**footnotetext: These authors contributed equally.

# Multi-population Opinion Dynamics Model

Tigran Bakaryan∗ University of Illinois at Urbana-Champaign, USA
<EMAIL_ADDRESS>,  Yuliang Gu∗ University of Illinois at Urbana-
Champaign, USA<EMAIL_ADDRESS>,  Naira Hovakimyan  University of
Illinois at Urbana-Champaign, USA<EMAIL_ADDRESS>,  Tarek Abdelzaher
University of Illinois at Urbana-Champaign, USA<EMAIL_ADDRESS>and
Christian Lebiere  Carnegie Mellon University, Pittsburgh, USA<EMAIL_ADDRESS>
###### Abstract.

We introduce multi-population opinion dynamics models linked to the bounded
confidence model, aiming to explore how interactions between individuals
contribute to the emergence of consensus, polarization, or fragmentation.
Existing models either neglect agent similarities, sacrificing accuracy for
scalability, or prioritize accuracy by introducing agent-wise connections,
constraining scalability. Our proposed model captures similarities between
agents in scalable matter.

In our setting, agents similarities are defined by their group affiliations.
Specifically, each sub-population is characterized by its distribution, and
the closeness between two sub-populations is measured by the Wasserstein
distance of their corresponding distributions. This leads to two mutually
connected dynamics: micro, the individual-based dynamics, and the macro, the
distribution-based one. The individual-wise interactions take into account the
population-wise interactions (similarities), and the population-wise
interactions are updated based on the individual-wise interactions.

We have proven the well-posedness of our models. Additionally, we conducted
several simulations to mimic certain complex social phenomena.

###### Key words and phrases:

Opinion Dynamics, Bounded Confidence, Multi-Population Model, Mean-Field
Limit, System of Continuity Equations.

This work was supported by DOD HQ00342110002.

##  1\. Introduction

The advancement of technologies, [1], increased the accessibility of online
social networks, such as Facebook, X (Twitter), WeChat, and LinkedIn, in our
daily lives. Hence, the evolution of opinions within these platforms has
become a crucial area of study. Understanding how individuals influence the
decisions of others in social networks has recently emerged as a significant
and widely discussed research area, [2, 3]. One of the main challenges of
working with online social networks (OSNs) is the huge number of agents
(users). In this paper, to study the opinion evolution in OSNs, we follow the
common approach of ”self-organized” models. In these models, the system’s
global behavior results from accumulating local interactions. Particularly, we
focus our study on the opinion dynamics models.

Opinion dynamics models aim to describe how opinions form and spread within a
population, [4, 5, 6, 7]. In these models, each individual (or ’agent’) of a
given population holds an opinion on a particular topic, quantified by a real
number. Furthermore, the models provide rules for how these individuals
interact with each other and how these interactions influence their opinions.

Before introducing our opinion dynamics models, we briefly overview the
bounded confidence (BC) models. The BC models were introduced in [7, 5], and
the motivation behind them comes from the concepts of ”biased assimilation”,
[8], and ”confirmation bias”, [9]. These theories suggest that individuals
tend to seek out sources that align with their initial opinions and are more
heavily influenced by others whose opinions are close to their own. In other
words, people have tendency to search for information that supports their
existing beliefs and ignore the ones that contradict them, [10]. This
phenomenon has been widely observed in the Web domain, see [11, 12] and the
references therein. Furthermore, the features of BC models are explored in
real-world studies of opinion formations see, for example, [13, 14]. Having
this, we focus our study on the models related to BC models.

The bounded confidence models were proposed to understand the opinion
formation of the population (to understand macroscopic behavior); that is, how
individual interactions lead to the formation of consensus, polarization, or
fragmentation. In the BC models, each individual within a group holds a
specific opinion, which the individual may adjust based on the interaction
with others, provided the difference between their viewpoints is within a
certain threshold (the ’bounded confidence’). Generally, the BC models (for
more details see Section 2) for a given population $\Lambda$ with the
population state $\mathbf{X}_{t}=\\{X^{i}_{t}\\}_{i\in\Lambda}$ at time $t$
can be described by the following dynamics

(1.1)
$\frac{\mathrm{d}X^{i}_{t}}{\mathrm{d}t}=\sum_{j\in\Lambda}f_{i}(\mathbf{X}_{t},\varepsilon_{i})X_{t}^{j},$

where $X^{i}_{t}$ is the state of agent $i$ at time $t$ and the function,
$f_{i}$, describes the interaction intensity of agent $i$ with other
individual within the population. In (1.1), the coefficients
$\varepsilon_{i}>0$ represent the bounded confidence; that is, agent $i$ does
not interact with agents who are outside its $\varepsilon_{i}$\- neighborhood.
Here, the micro-level interactions involve individuals adjusting their
opinions based on their peers’ opinions. The macro-level behavior that emerges
from these interactions may lead to global consensus or polarization,
depending on factors such as the initial distribution of opinions, and the
size of the confidence threshold of the individuals.

Simplified as they are, BC models still provide a powerful framework for
understanding how individual-level cognitive and social processes can lead to
macro-level patterns. However, known models often face trade off: either
scarify accuracy by simplifying and uniforming the agents interactions rule
(i.e. using the same simple $f_{i}$ in (1.1) for all agents) to ensure
scalability, see [7], or they define agent-wise interaction rule (i.e.
employing complex $f_{i}$ for all agents individually) ensuring accuracy at
the expense of scalability, see [15]. In this paper, we introduce opinion
dynamics models that balance between scalability and accuracy by capturing
similarities between agents in a scalable manner. Therefore, the proposed
models address the challenge of finding the right equilibrium between
scalability and accuracy, empowering our modeling ability. In Section 7, using
our models, we reproduce two interesting social phenomena.

Crucial social and cognitive principles that interface with BC models are
homophily [16] and cognitive dissonance [17]. The former refers to a tendency
of individuals to associate and bond with similar ones, and the latter refers
to a psychological state in which a person experiences discomfort or tension
due to holding two or more contradictory beliefs. Note that the impact of
these principles may manifest beyond the individual level, resulting in
intergroup interactions at a larger level, particularly when different groups
have incompatible beliefs or values [16, 17, 18]. In the context of opinion
dynamics models, it suggests that individuals may not always update their
opinions based purely on the opinions of others. Instead, their ”group
identities” and the opinions prevalent in their social groups may heavily
influence their opinion adjustments. Examples are social groups based on
various attributes such as ethnicity, nationality, gender, religion, political
affiliation, profession, etc.

Following the discussion above, we define similarities between agents based on
their group affiliations. Particularly, in our setting each group (population)
is described by its distribution and the closeness of two groups is defined by
Wasserstein distance between them (see Section 3). Relaying on this, we
introduce population biased or multi-population opinion dynamics models (for
more details see Sections 4-5). More precisely, suppose that the population
$\Lambda$ is divided into $U$ sub-populations and $\Lambda_{i}\in U$ is the
sub-population of agent $i$. Then, the dynamics in (1.1) can be written as
follows

(1.2)
$\frac{\mathrm{d}X^{i}_{t}}{\mathrm{d}t}=\sum_{j\in\Lambda}g_{i}(\Lambda_{i},\Lambda_{j})f_{i}(\mathbf{X}_{t},\varepsilon_{i})X_{t}^{j},$

where the function, $g_{i}$, is the population-wise interactions intensity
function. Throughout the paper, we consider three main classes of population
biased models. The first one is called identity invariant model and studied in
Section 4. In this case, group bias is defined only based on the orientation
of groups at the initial time (in (1.2) the functions $g_{i}$ depend only on
initial distance between sub-populations in $U$). This model is particularly
useful in scenarios where static identifiers, such as race, gender, or long-
standing cultural values, significantly influence group dynamics. In Section
5, we preset the second class, which is called time varying model. Here, the
group bias is changing over the time. This model is suitable in scenarios
where groups adapt or reconfigure in response to changing circumstances, such
as political shifts or social upheavals. Furthermore, the corresponding mean-
field limit is also considered (see Section 5.2), which leads to a system of
continuity equations. The third class of models, we consider in this paper is
called multi-identity multi-population model. Section 6 is devoted to study of
multi-identity model. In contract to previous two cases, in this model agent
may have several identities. That is, we assume that we are given, $V$,
partitions of the population $\Lambda$ and $\Lambda_{i}^{k}\in U_{k}$
corresponds to sub-population of agent $i$ with respect to $U_{k}\in V$
partition. This leads to the following generalization of (1.2)

$\frac{\mathrm{d}X^{i}_{t}}{\mathrm{d}t}=\sum_{j\in\Lambda}\Bigg{(}\sum_{k\in
V}g^{k}_{i}(\Lambda^{k}_{i},\Lambda^{k}_{j})\Bigg{)}f_{i}(\mathbf{X}_{t},\varepsilon_{i})X_{t}^{j},$

where now $g^{k}_{i}$, is the population-wise interactions intensity function
with respect to partition $U_{k}$. Here, the coefficients $g^{k}_{i}$ may be
defined according to both identity-invariant model or time-varying model. The
multi-identity multi-population model enables us to study complex social
phenomena where individuals may have several group affiliations. For example,
in the US, people’s opinions on the ’Abortion law’ are mostly influenced by
three major factors: gender, political orientation, and religion.

