimella  _et al._ (2018b); Cota  _et al._ (2019); Baumann  _et al._ (2020).

Figure 1: Polarized networks. (a) The Blogosphere network. Colors represent
political opinions, i.e., blue for liberals and red for conservatives. (b) A
social network generated by numerical simulations of our model for the
polarization phase.

To quantify the observed polarization, The scatter plot in Figure 2 depicts
the opinion distribution $P(s)$ for all three empirical datasets. We find that
the opinion distributions in two politics-related datasets (FB-HC and
Blogosphere) are U-shaped, suggesting a polarization phase. In contrast,
opinions in the FB-SC dataset are inverse U-shape distributed, indicating a
depolarization phase. To investigate the scaling relation between the
population size and opinion extremeness, we plot $P(s)$ as a function of $1\pm
s$ in the insets of Fig.2a-c, where $1\pm s$ measures the opinion deviation
from the most extreme $\pm 1$ ones. We find

$P(s)\sim\begin{cases}(1-s)^{\delta_{+}}&s\to 1\\\
(1+s)^{\delta_{-}}&s\to-1\\\ \end{cases},$ (1)

satisfying power laws, where $\delta_{\pm}$ characterizes the power-law
exponents when opinion $s$ approaches $\pm 1$. The negative exponent values
indicate that the population increases with the extremeness of their opinions
and diverges when the opinion score $s$ reaches limiting cases $\pm 1$. In
contrast, positive $\delta_{\pm}$ exponents indicate $P(s)$ is peaked around
$0$. Therefore, the exponents $\delta_{\pm}$ characterize polarization, i.e.,
$\delta_{\pm}<0$ and $>0$ for polarized and depolarized systems, respectively.

Figure 2: Empirical measures and theoretical predictions. (a)-(c) The
empirical (dots) and theoretical results (green curves) of opinion
distributions for FB-HC, FB-SC, and Blogosphere. The scatter plots in the
insets indicate that empirical opinion distributions follow power laws. The
solid lines represent our theoretical prediction from our model. (d) The
empirical results and theoretical predictions of the correlation between the
network’s degree and opinions for the Blogosphere. (e)-(f) Heatmaps for the
empirical and theoretical normalized joint-opinion distributions.

To explore the impact of the opinion’s polarization on the network structure,
we measured the average degree $\bar{k}$ as a function of the opinion $s$ for
the Blogosphere dataset. The scatter plot in Fig. 2d shows $\bar{k}$ increases
with the extremeness of opinion score $s$, indicating that users who hold
extreme political opinions are more likely to be the network’s hubs. We
further measured the joint opinion distribution $Q(s,s^{\prime})$ on each edge
of the Blogosphere network, which computes the number of edges connected from
an individual with opinion $s$ to another with $s^{\prime}$. Figure 2e plots
the normalized joint opinion distribution
$R(s,s^{\prime})=\frac{Q(s,s^{\prime})}{P(s)P(s^{\prime})}$, measuring the
deviations from the uncorrelated distribution Maslov and Sneppen (2002). We
found a strong enhancement of the number of two connecting agents with similar
opinions and suppression of the number of two connecting agents with
dissimilar opinions, indicating the existence of homophily clustering in the
real-world social network (Fig.1a).

Modeling framework. To account for the empirical observations, we consider a
coevolving network consisting of $N$ interactive agents, each with an opinion
$s$, varying continuously between $-1$ and $1$. Agents’ connection is
described by an adjacency matrix $A$, where the matrix element $A_{ij}=1$ or
$0$ represents the agent _i_ being connected or disconnected to _j_. Both
agents’ opinions and the connections among them evolve in continuous time $t$.
Moreover, we require the opinion dynamics and network evolution coupled
together, satisfying two intrinsic mechanisms: 1) Opinion homogenization, that
is, the change of an agent’s opinion is influenced by the opinions of its
neighbors and tends to change to similar views. 2) Homophily clustering, that
describes the social connections being more likely established if agents hold
similar views. The two mechanisms are coupled dynamically with the coevolution
of opinions and social connections. Below we discuss the proposed coevolving
dynamics in detail.