Note that in all these models, scalability is ensured due to the definition of
similarities between populations based on distributions. Furthermore, to
enhance the accuracy of the models, the intensity functions (i.e., $f_{i}$,
$g_{i}$, $g^{k}_{i}$) describing interactions between agents or populations
are determined through kernel functions (see Section 2). This approach
provides a comprehensive method to define individual or population-wise
similarity while preserving the main characteristics of the individual or
population.

##  2\. Local Kernel Model

This section provides insights into BC models and their general form.
Furthermore, following the idea of interaction potential in [19, 20]), we
introduce the local kernel-based BC model and prove the well-posedness of the
model, see Proposition 2.2.

Consider a group/population of $N$ agents, each of which is identified by an
index $i\in I=\\{1,2,...,N\\}$ and has state (opinion) $X^{i}_{t}$ at time $t$
taken value from the state space $\Omega=[-1,1]$. The associated distance
between states is the $L_{1}$ metric on ${\mathbb{R}}$; that is,

(2.1) $d(X^{i},X^{j})=|X^{i}-X^{j}|,\quad i,j\in I.$

We say a subset $\Lambda$ of $I$ is a sub-group/population and denote its
collective state by $\mathcal{X}^{\Lambda}_{t}=\\{X_{t}^{i}:i\in\Lambda\\}$
defined on the product space $\Omega^{|\Lambda|}\subset\Omega^{N}$. If
$\Lambda=I$, $\mathcal{X}^{\Lambda}_{t}$ stands for the entire group state at
time $t$. Each group is described by its distribution,
$\mu^{\Lambda}_{t}\in\mathcal{P}(\Omega)$. In other words, the opinion of the
agent $i$, $X_{t}^{i}\in\mathcal{X}^{\Lambda}_{t}$, of the group
$\mathcal{X}^{\Lambda}_{t}$ is an independent sample of the group
distribution, $\mu^{\Lambda}_{t}$. Later, we show (see Proposition 2.2) that
the opinion profile of the agents is defined on whole ${\mathbb{R}}$ with
support on $\Omega$, hence, the distribution, $\mu^{\Lambda}_{t}$, of the
group $\Lambda$ is defined on ${\mathbb{R}}$ and compactly supported on
$\Omega$. For our analysis, we use the latest setting.

###### Remark 2.1.

In the context of the opinion dynamics model, opinions are quantified on a
scale where $-1$ and $1$ represent the two polar extremes of a given
viewpoint. Specifically, a state of $-1$ corresponds to one extreme opinion,
while $1$ signifies its direct opposite. Values between these extremes
indicate intermediate opinions, capturing the spectrum of viewpoints on the
issue at hand. The interpretation of these numerical values heavily depends on
the specific topic and context being studied.

We start our discussion by examining the general framework of the opinion
dynamics model. In this model, agents continuously update their opinions, a
process that is heavily influenced by their interactions with other agents.
This dynamic can be mathematically represented as:

(2.2)
$\frac{\mathrm{d}X^{i}_{t}}{\mathrm{d}t}=-\alpha_{i}X^{i}_{t}+\alpha_{i}f(\mathcal{X}^{I}_{t}),$

where $f$ is the interaction rule among agents, and $\alpha_{i}\in[0,1]$
quantifies the degree of stubbornness of each agent, as discussed in various
studies [21, 22]. Many classical opinion dynamics models [4, 6, 5, 7] suggests
that the core mechanism driving these interactions is the similarity
difference (or alignment) between agents. Specifically, an agent’s response is
governed by its relative difference from the states of other agents. Within
this context, the fundamental structure outlined in Equation (2.2) can be
expanded into a weighted process as follows:

(2.3) $\frac{\mathrm{d}X^{i}_{t}}{\mathrm{d}t}=-\alpha
X^{i}_{t}+\alpha\sum_{j=1}^{N}w_{t}(X_{t}^{i},X_{t}^{j},\varepsilon)X_{t}^{j},$

where $w_{t}(\cdot)$ is a time-dependent coefficient quantifying the influence
exerted by agent $j$ on agent $i$. Despite the linear appearance of the
individual update rule in Equation (2.3), the function $w_{t}(\cdot)$ is
capable of introducing complex, non-linear dynamics into the opinion updating
process.

###  2.1. Local Kernel Methods

In this section, we present a general framework that integrates the widely-
accepted synchronous update rule within a bounded confidence interval. Bounded
confidence models, as elaborated in seminal works [7, 5], primarily feature
local interactions. In these models, an agent’s opinion is influenced only by
a specific subset of peers, often referred to as its neighbor set. This
concept is rooted in the theories of bounded rationality [23] and confirmation
bias [24], reflecting the cognitive and psychological complexities inherent in
the opinion-formation process [25].

Furthermore, the role of the metric, as defined in Equation (2.1), is critical
in shaping the opinion space and, consequently, the outcomes of the model.
While the use of Euclidean distance is a straightforward and intuitive
approach, it often falls short in accurately representing the multifaceted
nature of opinion dynamics in real-world scenarios. Particularly, the non-
linear nature of social interactions and opinion formation, influenced by
factors such as long-term and short-term memory, societal norms, and social
identity biases, demands a more nuanced metric. To better address these
complexities, we generalized the BC models based on the interacting potential
function (see for example [19, 26, 27, 20]) and introduce the local kernel
methods to characterize such dynamics. In our setting, the kernel function not
only depends on the distance between states (see (2.1)) but also on the state
of the considered agent.

Thus, based on the general framework (2.3), the update rule for individual
agent $i$ is defined by

(2.4) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z}\sum_{j=1}^{N}\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon^{i})X_{t}^{j}\\\
X^{i}_{0}\sim\mu^{i}_{0},\end{cases}$

where
$\kappa_{d}:{\mathbb{R}}\times{\mathbb{R}}\times{\mathbb{R}}^{+}\to[0,1]$ is a
kernel function with the Euclidean distance $d$ (2.1), $\mu^{i}_{0}$ is the
initial distribution of the population of agent $i$ with
$\operatorname{supp}(\mu^{i}_{0})\subseteq\Omega$, $\varepsilon_{i}$ is a
agent-wise confidence threshold defining agent $i$ neighborhood, $\alpha_{i}$
is level of stubbornness of the agent, and $Z$ is a normalization factor given
by

$Z=Z(X_{t}^{i},\varepsilon_{i})=\sum_{j=1}^{N}\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i}).$

In this particular context of bounded confidence, the kernel may have the
following form

(2.5)
$\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i})=\begin{cases}\kappa_{d}(X_{t}^{i},X_{t}^{j}),\quad
d(X_{t}^{i},X_{t}^{j})\leqslant\varepsilon_{i}\\\ 0,\quad\quad\quad
otherwise.\end{cases}$

The kernel employs locally within the state space $\Omega$. The kernel method
facilitates the modeling of intricate individual cognitive behaviors,
especially when dealing with large populations where an agent-by-agent
description of interactions is infeasible.

In the subsequent sections, we demonstrate that tailoring kernels based on
foundational principles from cognitive science and group dynamics can provide
a more adaptive and potent framework. Before that, we shall prove the well-
posedness of (2.4); that is, there exists a global solution to the system in
(2.4), and each component of the solution stays in $\Omega$.

###### Proposition 2.2.

Let the kernel $\kappa_{d}(\cdot,\cdot,\varepsilon)\to[0,1]$ is piece-wise
continuous in ${\mathbb{R}}^{2}$ and
$\kappa_{d}(x,x,\varepsilon)\geqslant\delta_{0}>0$ for all $x\in[-1,1]$. Then,
the ODEs system in (2.4) has a global solution, $\\{X_{t}^{i}\\}_{i=1}^{N}$.
Moreover, the solution $\\{X_{t}^{i}\\}_{i=1}^{N}$ belongs to $\Omega^{N}$.

###### Proof.

We write the ODEs system in (2.4) in following compact form

(2.6) $\begin{cases}\frac{{\rm d}\mathbb{X}_{t}}{{\rm
d}t}=\mathbb{F}(\mathbb{X}_{t})\\\
\mathbb{X}_{0}\sim\mathbb{\mu}_{0},\end{cases}$

where $\mathbb{\mu_{0}}=(\mu_{0}^{1},\dots,\mu_{0}^{N})$,
$\mathbb{X}_{t}=(X_{t}^{1},\dots,X_{t}^{N})$ and
$\mathbb{F}(\mathbb{X}_{t})=(F_{1}(\mathbb{X}_{t}),\dots,F_{N}(\mathbb{X}_{t}))$
with

(2.7)
$\begin{split}F_{i}(\mathbb{X}_{t})&=-\alpha_{i}X^{i}_{t}+\alpha_{i}\sum_{j=1}^{N}\frac{\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i})}{Z}X_{t}^{j}\\\
&=\alpha_{i}\sum_{j=1}^{N}\frac{\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i})}{Z}(X_{t}^{j}-X_{t}^{i}).\end{split}$

Note that the assumptions on the kernel, $\kappa_{d}$, imply that the function
$\mathbb{F}$ is a piece-wise continuous function. Therefore, because the
system of ODEs in (2.6) is autonomous by [28, Theorem 1, page 77 ] and [28,
Point 4, page 81] it follows that there exists a solution $\mathbb{X_{t}}$ to
(2.6).