Opinion homogenization. To model opinion homogenization, we assume the opinion
dynamic follows

$\mathrm{d}s_{i}=\mu(s_{i},\vec{s},\bm{A})\mathrm{d}t+\sigma(s_{i})\mathrm{d}W_{t},$
(2)

where $\vec{s}=(s_{1},s_{2},...s_{N})$ represents the set of opinions of all
agents, and $s_{i}$ is the opinion of agent $i$. $\bm{A}$ is the adjacency
matrix of the agents, and $W_{t}$ is the standard Wiener process. $\mu$ is the
drift term that controls the change of opinions on average, and $\sigma$ is
the diffusion term that controls the variance of opinion dynamics. Moreover,
we assume that the opinion drift $\mu$ depends on the opinions of agent $i$
and its neighbors, satisfying

$\mu_{i}=\sum_{\langle
i,j\rangle}F(s_{i},s_{j})=\sum_{j=1}^{N}A_{ij}F(s_{i},s_{j}),$ (3)

where $\langle i,j\rangle$ summing over interactions across all neighbors of
$i$ Holley and Liggett (1975); Clifford and Sudbury (1973); Deffuant  _et al._
(2000); DeGroot (1974); Friedkin and Johnsen (1990); Sood and Redner (2005).
The pairwise force $F(s_{i},s_{j})$ quantifies the interaction between
individuals $i$ and $j$. The opinion homogenization requires $F(s,s)=0$ and
$\partial_{s}F(s,s^{\prime}=s)<0$. On the other hand, the diffusion
$D(s)\equiv\sigma(s)^{2}/2$ depends only on each agent’s own opinion $s$. The
boundness of opinion requires the vanishing of the diffusion at the
boundaries, i.e., $D(s=\pm 1)=0$, implying a Taylor expansion,
$D(s)=\frac{\sigma(s)^{2}}{2}=\frac{\sigma_{0}^{2}}{2}(1-s^{2})+O((1-s^{2})^{2})$.
Solving Eq.(2) leads to the time evolution for single-agent opinion

$\displaystyle\frac{\partial P(s;t)}{\partial(\alpha t)}=$
$\displaystyle-\frac{\partial}{\partial
s}\int_{-1}^{1}F(s,s^{\prime})Q(s,s^{\prime};t)\,\mathrm{d}s^{\prime}$
$\displaystyle+\frac{\partial^{2}}{\partial s^{2}}\Big{[}D(s)P(s;t)\Big{]},$
(4)

where the integral captures the neighboring interactions, since
$Q(s,s^{\prime};t)$ depends on the underlying network $\bm{A}$ implicitly.

Homophily clustering. To model homophily clustering, we assume that agents
with similar opinions are more likely to be connected. Specifically, an agent
$i$ will connect to an unlinked agent $j$ to construct a new edge with a
probability rate $\gamma_{+}(s_{i},s_{j})/N$, where $s_{i}$ and $s_{j}$ are
the opinions for agents $i$ and $j$, respectively. The factor $1/N$ guarantees
the sparsity of the network. At the same time, an agent $i$ will disconnect
with a linked agent $j$ with probability rate $\gamma_{-}(s_{i},s_{j})$,
leading to annihilating an existing edge. That is,

$\mathrm{d}P[A_{ij}(0\to 1)]=\frac{\gamma_{+}(s_{i},s_{j})}{N}\mathrm{d}t$
(5a) $\mathrm{d}P[A_{ij}(1\to 0)]=\gamma_{-}(s_{i},s_{j})\mathrm{d}t.$ (5b)

To capture the homophily clustering, we assume that $\gamma_{\pm}$ depends on
the opinions of the agent $i$’ and its neighbors $j$. This dependency leads to
the coupling of network evolution with opinion dynamics.