Next, we prove that any solution, $\mathbb{X_{t}}$, to (2.6) remains within
$\Omega^{N}$. Let $S(t)$ be the convex hull of
$\mathbb{X}_{t}=(X_{t}^{1},\dots,X_{t}^{N})$ and
$s_{min}(t)=\min\\{X_{t}^{1},\dots,X_{t}^{N}\\}$,
$s_{max}(t)=\max\\{X_{t}^{1},\dots,X_{t}^{N}\\}$ are extremal points of
$S(t)$. Because $S(0)\subset\Omega$ to prove that the solution,
$\mathbb{X_{t}}$, to (2.6) belongs to $\Omega^{N}$, it is enough to prove that
the set $S(t)$ is non-increasing. This is equivalent to $s_{min}(\cdot)$ being
monotone non-decreasing and $s_{max}(\cdot)$ being monotone non-increasing.

Here, we only prove that $s_{min}(\cdot)$ is monotone non-decreasing, the
proof for $s_{max}(\cdot)$ being monotone non-increasing is similar, so we
omit it. Let at time $t$ $s_{min}(t)=X_{t}^{k}$. According to (2.6) and (2.7)
$X_{t}^{k}$ solves

(2.8) $\frac{{\rm d}X^{k}_{t}}{{\rm
d}t}=\alpha_{k}\sum_{j=1}^{N}\frac{\kappa_{d}(X_{t}^{k},X_{t}^{j},\varepsilon_{i})}{Z}(X_{t}^{j}-X_{t}^{k}).$

Recalling that $X_{t}^{i}\in{\mathbb{R}}$, we note that $y\in S(t)$ if and
only if $s_{min}(t)\leqslant y\leqslant s_{max}(t)$. Hence,
$X_{t}^{k}=s_{min}(t)\leqslant X_{t}^{j}\in S(t)$. Using this and the fact
that $\alpha_{k}>0$ and
$\frac{\kappa_{d}(X_{t}^{k},X_{t}^{j},\varepsilon_{i})}{Z}\geqslant 0$ from
(2.8) follows that $\frac{{\rm d}X^{k}_{t}}{{\rm d}t}\geqslant 0$. Thus,
$s_{min}(t)$ is non-decreasing. Consequently, any solution, $\mathbb{X_{t}}$,
to (2.6) stays in $\Omega^{N}$. This with the standard extending solution
arguments (see, for example, [29, Theorem 3.3]) implies the global existence
of solutions. ∎

###### Remark 2.3.

If to the assumptions of Proposition 2.2, we also add locally Lipschitzness of
the kernel, $\kappa_{d}(\cdot,\cdot,\varepsilon)$, then, the solution to (2.6)
is also unique.

###  2.2. Examples of Local Kernel Functions

To illustrate the versatility of the general framework described by the
equation (2.4), we present several concrete examples of kernel functions.
These examples showcase how different kernels can be applied to model varying
interaction dynamics:

  * •

Uniform kernel: This kernel assigns equal weight to all agents within the
confidence interval, disregarding the exact distance between opinions.

(2.9) $\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)=\begin{cases}1,&\text{if
}d(X_{t}^{i},X_{t}^{j})\leqslant\varepsilon,\\\
0,&\text{otherwise}.\end{cases}$

  * •

Triangular kernel: The influence decreases linearly with the increase in
distance between opinions, reaching zero at the edge of the confidence
interval.

$\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)=\begin{cases}\varepsilon-d(X_{t}^{i},X_{t}^{j}),&\text{if
}d(X_{t}^{i},X_{t}^{j})\leqslant\varepsilon,\\\
0,&\text{otherwise}.\end{cases}$

  * •

Exponential kernel: This kernel introduces a non-linear decay of influence
with distance, controlled by the parameters $\gamma$ and $\alpha$:

(2.10)
$\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)=\begin{cases}\exp\left(-\gamma
d(X_{t}^{i},X_{t}^{j})^{\alpha}\right),&\text{if
}d(X_{t}^{i},X_{t}^{j})\leqslant\varepsilon,\\\
0,&\text{otherwise},\end{cases}$

where $\gamma,\alpha>0$ are parameters that modulate the decay rate and
smoothness of the influence. Notably, when $\alpha=2$, it’s a squared
exponential kernel (a type of radial basis function kernel); when $\alpha=1$,
it takes the form of an absolute exponential kernel (Ornstein–Uhlenbeck
kernel).

These kernel functions are integral in modeling the nuanced interactions
within the opinion dynamics, allowing for a flexible representation of how
agents influence each other based on the proximity of their opinions.

(a) Basic Kernel Functions.

(b) Exponential Kernels with Varying $\gamma$.

Fig. 1. Illustration of different local kernel functions: for all simulations,
the initial opinion profiles are sampled from a uniform distribution, and
model-related parameters, except for the kernels, are kept constant (i.e.,
bounded confidence $\varepsilon=0.6$, level of stubbornness $\alpha=0.1$,
number of agents $N=500$).

Figures 1(a) and 1(b) show the trajectories of opinion profiles over time
under the influence of different local kernels. The results illustrate how the
shape of the kernels affects the collective dynamics of opinions and the
resulting patterns in the steady state. The uniform-like shape of the kernels
(as seen in the first plots of Figures 1(a) and 1(b)) indicates a higher
openness of agents to their neighbors, while a normal-like shape suggests
greater stubbornness (as seen in the third plots of Figures 1(a) and 1(b)). A
deviation from uniform-like to normal-like shape encourages heterogeneous
opinion configurations (opinion fragmentation), because the intensity and
range of influence among agents become more localized and selective.
Furthermore, the nuanced differences between kernels of similar shapes affect
the dynamics, particularly in terms of reaching a steady state. For instance,
slight variations in the steepness or spread of a normal-like kernel could
either hasten the formation of consensus or deepen the divides, influencing
how quickly and to what extent a steady state of opinion distribution is
reached

Observe that several kernel functions, notably the exponential kernels,
inherently possess a locality characteristic: the intensity of interactions
diminishes with increasing distance across the state space. In our model, this
local behavior is further emphasized by the parameter $\varepsilon$. This
approach not only distinguishes our dynamics from models where such locality
is an intrinsic property of the kernel functions but also introduces a hard
cut-off in interactions. Such a truncation sets the stage for the group-level
interaction dynamics, which we shall delve into later in this paper.

###### Remark 2.4.

If $\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon^{i})$ is a uniform kernel (see
equation in (2.9)), the dynamics described by (2.4) resembles the classical
Hegselmann-Krause opinion dynamics model [7]. The derivation of this
relationship is straightforward and is omitted here for brevity.

###  2.3. General Kernel Functions

Until now, we’ve restricted our focus on isotropic kernel functions, i.e.,
kernels that depend solely on the distance between agents, i.e.
$\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)=\kappa(d(X_{t}^{i},X_{t}^{j}),\varepsilon)$.
These kernels offer several appealing properties: they are inherently simple
and intuitive within the context of opinion dynamics, and they maintain the
essential character of local interaction, particularly beneficial when dealing
with a large number of agents. In many established bounded confidence models
of opinion dynamics and their variants, this isotropy is a fundamental
assumption, primarily to model Homophily – the tendency of individuals to
connect due to similar characteristics – as observed in social networks [16].

In our formulation (see (2.5)), we broaden the scope beyond this isotropic
constraint, allowing $\kappa_{d}(\cdot)$ to be influenced not only by the
distance $d(X_{t}^{i},X_{t}^{j})$ but also by the specific states of
$X_{t}^{i}$ and $X_{t}^{j}$, and the parameter $\varepsilon$. This adjustment
retains the model’s scalability while enhancing its capacity to replicate more
complex social dynamics. Next, we will illustrate with a concrete example how
this generalization of kernel functions enriches our model, particularly in
dynamically capturing nuances of interpersonal attraction.

###### Example 2.5.

Resistance in Extreme Opinion Regions. Research indicates that individuals
with extreme opinions tend to be more resistant to change and less open to
alternative viewpoints, a phenomenon attributed to Confirmation Bias [24] and
Selective Exposure [30]. Conversely, individuals with neutral opinions or
those newly exposed to a subject are generally more receptive to persuasion
and often experience ambivalence when confronted with conflicting arguments
[31, 32]. To model these divergent tendencies in opinion dynamics, we employ
the following kernel function:

(2.11)
$\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)=\begin{cases}\exp\Big{(}-\gamma(|X_{t}^{i}|)d(X_{t}^{i},X_{t}^{j})^{\alpha}\Big{)},&\text{if
}d(X_{t}^{i},X_{t}^{j})\leqslant\varepsilon,\\\
0,&\text{otherwise},\end{cases}$

where $\gamma(\cdot)$ is a monotonically increasing function of $|X_{t}^{i}|$,
with $\gamma(0)=0$. Figure 2 demonstrates the evolution of $\kappa_{d}(\cdot)$
with respect to the state over $[-1,1]$, for $\gamma(x)=x$.