However, we notice that in most social media, opinion changes are much slower
than network evolution. To quantify their relative time scales, we rescale the
opinion drift $F\to\alpha F$ and diffusion $D\to\alpha D$ by a factor $\alpha$
that characterizes the opinion update rate relative to the network evolution.
In terms of the power expansion of $\alpha$, we find the time evolution of the
joint opinion distribution $Q(s,s^{\prime};t)$ satisfies

$\displaystyle\frac{\partial Q(s,s^{\prime};t)}{\partial t}=$
$\displaystyle\gamma_{+}(s,s^{\prime})P(s;t)P(s^{\prime};t)$ $\displaystyle-$
$\displaystyle\gamma_{-}(s,s^{\prime})Q(s,s^{\prime};t)+O(\alpha),$ (6)

where $O(\alpha)$ term captures higher-order corrections. We will keep our
discussion below at the limit $\alpha\to 0$, i.e., the network evolves
adiabatically, and $O(\alpha)$ will be omitted under this adiabatic
approximation.

Stationary solution. Below we will focus on the stationary solution of the
proposed framework. Equation (Emergence of Polarization in Coevolving
Networks) leads to the stationary joint opinion distribution,
$Q_{st}(s,s^{\prime})=\frac{\gamma_{+}(s,s^{\prime})}{\gamma_{-}(s,s^{\prime})}P_{st}(s)P_{st}(s^{\prime})$,
where $P_{st}(s)$ is the stationary opinion distribution. Substituting Eq.
(Emergence of Polarization in Coevolving Networks) leads to
$\int_{-1}^{1}K(s,s^{\prime})P_{st}(s^{\prime})\mathrm{d}s^{\prime}=\ln(D(s)P_{st}(s))$,
where the kernel
$K(s,s^{\prime})\equiv\int\frac{\gamma_{+}(s,s^{\prime})F(s,s^{\prime})}{\gamma_{-}(s,s^{\prime})D(s)}\mathrm{d}s$.
When $s\to\pm 1$, we find
$\left(\int_{-1}^{1}\kappa_{\pm}(s^{\prime})P_{st}(s^{\prime})\mathrm{d}s^{\prime}\right)\ln(1\mp
s)\sim\ln[(1\mp s)P_{st}(s)]$, where
$\kappa_{\pm}(s^{\prime})\equiv\frac{\gamma_{+}(\pm 1,s^{\prime})F(\pm
1,s^{\prime})}{\gamma_{-}(\pm 1,s^{\prime})\sigma_{0}^{2}}$. Therefore, our
theory predicts the universal scaling law $P_{st}(s)\sim(1\mp
s)^{\delta_{\pm}}$, where the exponents
$\delta_{\pm}=\int_{-1}^{1}\kappa_{\pm}(s^{\prime})P_{st}(s^{\prime})\mathrm{d}s^{\prime}-1$,
in line with the scaling law (1) discovered in real-world networks (see SM
Analytical Solutions). This finding suggests that the universal scaling law is
rather generic for coevolving networks and largely independent of microscopic
details.

Opinion topology correlation. Equation (Emergence of Polarization in
Coevolving Networks) indicates the underlying correlations between the network
structures and opinions (see SM Analytical Solutions). Incorporating Eqs.
(Emergence of Polarization in Coevolving Networks–5) with Eq. (Emergence of
Polarization in Coevolving Networks) leads to the normalized joint opinion
distribution
$R(s,s^{\prime})=\frac{Q(s,s^{\prime})}{P(s)P(s^{\prime})}=\frac{\gamma_{+}(s,s^{\prime})}{\gamma_{-}(s,s^{\prime})}$.
Since the joint opinion distribution $Q(s,s^{\prime})$ counts for the number
of edges connected by two agents with opinions $s$ and $s^{\prime}$. Therefore
the expected degree of an agent with opinion $s$,
$\bar{k}(s)=\int_{-1}^{1}Q(s^{\prime}|s)ds^{\prime}=\int_{-1}^{1}\frac{Q(s,s^{\prime})}{P(s)}ds^{\prime}=\int_{-1}^{1}\mathrm{d}s^{\prime}P_{st}(s^{\prime})\frac{\gamma_{+}(s^{\prime},s)}{\gamma_{-}(s^{\prime},s)}$,
allowing us to predict the correlation between network measure, $\bar{k}$, and
opinion $s$.