Fig. 2. State-Dependent Kernel Function.

Under this kernel, agents with neutral opinions exhibit more openness to their
neighbors, akin to uniform weighting within the bounded confidence interval.
In contrast, agents with extreme opinions (near $-1$ or $1$) display more
stubborn behavior. Figure 3 illustrates the opinion density evolution under
such state-dependent kernel functions. Notice the formation of high-density
opinion clusters in extreme regions. As agents enter these extreme states due
to interpersonal attraction, they tend to remain there due to increased
resistance. This tendency facilitates concentration: clusters with more agents
exert a strong collective attraction force on others outside the region. Such
feedback effects provide a potential explanation for the formation of Echo
Chambers and public opinion polarization.

Fig. 3. Opinion Density Evolution under State-Dependent Kernel, equation in
(2.11): $\mu_{0}=Unif[\Omega]$, $N=300$, $\alpha=0.1$, $\varepsilon=0.6$,
$\textit{time steps}=20$, and $\textit{total number of trials}=50$.

###  2.4. Extending to Multi-Population

The local kernel methods naturally extend to a multi-population (multi-group)
setting. Consider a population that can be divided into sub-populations based
on their social categories and identities, such as age, gender, race,
education level, shared goals, and objectives. In the opinion dynamics
setting, the shared characteristics of each sub-population may exhibit a
collective interaction mode that differs across various populations. This
variation in interaction modes can be captured using different local kernel
methods. These local kernels are tailored to capture the specificities of
interactions, including the propensity to influence or be influenced, the
intensity of interactions, and the nature of information exchange within each
group.

Mathematically, the agents set $I$ of size $N$ is partitioned into $p$
disjoint subsets so that each represents a sub-populations; that is,
$P=\\{1,2,...,p\\}$, $I=\cup_{k\in P}\Lambda^{k}=\cup_{k=1}^{p}\Lambda^{k}$,
$|\Lambda^{k}|=n_{k}$ and $\sum_{k=1}^{p}n_{k}=N$. Following the discussion
above, for agent $i$ at time $t$, we denote its population and corresponding
opinion profile by $\Lambda(i)$ and $\mathcal{X}_{t}^{\Lambda(i)}$. Using
these notations, the general update rule for agent $i$ is defined by

(2.12) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\alpha_{i}\sum_{j=1}^{N}\frac{\kappa_{d}^{\Lambda(i)}(X_{t}^{i},X_{t}^{j},\varepsilon_{i})}{Z}X_{t}^{j}\\\
X^{i}_{0}\sim\mu^{i}_{0},\end{cases}$

where $\kappa_{d}^{\Lambda(i)}(\cdot)$ is a population-specific kernel
functions.

In this extended framework, we have preliminarily explored the multi-
population setting where the population-dependent bias is static. The group
identities do not change over time, resulting in a static interaction mode for
each population. Next, we will formally introduce multi-population opinion
dynamics models, where the group bias can be dynamic as well and defined
concurrently with the opinion formation. This approach helps us study various
opinion formation phenomena, such as the formation of echo chambers, the
emergence of new groups, etc. Before that, we first establish a proper and
uniform metric at the group level in the subsequent section.

##  3\. Population Similarity Measure

In this section, we define population similarities and present the motivation
behind them. Furthermore, we illustrate certain properties of Wasserstein
distance to motivate our study based on the population-wise similarities given
by the Wasserstein metric.

We begin by noting that the distribution of opinions within groups of
differing social identities can often reflect the defining characteristics of
those identities. For instance, in a survey experiment conducted by [33],
participants with varying political beliefs expressed their views on the focal
issue of wealth redistribution. The data revealed stark differences in the
opinion spectra between self-identified Democrats and Republicans.
Specifically, the opinions among Republican participants followed a
distribution resembling a normal distribution, and their views, on average,
were against wealth redistribution, but did exhibit heterogeneity. In
contrast, the opinions among Democrats followed a Gamma-like distribution,
with an overwhelming majority supporting wealth redistribution. This
illustrative example underscores how group identities are encoded in the
distribution of opinions. It also highlights why using distribution functions
provides a more comprehensive and generalizable description compared to
relying solely on descriptive statistics such as mean, variance, mode, etc.

Thus, motivated by the previous example and following the well established
theories like particle physics and mean-field games, we quantify the identity
of a given set of agents on the state space $\Omega$ by its distribution. More
precisely, for a given sub-population $\Lambda$ with corresponding states of
agents $\mathcal{X}^{\Lambda}_{t}=\\{X_{t}^{i}:i\in\Lambda\\}$, we describe
the collective state or the sub-population $\Lambda$ by distribution function
$\mu^{\Lambda}\in\mathcal{P}({\mathbb{R}})$ with
$\operatorname{supp}(\mu^{\Lambda})\subseteq\Omega$. So, the opinion of the
agent $i$, $X_{t}^{i}\in\mathcal{X}^{\Lambda}_{t}$, of the group
$\mathcal{X}^{\Lambda}_{t}$ is an independent sample of the group
distribution, $\mu^{\Lambda}_{t}$. Later, we consider two different approaches
for the determination of population distribution (see Sections 5.1 and 5.2).

The preceding probabilistic modelling also allows for a mathematical
quantification of opinion shifts over time or among different populations. We
measure the discrepancies in opinion distributions across various groups by
the statistical distance. Consider two distinct sets of agents $\Lambda^{k}$,
$\Lambda^{q}$, their opinion discrepancy
dist$(\mathcal{X}^{\Lambda^{k}}_{t},\mathcal{X}^{\Lambda^{q}}_{t})$ is

$\texttt{dist}(\mathcal{X}^{\Lambda^{k}}_{t},\mathcal{X}^{\Lambda^{q}}_{t})=d(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t}),$

where $d(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t})$ is the statistical
distance between probability distributions
$\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t}$ at time $t$. The notion of
statistical distance is used here in a broad sense. Depending on the context,
different measures of distance or divergence can be employed, such as Total
Variational Distance, $p$-Wasserstein Distance, or $f$-Divergences (a
comparison of these distances is demanding in Figure 4). However, for the
purposes of our models and analysis, we specifically employ the 1-Wasserstein
Distance. We chose this metric because it offers desirable properties from
both an interpretive and computational standpoint.

Wasserstein Distance as Population-Similarity Measure. Consider two distinct
sets of agents $\Lambda^{k}$, $\Lambda^{q}$, their distance
$d(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t})$ is:

(3.1)
$d(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t})=W_{p}(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t}),$

where $W_{p}(\cdot)$ is the Wasserstein distance of order $p$ (also commonly
called the Kantorovich-Rubinstein distance, when $p=1$). Recalling that we are
in one-dimension, the distance has an analytic form

$W_{p}(\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t})=\Big{(}\int_{\Omega}\big{|}F_{\Lambda^{k}}(x)-F_{\Lambda^{q}}(x)\big{|}^{p}dx\Big{)}^{1/p},$

where $F_{\Lambda^{k}}$, $F_{\Lambda^{q}}$ are the cumulative distribution
functions (CDF) of $\mu^{\Lambda^{k}}_{t},\mu^{\Lambda^{q}}_{t}$ respectively,
for $p=1$ and are quantiles, when $p>1$.

Fig. 4. Statistical Distance under Displacement and Dispersion Variation.

1-Wasserstein distance is well adapted to the opinion dynamics setting with
the model we considered (2.4) because:

  * •

Metric Structure: Unlike pseudo-metrics such as divergences, the Wasserstein
distance is a true metric. It satisfies important properties like non-
negativity, identity of indiscernibles, symmetry, and the triangle inequality.
These properties are crucial for quantifying differences in a focal issue
between populations in a discernible way. For instance, the identity of
indiscernibles ensures that identical opinion spectra have a zero distance
between them, providing a baseline reference point for comparisons.

  * •

Norm Feature: The Wasserstein distance of order 1 has a specific norm feature
(for more details see [34]) that makes it compatible with the absolute
distance $d(\cdot)$ in the original state space $\Omega$, i.e.
$\|\delta_{x}-\delta_{y}\|=d(x,y)$ for all $x,y\in\Omega$, where $\delta_{x}$
is Dirac delta function. In context, this compatibility allows us to naturally
extend distances between individuals to distances between entire groups. This
generalization from the individual level to the group level effectively
provides an explicit isometric embedding of the original state space $\Omega$
in a Banach space (complete normed vector space), where the dynamics between
groups take place. This paves the way for us to define interaction rules on
the mean behavior of large group later in the paper.