Minimal Model. To compare our theory with the empirical data, below we will
focus on a minimal realization of the proposed framework. To be specific, we
use a linear opinion dynamics model Holley and Liggett (1975); Clifford and
Sudbury (1973), $F(s,s^{\prime})=\lambda(s^{\prime}-s)$, where $\lambda$ is a
constant controlling the opinion change rate. We also assume the edge birth
and death rates satisfy
$\gamma_{\pm}(s,s^{\prime})=r_{\pm}(1+J_{\pm}ss^{\prime})$, where
$|J_{\pm}|\leq 1$ quantify the strength of homophily clustering, that is,
similar individuals to establish their relationships. We find the explicit
form of the kernel
$K(s,s^{\prime})=g\int\frac{1+J_{+}ss^{\prime}}{1+J_{-}ss^{\prime}}\frac{2(s^{\prime}-s)}{1-s^{2}}\mathrm{d}s$
for the minimal model, where parameter
$g\equiv\frac{\lambda}{\sigma_{0}^{2}}\frac{r_{+}}{r_{-}}$. Here
$\frac{\lambda}{\sigma_{0}^{2}}$ represents the opinion homogenization rate
and $\frac{r_{+}}{r_{-}}$ is the birth-death ratio for network connections.
The parameter $g$ integrates the interaction strengths of both opinion and
network dynamics. Together with Eqs. (Emergence of Polarization in Coevolving
Networks & Emergence of Polarization in Coevolving Networks), we solve $P(s)$
for the minimal model ($\langle s\rangle\neq 0$), finding our predictions
agree with the empirical data (Fig. 2a-c, see SM Minimal Model). Note that the
minimal model can also generate asymmetric $P(s)$ distribution as shown in
Fig. 2a-c. However, the average opinion $\langle s\rangle$ measured on the
empirical data is very close to zero, implying the corresponding $P(s)$ is
well approximated by a symmetric distribution. For simplicity, we will focus
only on the symmetric case ($\langle s\rangle=0$) for the discussion below.

To explore different phases, we perform the numerical simulations for various
$g$ with fixed $J_{+}$ and $J_{-}$. Figure 1b demonstrates a simulated minimal
model of $2,000$ nodes, showing a qualitative similarity with the empirical
network shown in Fig. 1a. Figure 3a shows that a bipolarized U-shaped opinion
distribution $P(s)$ emerges at a smaller $g$ ($g\approx 0.9$). By gradually
increasing the opinion change rate, we find that $P(s)$ transits from the
U-shape to M-shape, indicating partial polarization. For sufficiently large
$g$ values ($g\approx 2.4$), $P(s)$ turns into a depolarized inverse U-shaped
distribution. To quantify different phases, we define an order parameter
$s_{max}=|\arg\max_{s}P(s)|$ as the most probable opinion. We find $s_{max}=0$
for the depolarized phase (Fig. 3c), and $s_{max}=1$ for the polarized phase
(Fig. 3a). For the partially polarized phase, $s_{max}$ are ranging between
$0$ and $1$ (Fig. 3b). Figure 3d plots $s_{max}$ as a function of $g$,
indicating two phase transitions exist. The first phase transition occurs at
$g_{*}\approx 1.3$, when $s_{max}$ departs from $1$, indicating that the
system transits from polarization to partial polarization. The second phase
transition occurs at $g_{**}\approx 1.78$ when $s_{max}$ vanishes, indicating
the transition between partial polarization and depolarization. Overall,
decreasing $g$ moves the system towards polarization. Indeed, a smaller
opinion homogenization rate drives opinions far from homogenization, heading
to polarization. On the other hand, a larger birth-death ratio leads to a
lower chance for dissimilar agents to be connected; hence, similar agents tend
to cluster with their opinions homogenized, leading to polarization.

Figure 3: Phase transitions. (a)-(c) The phase transition of opinion
distributions with fixed $J_{+}=0.7$ and $J_{-}=-0.8$. These values are
determined by empirical measurements. (a) Polarization: $g=0.9$; (b) Partial
polarization: $g=1.5$; and (c) Depolarization: $g=2.4$). (d) $s_{max}$ against
the parameter $g$. ($g_{*}$: polarization (yellow) to partial polarization
(green); and $g_{**}$: partial polarization to depolarization (blue)).

To explore the opinion topology correlation, we plot predicted normalized
joint opinion distribution
$R(s,s^{\prime})=\frac{\gamma_{+}(s,s^{\prime})}{\gamma_{-}(s,s^{\prime})}=\frac{r_{+}(1+J_{+}ss^{\prime})}{r_{-}(1+J_{-}ss^{\prime})}$
in Fig. 2f, showing two opposite clustered domains around the bottom-left and
up-right corners, in line with the empirical observation (2e). Figure 2d
depicts the analytical prediction of $\bar{k}(s)$, agreeing with the empirical
measurement. Note that the empirical data shows a small peak for neutral
individuals, implying that these users have expected a large degree compared
to our theory. This discrepancy might be due to a small group of independent
media that are politically neutral but attract a large amount of connectivity
from both sides. We will leave the investigation of this phenomenon for future
studies. This prediction suggests that our model generates two highly
polarized clusters (Fig.1b), whose hubs show significant extremeness, in line
with real-world polarized networks (Fig. 1a).

Phase diagram. As our theory successfully captures polarized and depolarized
phases, one may wonder how the modeling parameters control different phases.
We focus only on symmetric cases for simplicity, i.e., $\langle s\rangle=0$.
As we discussed above, the exponent $\delta<0$ is for the polarized phase,
whereas $\delta>0$ is for both partial-polarized and depolarized phases.
Therefore, the transition between polarization and partial-polarization
emerges when the exponent $\delta=0$, leading to
${g_{*}}^{-1}+(J_{+}-J_{-})f(J_{-})=1$, where
$f(J_{-})=(1+J_{-})\int^{1}_{-1}\mathrm{d}s^{\prime}\frac{s^{\prime
2}}{(1-J_{-}^{2}s^{\prime 2})}P_{st}(s^{\prime})$.

For the transition between the partially polarized and depolarized phases, the
first derivative at the $s=0$ vanishes, i.e., $P^{\prime}(0)=0$ because of the
symmetry. However, for the M-shaped $P(s)$ the second derivative
$P_{st}^{\prime\prime}(0)>0$, whereas $P_{st}^{\prime\prime}(0)<0$ for inverse
U-shaped $P(s)$. Therefore, the transition occurs when the second derivative
vanishes, i.e., $P_{st}^{\prime\prime}(0)=0$. Substituting
$P_{st}^{\prime}(0)=0$ and $P_{st}^{\prime\prime}(0)=0$ into Eq. (Emergence of
Polarization in Coevolving Networks), we obtain
${g_{**}}^{-1}+(J_{+}-J_{-})\langle s^{2}\rangle=1$, where $\langle
s^{2}\rangle$ is the variance of the opinion distribution. Figure 4 plots the
phase diagram predicted by our model. The solid curves separate the domains
corresponding to different phases. We mark the empirical datasets in the
diagram based on the modeling parameters fitting from the data. The plot shows
that Blogosphere and FB-HC are located at the polarization phase, whereas FB-
SC is located at the depolarization phase.

Figure 4: Phase diagram. This figure shows the phase transition of opinion
distributions with parameters $g$ and $J_{\pm}$. The solid lines are the exact
solutions of the phase transition lines between depolarization (blue), partial
polarization (green), and polarization (yellow). The dash lines are the
analytical approximations. The red and purple squares represent the real data
for FB-HC and Blogosphere respectively, whereas the red triangle represents
the data of FB-SC.

In conclusion, we discover a universal scaling law for opinion distributions
empirically, characterized by a set of scaling exponents, allowing us to
quantify different polarizing phases of the real social system. We propose a
generic framework for polarization dynamics of coevolving networks where
opinion dynamics and network evolution are coupled based on two ingredients.
Compared to the existing RM model, our model predicts stable bipolarized
phases and a depolarization phase. In particular, our theory finds the
bipolarized opinion distributions and network structures analytically, in line
with empirical observations. Moreover, our framework offers the exact solution
to coevolving network dynamics, which not only counts for the observed scaling
law but also predicts the corresponding phase diagram with three different
phases.