  * •

Robustness: The Wasserstein distance is stable under mild perturbations of the
probability distributions. This is particularly useful when dealing with
empirical distributions, which might be subject to noise or sampling errors.
In our models where agents have beliefs represented by probability
distributions, stability under mild perturbations ensures that small changes
in individual beliefs do not lead to disproportionate changes in the overall
belief landscape (See Figure 4).

The Wasserstein distance provides a comprehensive metric for dynamically
quantifying the differences in opinion profiles across diverse populations. We
have established its suitability for analyzing complex opinion dynamics within
varied social identities. In the next sections, we will delve into the
formulation of population-level interaction functions grounded in the
Wasserstein distance. Our focus will particularly be on two distinct
scenarios: static group identities, which reflect consistent societal
structures, and dynamic group identities, which capture the evolving nature of
social affiliations and perspectives.

##  4\. Identity-Invariant Multi-Population Model

In this section, we introduce the identity-invariant multi-population model,
an advanced extension of the opinion dynamics framework initially presented in
Equation 2.4 and discussed in Section 2 (see Subsection 2.4). Our model
incorporates the dynamics of group identity into the process of opinion
formation. It specifically focuses on the probabilistic structural dynamics
that are inherent in this phenomenon, providing a more nuanced understanding
of how opinions evolve within groups.

The identity-invariant multi-population model is designed to explore the
effects of static identifiers such as race, gender, or long-standing cultural
values on group opinion dynamics. By integrating these aspects of group
identity, the model offers a unique perspective on the interplay between
individual opinions and collective identity.

At the core of our model is the application of the Wasserstein distance
(Equation 3.1) as a uniform metric within the collective opinion space.
Concretely, we introduce a population-wise local kernel method in the
Wasserstein space. This kernel function, $K$, functions as an interaction
functional between sub-populations, defined by their respective probability
distribution functions. Formally, the kernel function is defined as
$K:\mathcal{P}_{1}(\mathbb{R})\times\mathcal{P}_{1}(\mathbb{R})\times\mathbb{R}^{+}\to\mathbb{R}^{+}$,
where $\mathcal{P}_{1}(\mathbb{R})$ represents the space of probability
distributions with finite first moments. The function
$K(\mu,\nu,\sigma)=K(W_{p}(\mu,\nu),\sigma)$ for
$\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$, and $\sigma\in\mathbb{R}^{+}$.

Now, based on the discussions above, we present our first multi-population
model, where the population bias has a simple form and it has a
straightforward interpretation. We extend the model in (2.12) to the multi-
population level:

(4.1) $\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z_{\beta}}\sum_{\Lambda(j)\in
I}\beta_{i}(t)K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})\sum_{j\in\Lambda(j)}\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X_{t}^{j},$

where the group interaction function $K(\cdot)$ only depends on the initial
distributions, $\beta_{i}$ is decaying factor and $Z_{\beta}$ is normalization
constant given by

$Z_{\beta}=Z_{\beta}(\mathbf{\mu}^{\Lambda}_{0},\mathbf{X}_{t},\sigma_{i},\varepsilon_{i})=\sum_{j=1}^{N}K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i}).$

In (4.1), the function $K(\cdot)$ defines the interaction between populations
(an analog of kernel functions between individuals), and the decaying factor
$\beta_{i}$ describes its intensity change over time. The well-definiteness of
the ODEs system in (4.1) follows from a more general result proved in the next
section (see Proposition 5.2).

Contrasting with individual-level dynamics common in single-population models,
our framework emphasizes group-level interactions. This distinction is
critical as it allows for a more complex interaction function $K(\cdot)$. The
function is engineered to encapsulate a broad spectrum of intergroup
relations, including social, demographic, and ideological aspects, thus
offering a richer model that mirrors the complexities of real-world dynamics.
Next, we give a few concrete examples of group interactions based on the
dynamics given in (4.1).

###### Example 4.1.

In-Group Favoritism. In-group favoritism refers to the tendency for
individuals to favor or give preferential treatment to members of their own
group over those of an out-group[35, 36]. This can be captured by the
following kernel function

$K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})=\begin{cases}\exp(-\Gamma
W_{p}(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0})),\quad&W_{p}(\mu_{\mathcal{X}^{i}_{t}},\mu_{\mathcal{X}^{j}_{t}})\leqslant\sigma_{i}\\\
0,\quad&\text{otherwise,}\end{cases}$

with the scaling parameter $\Gamma>0$ that controls the degree of group bias
(i.e., a small $\Gamma$ represents a strong preferential bias between two
populations, and vice versa). The simulation results in Figure 5 illustrate
the impact of in-group bias (plots with decreasing $\Gamma$) on the dynamics
of opinion formation over time. Three distinct conditions were simulated: no
group bias, moderate group bias, and strong group bias. In the absence of
group bias, opinions from different groups displayed considerable convergence,
suggesting a scenario where inter-group dialogue might lead to a consensus. As
the group bias increased to a moderate level, opinions within each group
showed signs of internal alignment, with a clear but diminishing inter-group
influence. In the case of strong group bias, the groups’ opinions diverged
sharply, converging strongly within their respective groups and indicating the
formation of echo chambers or polarization.

Fig. 5. In-Group Bias: initial opinion profiles $\mu_{0}^{p1}$ and
$\mu_{0}^{p2}$ are sampled from truncated Gaussian distribution, $N=400,\,\
\varepsilon=0.6,\,\ \alpha=0.5,\,\ \kappa_{d}=uniform\ kernel$, (2.9).

###### Example 4.2.

Asymmetric Group Bias. The phenomenon of asymmetric group bias is particularly
relevant in settings where one group may have a preferential view or favorable
bias towards another group, which is not equally reciprocated. To capture this
phenomenon in opinion dynamics context, we consider the following kernel
function based on the previous example (4.1) (for simplicity, a two-population
case is studied):

$K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})=\begin{cases}\exp(-\Gamma
W_{p}(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0})),\quad&W_{p}(\mu_{\mathcal{X}^{i}_{t}},\mu_{\mathcal{X}^{j}_{t}})\leqslant\sigma_{i}\
\text{and}\ i\in p_{1}\\\ 0,\quad&\text{otherwise,}\end{cases}$

with the scaling parameter $\Gamma>0$ that controls the degree of group bias
and the condition, $i\in p_{1}$, exerts this group bias asymmetrically on the
first population. In other words, this bias occurs when one group $p_{1}$ has
a preferential view or favorable bias that is not reciprocated by another
group, $p_{2}$, leading to an imbalanced influence on collective opinion
formation. As shown in Figure 6, with no bias, opinions are only exchanged
within groups, suggesting polarization. With moderate bias, group $p_{1}$
starts to exert more influence and group $p_{2}$’s opinions begin to disperse,
suggesting a shift towards the dominant group’s stance. Under strong bias,
group $p_{1}$’s opinions dominate the narrative, pulling group $p_{2}$’s
opinions towards them. This leads to a significant alignment of group
$p_{2}$’s opinions with that of group $p_{1}$.

Fig. 6. Asymmetric Group Bias: $\mu_{0}^{p1}$ and $\mu_{0}^{p2}$ are sampled
from truncated Gaussian distribution, $N=400,\,\ \varepsilon=0.6,\,\
\alpha=0.5,\,\ \kappa_{d}=uniform\ kernel$, (2.9).

In this section, we have examined the static nature of group opinion dynamics
through the lens of the identity-invariant multi-population model. Our
exploration encompassed scenarios with both in-group favoritism and asymmetric
group bias, offering insights into how static identifiers and inherent biases
shape the landscape of opinion formation within and across various groups. The
application of the Wasserstein distance and the introduction of a population-
wise local kernel method have enabled us to quantify and visualize the nuances
of these interactions in a structured and scalable manner.

In the upcoming section, we aim to explore how groups adapt or reconfigure in
response to changing circumstances, such as political shifts or social
upheavals. This shift in focus will allow us to extend our current model to
dynamically evolving environments, where group identities and opinions are not
fixed but are fluid and responsive to external stimuli.

##  5\. Time-Variant Multi-Population Model

In this section, we introduce time-varying multi-population models. In these
models, each agent from every sub-population updates its opinion according to
a given rule (dynamics). Because of this, the collective state of each sub-
population is updated, and consequently, their distributions are updated as
well. Therefore, the interaction between sub-populations changes. Based on
this, we define a time-varying multi-population opinion dynamics model, where
’time-varying’ refers to the population bias defined by the changing over time
distributions. We considered two types of time-variant models. The first one
is called the isolated one. In this case, the population distribution is
defined by the considered agents, empirical distribution. Subsection 5.1 is
devoted to the isolated model, where we prove its well-posedness, see
Proposition 5.2. The second class of model is called the integrated multi-
population model, which is introduced in Subsection 5.2. Here, agents have
access to the actual distribution of their population, which is governed by
the system of continuity equations. Theorem 5.5

Now, following the discussions above, we introduce time-varying multi-
population model

(5.1) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z_{p}}\sum_{j=1}^{N}K(\mu^{\Lambda(i)}_{t},\mu^{\Lambda(j)}_{t},\sigma_{i})\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X^{j}_{t}\\\
X^{i}_{0}\sim\mu^{i}_{0},\end{cases}$

where $\mu^{\Lambda}_{t}$ is the distribution of the sub-population $\Lambda$
at time $t$, $K(\cdot)$ is an interaction function that defines the update
rule between populations (an analog of kernel functions between individuals)
and $Z_{p}$ is a normalization factor given by

$Z_{p}=Z_{p}(\mu^{\Lambda(i)}_{t},X_{t}^{i},\sigma_{i},\varepsilon_{i})=\sum_{j=1}^{N}K(\mu^{\Lambda(i)}_{t},\mu^{\Lambda(j)}_{t},\sigma_{i})\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon_{i}).$

In this formulation, the opinion of an individual is seen to be influenced by
a combination of factors related to both the specific group $\Lambda(i)$ to
which they belong to and the broader population.

Observe that the interactions between groups and individuals are decomposed in
(5.1). So, we can rewrite the dynamics in the following form:

(5.2) $\frac{{\rm d}X^{i}_{t}}{{\rm d}t}=-\alpha
X^{i}_{t}+\frac{\alpha}{Z_{p}}\underbrace{\sum_{k=1}^{p}K(\mu^{\Lambda(i)}_{t},\mu^{\Lambda^{k}}_{t},\sigma_{i})}_{\text{intergroup
interaction}}\underbrace{\sum_{j\in\Lambda^{k}}\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X_{t}^{j}}_{\text{in-
group interaction}}.$

In Sections 5.1 and 5.2, we present two different approaches for the
determination of the sub-populations time-varying distributions in (5.1) or
(5.2). Before going to that specifications, next, we present three different
classes of population-wise kernels, which provides some insight into the
modeling capabilities the considered model.

Examples of Population-wise Kernel Functions. Here, we provide examples of
population-wise kernel functions with different use of threshold $\sigma$.

  * •

No threshold: In this type of kernels the representatives of different
populations always interact even though the distance between them
($W_{p}(\mu,\nu)>>1$) is too big. An interesting kernel of this type is

$K(\mu,\nu,\sigma)=K(\mu,\nu)=e^{-W_{p}(\mu,\nu)}.$

  * •

Threshold from above: This type of kernels does not allow interaction between
the representatives of the different populations if the distance between them
is too big ($W_{p}(\mu,\nu)>\sigma$). The general form of this type of kernels
is

$K(\mu,\nu,\sigma)=\begin{cases}K_{d}(W_{p}(\mu,\nu)),\quad&W_{p}(\mu,\nu)\leqslant\sigma\\\
0,\quad&\text{otherwise}.\end{cases}$

  * •

Threshold from below: The following class of kernels prevent the convergence
of different populations; that is, when the populations are getting to close,
$W_{p}(\mu,\nu)<\sigma$, they stop interacting. These kernel are given by

$K(\mu,\nu,\sigma)=\begin{cases}0,\quad&W_{p}(\mu,\nu)\leqslant\sigma\\\
K_{d}(W_{p}(\mu,\nu)),\quad&\text{otherwise}.\end{cases}$

Note that when the function $K(\cdot)$ is from the ”Threshold from above”
class, the threshold, $\sigma$, is intemperate as a population-wise bounded
confidence level. The population-wise bounded confidence is a natural
generalization of the individual bounded confidence but gives rich and
distinct interpretation in the context of group dynamics. It imposes a hard
cut-off on the interactions between individuals based on their group
identities. An immediate result of such a threshold in the multi-population
opinion dynamics is that agents might not be influenced by their close out-of-
group neighbors. Unlike ”Threshold from above” the threshold, $\sigma$, for
the class of ”Threshold from below” does not have its counterpart in the
individual based models. However, its appearance in the population-dependent
model is natural.

###### Example 5.1.

Group Cohesion. Group cohesion refers to the sense of solidarity, unity, and
bonding that members feel toward their group. This can manifest through shared
goals, values, or identities[37]. High group cohesion and low group cohesion
represent two ends of a spectrum that can significantly influence the
functioning, performance, and well-being of a group. In the context of opinion
dynamics, groups with weak group cohesion are more susceptible to external
opinions because there’s less internal pressure to maintain a consistent group
viewpoint. These effects can be captured by the following kernel function:

$K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})=\begin{cases}\exp(-\Gamma
W_{p}(\mu^{\Lambda(i)},\mu^{\Lambda(j)})),\quad&W_{p}(\mu^{\Lambda(i)},\mu^{\Lambda(j)})\leqslant\sigma_{i}\\\
0,\quad&\text{otherwise,}\end{cases}$

where parameter $\Gamma>0$ determines the degree of group cohesion. The
Wasserstein distance between the opinion distributions of two groups to
determine the level of interaction or influence they have on each other,
modulated by the parameter $\Gamma$. A smaller $\Gamma$ suggests a stronger
cohesive force within the group, making it less likely to be influenced by
external opinions as long as the distance between group opinions is less than
the threshold.

Fig. 7. Group Cohesion: initial opinion profiles $\mu_{0}^{p1}$ and
$\mu_{0}^{p2}$ are sampled from truncated Gaussian distribution,
$N=400,\alpha=0.4,\varepsilon=0.4,\kappa_{d}=exp\ kernel$, see (2.10).

The simulation results in Figure 7 showcase the impact of group cohesion on
opinion dynamics. In scenarios of strong group cohesion, opinions remain
tightly clustered in group, indicating a group’s resistance to external
influence and a high degree of internal consensus. For moderate group
cohesion, there is a noticeable spread in opinions in the early stage (time
step 4 to 6) and consensus in the final stage, suggesting some openness to
external ideas while retaining a level of common agreement. In contrast, weak
group cohesion leads to straight concensus, reflecting a significant openness
to external influences and a lack of group stance. The corresponding
Wasserstein Distance plots affirm these observations. Note that this example
is different from Example 4.1, where in-group favoritism was given only based
on the initial configurations.

###  5.1. Isolated Multi-Population Model

Recall that a given population of $N$ agents, $I$, is divided into $p$ sub-
populations. That is, for $k=1,2,...,p$, $I=\cup_{k=1}^{p}\Lambda^{k}$,
$|\Lambda^{k}|=n_{k}$ and $\sum_{k=1}^{p}n_{k}=N$. As before, for agent $i$ at
time $t$, we denote its population and corresponding opinion profile by
$\Lambda(i)$ and $\mathcal{X}_{t}^{\Lambda(i)}$. In this subsection, we assume
that the sub-population $\Lambda(i)$ is only defined by the agents from the
considered population $I$. This means that agents don’t know the actual state
of their sub-population (actual distribution), and they set their sub-
population state based only on their peers in isolated population, $I$. In
other words, the state or distribution of sub-population $\Lambda(i)$ is
defined by the empirical distribution of agents’ from $\Lambda(i)$. Thus, in
the multi-population opinion dynamics in (5.1), population-dependent weights
are updated based on empirical distributions. Particularly, the dynamics is
given

(5.3) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z_{p}}\sum_{j=1}^{N}K_{d}(\bar{p}_{ij}(t),\sigma_{i})\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X^{j}_{t}\\\
X^{i}_{0}\sim\bar{\mu}^{i}_{0},\end{cases}$

where

$\bar{p}_{ij}(t)=W_{p}(\bar{\mu}^{\Lambda(i)}_{t},\bar{\mu}^{\Lambda(j)}_{t}),$

and

(5.4)
$\bar{\mu}^{\Lambda(j)}_{t}(x)=\frac{1}{|\Lambda(i)|}\sum_{X_{t}^{j}\in\Lambda(i)}\delta_{X_{t}^{j}}(x),$

is empirical distribution of the population $\Lambda(j)$.

###### Proposition 5.2.

Let $p=1$ and
$K_{d}:{\mathbb{R}}^{+}_{0}\times{\mathbb{R}}^{+}_{0}\to{\mathbb{R}}^{+}_{0}$,
$\kappa_{d}:{\mathbb{R}}\times{\mathbb{R}}\times{\mathbb{R}}^{+}_{0}\to{\mathbb{R}}^{+}_{0}$
be piece-wise continuous functions. Suppose that there exits $\delta_{0}>0$
such that $K_{d}(0,\sigma_{i})\geqslant\delta_{0}$ and
$\kappa_{d}(x,x,\varepsilon_{i})\geqslant\delta_{0}$ for all $x\in[-1,1]$.
Then, the ODEs system in (5.3) has a global solution,
$\\{X_{t}^{i}\\}_{i=1}^{N}$, and $\\{X_{t}^{i}\\}_{i=1}^{N}\subset\Omega^{N}$.