On the other hand, our analytic solution has been found under the adiabatic
approximation. We will leave the seek of a general solution for future
investigations. Our theory provides a generic framework that can be applied to
other areas. For instance, by introducing $s_{i}=\cos(\theta_{i})$ the
framework is capable of modeling non-linearly coupled oscillators involving
background changes Kuramoto (1975); Sepulchre  _et al._ (2007); Wiesenfeld
_et al._ (1998). Most importantly, our results potentially impact the
understanding of human society across disciplines including social and
political science.

###### Acknowledgements.

C.S. was supported partially by the National Science Foundation under Grants
2150830 and IBSS-1620294, the Institute of Education Sciences under Grant
R324A180203, and the National Institutes of Health under Grant R01DC018542.
N.F.J. was funded by AFOSR grants FA9550-20-1-0382 and FA9550-20-1-0383. J.L.
and S.H. contributed equally to the work.

## References

  * Bramson  _et al._ (2016) A. Bramson, P. Grim, D. J. Singer, S. Fisher, W. Berger, G. Sack, and C. Flocken, The Journal of Mathematical Sociology 40, 80 (2016).
  * DiMaggio  _et al._ (1996) P. DiMaggio, J. Evans, and B. Bryson, American journal of Sociology 102, 690 (1996).
  * Matakos  _et al._ (2017) A. Matakos, E. Terzi, and P. Tsaparas, Data Mining and Knowledge Discovery 31, 1480 (2017).
  * Manrique and Johnson (2018) P. D. Manrique and N. F. Johnson, Physical Review E 97, 032311 (2018).
  * Bakshy  _et al._ (2015) E. Bakshy, S. Messing, and L. A. Adamic, Science 348, 1130 (2015).
  * Cinelli  _et al._ (2021) M. Cinelli, G. D. F. Morales, A. Galeazzi, W. Quattrociocchi, and M. Starnini, Proceedings of the National Academy of Sciences 118 (2021).
  * Del Vicario  _et al._ (2016a) M. Del Vicario, A. Bessi, F. Zollo, F. Petroni, A. Scala, G. Caldarelli, H. E. Stanley, and W. Quattrociocchi, Proceedings of the National Academy of Sciences 113, 554 (2016a).
  * Garimella  _et al._ (2018a) K. Garimella, G. D. F. Morales, A. Gionis, and M. Mathioudakis, ACM Transactions on Social Computing 1, 1 (2018a).
  * Del Vicario  _et al._ (2016b) M. Del Vicario, G. Vivaldo, A. Bessi, F. Zollo, A. Scala, G. Caldarelli, and W. Quattrociocchi, Scientific reports 6, 1 (2016b).
  * Borge-Holthoefer  _et al._ (2015) J. Borge-Holthoefer, W. Magdy, K. Darwish, and I. Weber, in  _Proceedings of the 18th ACM Conference on Computer Supported Cooperative Work & Social Computing_ (2015) pp. 700–711.
  * Garimella  _et al._ (2018b) K. Garimella, G. De Francisci Morales, A. Gionis, and M. Mathioudakis, in  _Proceedings of the 2018 world wide web conference_ (2018) pp. 913–922.
  * Cota  _et al._ (2019) W. Cota, S. C. Ferreira, R. Pastor-Satorras, and M. Starnini, EPJ Data Science 8, 1 (2019).
  * Lazer  _et al._ (2009) D. Lazer, A. Pentland, L. Adamic, S. Aral, A.-L. Barabási, D. Brewer, N. Christakis, N. Contractor, J. Fowler, M. Gutmann, _et al._ , Science 323, 721 (2009).
  * Holme and Newman (2006) P. Holme and M. E. J. Newman, Physical Review E 74, 056108 (2006).
  * Iniguez  _et al._ (2009) G. Iniguez, J. Kertész, K. K. Kaski, and R. A. Barrio, Physical Review E 80, 066119 (2009).
  * Vazquez  _et al._ (2007) F. Vazquez, J. C. González-Avella, V. M. Eguíluz, and M. San Miguel, Physical Review E 76, 046120 (2007).
  * Flache and Macy (2011) A. Flache and M. W. Macy, The Journal of Mathematical Sociology 35, 146 (2011).
  * Baumann  _et al._ (2020) F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, Physical Review Letters 124, 048301 (2020).
  * Baumann  _et al._ (2021) F. Baumann, P. Lorenz-Spreen, I. M. Sokolov, and M. Starnini, Physical Review X 11, 011012 (2021).
  * Starnini  _et al._ (2016) M. Starnini, M. Frasca, and A. Baronchelli, Scientific reports 6, 1 (2016).
  * Sasahara  _et al._ (2021) K. Sasahara, W. Chen, H. Peng, G. L. Ciampaglia, A. Flammini, and F. Menczer, Journal of Computational Social Science 4, 381 (2021).
  * Clifford and Sudbury (1973) P. Clifford and A. Sudbury, Biometrika 60, 581 (1973).
  * Holley and Liggett (1975) R. A. Holley and T. M. Liggett, The annals of probability , 643 (1975).
  * Black (1948) D. Black, Journal of political economy 56, 23 (1948).
  * Adamic and Glance (2005) L. A. Adamic and N. Glance, in  _Proceedings of the 3rd international workshop on Link discovery_ (2005) pp. 36–43.
  * Maslov and Sneppen (2002) S. Maslov and K. Sneppen, Science 296, 910 (2002).
  * Deffuant  _et al._ (2000) G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, Advances in Complex Systems 3, 87 (2000).
  * DeGroot (1974) M. H. DeGroot, Journal of the American Statistical Association 69, 118 (1974).
  * Friedkin and Johnsen (1990) N. E. Friedkin and E. C. Johnsen, Journal of Mathematical Sociology 15, 193 (1990).
  * Sood and Redner (2005) V. Sood and S. Redner, Physical review letters 94, 178701 (2005).
  * Kuramoto (1975) Y. Kuramoto, in  _International symposium on mathematical problems in theoretical physics_ (Springer, 1975) pp. 420–422.
  * Sepulchre  _et al._ (2007) R. Sepulchre, D. A. Paley, and N. E. Leonard, IEEE Transactions on automatic control 52, 811 (2007).
  * Wiesenfeld  _et al._ (1998) K. Wiesenfeld, P. Colet, and S. H. Strogatz, Physical Review E 57, 1563 (1998).