###### Proof.

Note that the cumulative distribution function (CDF) of the sub-population
$\Lambda(j)$ is a piece-wise continuous function

$F^{\Lambda(j)}(y)=\frac{1}{|\Lambda(j)|}\sum_{k=1}^{|\Lambda(j)|}\mathbb{1}_{X_{t}^{k}\leqslant
y}.$

Because we are in one dimension and $p=1$, we have

$W_{1}(\bar{\mu}^{\Lambda(i)}_{t},\bar{\mu}^{\Lambda(j)}_{t})=\int_{\Omega}\big{|}F^{\Lambda(i)}(y)-F^{\Lambda(j)}(y)\big{|}{\rm
d}y.$

Therefore,

$G^{i,j}((X_{1},\dots,X_{n_{i}}),(X_{1},\dots,X_{n_{j}}))=K_{d}(W_{p}(\bar{\mu}^{\Lambda(i)}_{t},\bar{\mu}^{\Lambda(j)}_{t}),\sigma_{i}),$

is a piece-wise continuous function, hence, the right-hand side of (5.3) as
well. Thus, arguing as in Proposition 2.2, we complete the proof. ∎

Note that in (5.3), we assume that agents are aware of the whole population
state; that is, each agent knows the distributions (empirical distributions)
of all sub-populations. So, to update their opinions, agents use that
information. However, there are systems where agents do have only local
information. Next, in detail, we explain what we mean by local information.
Recall that the bounded confidence level, $\varepsilon_{i}$, of agent $i$
defines the radius of influence on agent $i$. By a scope of the agent, we
understand the maximal neighborhood of the agent that it has access. Let
$s_{i}\geqslant\varepsilon_{i}$ be the scope of agent $i$. Agent $j$ is in the
scope of agent $i$ at time $t$ if $|X_{t}^{i}-X_{t}^{j}|\leqslant s_{i}$ and
agent $i$ has access to the agent $j$ state $X^{j}_{t}$. If in addition
$|X_{t}^{i}-X_{t}^{j}|\leqslant\varepsilon_{i}\leqslant s_{i}$, then, agent
$j$ effect $i$ and the dynamics is given by

(5.5) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z_{p}}\sum_{j=1}^{N}K_{d}(\bar{p}^{s_{i}}_{ij},\sigma_{i})\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X^{j}_{t}\\\
X^{i}_{0}\sim\bar{\mu}^{i}_{0},\end{cases}$

where
$\bar{p}^{s_{i}}_{ij}(t)=W_{p}(\bar{\mu}^{\Lambda_{s_{i}}(i)}_{t},\bar{\mu}^{\Lambda_{s_{i}}(j)}_{t})$
with $\Lambda_{s_{i}}(j)$ being the set of all agents from sub-population
$\Lambda(j)$ in the scope of agent $i$ and
$\bar{\mu}^{\Lambda_{s_{i}}(j)}_{t}$ is the empirical distribution of the
population $\Lambda(j)$ in the scope agent $i$ or it is the empirical
distribution defined by $\Lambda_{s_{i}}(j)$. Note that Proposition 5.2 also
holds for (5.5).

###  5.2. Integrated Multi-Population Model

Here, we assume that the number of agents is very large (infinity) and to
describe the opinion evolution of the population, we follow the mean-field
models in [38, 39, 40] and consider the mean-field limit of (5.3). In contrast
to the isolated model, in this case, the agents have access to the actual
distribution of their population. That is, each agent is integrated with its
population.

We start our discussion with a single population case. For a given single
population with $N$ agents, we consider the dynamics in (5.1) (or (5.3) for a
single population)

(5.6) $\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm d}t}=-\alpha
X^{i}_{t}+\frac{\alpha}{Z(X_{t}^{i},\varepsilon)}\sum_{j=1}^{N}\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon)X_{t}^{j}\\\
X^{i}_{0}\sim\mu_{0},\end{cases}$

with

$Z(X_{t}^{i},\varepsilon)=\sum_{j=1}^{N}\kappa_{d}(X_{t}^{i},X_{t}^{j},\varepsilon).$

Note that $\delta_{X_{t}^{i}}(X_{t}^{j})=0$ if $i\neq j$ and
$\delta_{X_{t}^{i}}(X_{t}^{i})=1$. Using this and recalling the definition in
(5.4), heuristically, multiplying and dividing the right-hand sides in (5.6)
by $N$ and letting $N\to\infty$, we get

(5.7) $\begin{cases}\frac{{\rm d}X_{t}}{{\rm d}t}=-\alpha
X_{t}+\frac{\alpha}{\varphi(X_{t},\mu)}\int_{\Omega}\kappa_{d}(X_{t},y,\varepsilon)y~{}{\rm
d}\mu(y)\\\ X_{0}\sim\mu_{0},\end{cases}$

where $\mu(t)$ is distribution of the population and

$\varphi(X_{t},\mu)=\int_{\Omega}\kappa_{d}(X_{t},\tau,\varepsilon)~{}{\rm
d}\mu(\tau).$

For more details on the limiting argument, see [41, 20].

By letting $N\to\infty$ in (5.6), we describe (approximate) the whole
population behavior by the representative agent, which dynamics is given by
(5.7).

Next, we prove the well-posedness of (5.7) and its solution connection with
the solutions of a continuity equation. To do so, we need the following
assumption.

###### Assumption 1.

Let $\varepsilon>0$, $\kappa_{d}(\cdot,\varepsilon)\in
C^{1}({\mathbb{R}}^{2})\in$ and
$\kappa^{e}_{d}(x,y,\varepsilon)=\kappa_{d}(x,y,\varepsilon)y$. Suppose that
$\kappa_{d}(\cdot,\varepsilon)$ and $\kappa^{e}_{d}(\cdot,\varepsilon)$ are
Lipschitz in ${\mathbb{R}}^{2}$. Furthermore, assume that there exits $c>0$
such that $c\leqslant\kappa_{d}(x,y,\varepsilon)\leqslant 1$ for all
$x,y\in{\mathbb{R}}$.

The following is an example of a class of functions that satisfies the
conditions in Assumption 1.

###### Example 5.3.

Let $\varepsilon>0$ and the function $\kappa_{d}(\cdot,\varepsilon)$ is
Lipschitz in $[-2,2]^{2}$. Suppose that there exits $c>0$ such that
$c=\kappa_{d}(x,y,\varepsilon)$ for all $x,y\in{\mathbb{R}}\setminus[-2,2]$
and $c\leqslant\kappa_{d}(x,y,\varepsilon)\leqslant 1$ for all
$x,y\in{\mathbb{R}}$.

Note that the condition that the function, $\kappa^{x}_{d}(\cdot)$, is
Lipschitz in ${\mathbb{R}}$ (or in Example 5.3 the condition,
$c=\kappa_{d}(x,y,\varepsilon)$ for all $x,y\in{\mathbb{R}}\setminus[-2,2]$)
in Assumption 1 is not shrinking the set of choice of the kernel function,
$k_{d}$, because in our setting the models are determined by the values of
$k_{d}$ only defined on $\Omega$. The mentioned condition is only needed for
technical reasons.

The well-posedness of (5.7) is given in the following proposition.

###### Theorem 5.4.

Let $T,\varepsilon>0$ and suppose that Assumption 1 holds. Then, there exists
unique $X_{t}\in C^{1}[0,T]$ solving (5.7). Moreover, the population
distribution, $\mu(t,\cdot)$, solves the continuity equation

(5.8) $\begin{cases}\mu_{t}+\operatorname{div}\Big{(}\Big{(}-\alpha
x+\frac{\alpha}{\varphi(x)}\int_{\Omega}\kappa_{d}(x,y,\varepsilon)y~{}{\rm
d}\mu(t,y)\Big{)}\mu(t,x)\Big{)}=0\\\ \mu(0,x)=\mu_{0},\end{cases}$

in the sense of distributions.

provides the existence and uniqueness of solutions to the system of continuity
equations and its connection to our integrated opinion dynamics model.

###### Proof.

Consider the following equation in whole ${\mathbb{R}}$; that is,

(5.9) $\begin{cases}\frac{{\rm d}X_{t}}{{\rm d}t}=-\alpha
X_{t}+\frac{\alpha}{\varphi_{R}(X_{t},\mu)}\int_{{\mathbb{R}}}\bar{\kappa}_{d}(X_{t},y,\varepsilon)y~{}{\rm
d}\mu(y)=b(X_{t},\mu)\\\ X_{0}\sim\mu_{0},\end{cases}$

where

(5.10)
$\varphi_{R}(x,\mu)=\int_{{\mathbb{R}}}\kappa_{d}(x,\tau,\varepsilon)~{}{\rm
d}\mu(\tau),$

and $\bar{k_{d}}(\cdot,\varepsilon)\in C^{1}({\mathbb{R}}^{2})$ with

$\bar{k}_{d}(x,y,\varepsilon)=\begin{cases}\kappa_{d}(x,y,\varepsilon),\quad(x,y)\in[-2,2]^{2}\\\
0,\quad(x,y)\in{\mathbb{R}}^{2}\setminus[-3,3]^{2}.\end{cases}$

First, we prove that (5.9) and its corresponding continuity equation (see
(5.15)) has unique solution. Then, we prove that for the solution the
equations in (5.9) and (5.7) coincides. Denoting