* Zhao et al. (2022) Sen Zhao, Wei Wei, Ding Zou, and Xianling Mao. 2022\.  Multi-View Intent Disentangle Graph Networks for Bundle Recommendation.  _Proceedings of the AAAI Conference on Artificial Intelligence_ 36, 4 (June 2022), 4379–4387.  https://doi.org/10.1609/aaai.v36i4.20359
  * Zhu et al. (2020) Nengjun Zhu, Jian Cao, Yanchi Liu, Yang Yang, Haochao Ying, and Hui Xiong. 2020\.  Sequential Modeling of Hierarchical User Intention and Preference for Next-item Recommendation. In _Proceedings of the 13th International Conference on Web Search and Data Mining_. 807–815.  https://doi.org/10.1145/3336191.3371840
  * Zhu et al. (2022) Nengjun Zhu, Jian Cao, Xinjiang Lu, and Hui Xiong. 2022\.  Learning a Hierarchical Intent Model for Next-Item Recommendation.  _ACM Transactions on Information Systems_ 40, 2 (April 2022), 1–28.  https://doi.org/10.1145/3473972
  * Zhu et al. (2023a) X. Zhu, C. Bu, B. Dong, S. Ji, Y. He, and X. Wu. 2023a.  IKGN: Intention-aware Knowledge Graph Network for POI Recommendation. In _2023 IEEE International Conference on Data Mining (ICDM)_. 908–917.  https://doi.org/10.1109/ICDM58522.2023.00100
  * Zhu et al. (2023b) Xiaoyu Zhu, Chenyang Bu, Bingbing Dong, Shengwei Ji, Yi He, and Xindong Wu. 2023b.  IKGN: Intention-aware Knowledge Graph Network for POI Recommendation. In _2023 IEEE International Conference on Data Mining (ICDM)_. 908–917.  https://doi.org/10.1109/ICDM58522.2023.00100