(5.11)
$J(x,\mu)=\int_{{\mathbb{R}}}\bar{\kappa}_{d}(X_{t},\tau,\varepsilon)\tau~{}{\rm
d}\mu(\tau),$

using the definition of $\bar{k}_{d}$ and Assumption 1 for
$x_{1},x_{2}\in{\mathbb{R}}$ and
$\mu_{1},\mu_{2}\in\mathcal{P}_{1}({\mathbb{R}})$, we get

(5.12)
$\begin{split}|J(x_{1},\mu_{1})-J(x_{2},\mu_{2})|&=\Bigg{|}\int_{{\mathbb{R}}}\bar{\kappa}_{d}(x_{1},\tau,\varepsilon)\tau~{}{\rm
d}\mu_{1}(\tau)-\int_{{\mathbb{R}}}\bar{\kappa}_{d}(x_{2},\tau,\varepsilon)\tau~{}{\rm
d}\mu_{2}(\tau)\Bigg{|}\\\
&\leqslant\int_{{\mathbb{R}}}|(\bar{\kappa}_{d}(x_{1},\tau,\varepsilon)-\bar{\kappa}_{d}(x_{2},\tau,\varepsilon))\tau|~{}{\rm
d}\mu_{2}(\tau)\\\
&+\Bigg{|}\int_{{\mathbb{R}}}\bar{\kappa}_{d}(x_{1},\tau,\varepsilon)\tau~{}{\rm
d}(\mu_{1}(\tau)-\mu_{2}(\tau))\Bigg{|}\\\ &\leqslant
C\Big{(}|x_{1}-x_{2}|+W_{1}(\mu_{1},\mu_{2})\Big{)},\end{split}$

where the constant $C$ does not depend on $x_{1},x_{2}$ and $\mu_{1},\mu_{2}$.
Similarly, we prove that

(5.13)
$\Big{|}\varphi_{R}(x_{1},\mu_{1})-\varphi_{R}(x_{2},\mu_{2})\Big{|}\leqslant
C\Big{(}|x_{1}-x_{2}|+W_{1}(\mu_{1},\mu_{2})\Big{)}.$

Using the preceding estimate, (5.12), Assumption 1 and the inequalities
$\frac{1}{\varphi_{R}(x,\mu)}\leqslant\frac{1}{c}$,
$\varphi_{R}(x,\mu)\leqslant c$, $|J(x,\mu)|\leqslant C$, we obtain

$\begin{split}\Bigg{|}\frac{J(x_{1},\mu_{1})}{\varphi_{R}(x_{1},\mu_{1})}-\frac{J(x_{2},\mu_{2})}{\varphi_{R}(x_{2},\mu_{2})}\Bigg{|}&\leqslant\frac{|J(x_{1},\mu_{1})\varphi_{R}(x_{2},\mu_{2})-J(x_{2},\mu_{2})\varphi_{R}(x_{1},\mu_{1})|}{\varphi_{R}(x_{1},\mu_{1})\varphi_{R}(x_{2},\mu_{2})}\\\
&\leqslant\frac{1}{c^{2}}\Bigg{(}\Big{|}\varphi_{R}(x_{2},\mu_{2})(J(x_{1},\mu_{1})-J(x_{2},\mu_{2}))\Big{|}\Bigg{)}\\\
&+\frac{1}{c^{2}}\Bigg{(}\Big{|}J(x_{2},\mu_{2})(\varphi_{R}(x_{2},\mu_{2})-\varphi_{R}(x_{1},\mu_{1}))\Big{|}\Bigg{)}\\\
&\leqslant C\Big{(}|x_{1}-x_{2}|+W_{1}(\mu_{1},\mu_{2})\Big{)}.\end{split}$

Hence, the vector field $b$ is globally Lipschitz; that is,

(5.14) $(b(x_{1},\mu_{1})-b(x_{2},\mu_{2}))\leqslant
C\Big{(}|x_{1}-x_{2}|+W_{1}(\mu_{1},\mu_{2})\Big{)}.$

Therefore, by Theorems 2 and 3 in [42] the continuity equation corresponding
to (5.9)

(5.15)
$\begin{cases}\mu_{t}+\operatorname{div}\Big{(}b(x,\mu)\mu(t,x)\Big{)}=0\\\
\mu(0,x)=\mu_{0},\end{cases}$

has unique solution $\mu(t)\in C([0,T],\mathcal{P}_{c}^{Ac})$. Thus,
$b(x,\mu)=\mathbf{b}(t,x)$ is in time, and by (5.14) it is globally Lipschitz
in $x$ as well. Therefore, by the Picard–Lindelöf (Carathéodory) existence and
uniqueness result (see Theorem 2.2 in [43]), the ODE in (5.9) has a unique
solution, $X_{t}\in C^{1}[0,T]$.

Next, we prove that $X_{t}\in\Omega$. Note that because
$\mu(t)\in\mathcal{P}({\mathbb{R}})$, we rewrite the first equation in (5.9)
as follows

(5.16) $\frac{{\rm d}X_{t}}{{\rm
d}t}=-\frac{\alpha}{\varphi(X_{t},\mu)}\Bigg{(}\int_{{\mathbb{R}}}\bar{\kappa}_{d}(X_{t},\tau,\varepsilon)\Big{(}X_{t}-\tau\Big{)}~{}{\rm
d}\mu(\tau)\Bigg{)}.$

Recalling that $\operatorname{supp}(\mu(0))\subseteq\Omega=[-1,1]$, and using
the continuity of the solution to (5.9), we denote by $0\leqslant
t_{e}\leqslant T$ the smallest time for which $X_{t_{e}}=1$. If such $t_{e}$
does not exist or $t_{e}=T$, the proof is completed. Now, suppose that
$0\leqslant t_{e}<T$. Note that because the solution to (5.15), $\mu(t)$, is
the push of $\mu(0)$ by $X_{t}$, then,
$\operatorname{supp}(\mu(t))\subseteq\Omega$ for all $t\in[0,t_{e}]$. Hence,
we have

$\int_{{\mathbb{R}}}\bar{\kappa}_{d}(X_{t_{e}},\tau,\varepsilon)\Big{(}X_{t_{e}}-\tau\Big{)}~{}{\rm
d}\mu(\tau,t_{e})=\int_{\Omega}\bar{\kappa}_{d}(1,\tau,\varepsilon)\Big{(}1-\tau\Big{)}~{}{\rm
d}\mu(\tau,t_{e})\geqslant 0.$

Therefore, from (5.16) it follows that $\dot{X}_{t_{e}}\leqslant 0$, which
implies that there exists $\Delta t$ such that $X_{t}\leqslant 1$, for all and
$t\in[0,t_{e}+\Delta t]$. Repeating the same arguments finite many times, we
conclude that $X_{t}\leqslant 1$, for all $t\in[0,T]$. Similarly, we prove
that $X_{t}\geqslant-1$, for all $t\in[0,T]$. Thus, we have proved that
$\operatorname{supp}(\mu(t))\subseteq\Omega$. Hence, (5.9) coincides with
(5.7). This completes the first part of the proof.

Noticing that $\mu(t)$ is the push forward of $\mu_{0}$ by the diffeomorphisms
defined by (5.7) (for more details see Theorem 2.7 in [44]), we deduce that
$\mu$ solves (5.8). ∎

Now, we are ready to discuss the multi-population case. Let there are $P$ sub-
populations and assume that the number of agents in each sub-population is
very large and they are in the same order as the whole population; that is,
there exists $\lambda_{1},\dots,\lambda_{P}\in(0,1)$ such that
$\sum_{k=1}^{P}\lambda_{k}=1$ and $|V^{k}|=n_{k}=\lambda_{k}N$. Therefore,
similar to the previous case, by multiplying and dividing the second term in
(5.2) by $N$ and letting $N\to\infty$, we get

(5.17) $\begin{cases}\frac{{\rm d}X^{k}_{t}}{{\rm
d}t}=-\alpha_{k}X^{k}_{t}+\frac{\alpha_{k}}{\psi(X_{t}^{k},\mu^{k})}\sum_{r=1}^{P}\lambda_{r}K_{d}(p_{kr}(t),\sigma_{k})\int_{\Omega}\kappa^{k}_{d}(X^{k}_{t},y,\varepsilon_{k})y~{}{\rm
d}\mu^{r}(y)\\\ X^{k}_{0}\sim\mu^{k}_{0},\end{cases}$

where $p_{kr}(t)=W_{p}(\mu^{\Lambda_{k}}_{t},\mu^{\Lambda_{r}}_{t})$,
$\mu^{k}(t)$ is the distribution of the sub-population $k$ and

(5.18)
$\psi(X^{k}_{t},\mu^{k})=\sum_{r=1}^{P}\lambda_{r}K_{d}(\mu^{k}(t),\mu^{r}(t),\sigma_{k})\int_{\Omega}\kappa^{k}_{d}(X^{k}_{t},y,\varepsilon_{k})~{}{\rm
d}\mu^{r}(y).$

As in the case of a single population, here, as well, the sub-populations are
substituted by their representative agents (e.g. $X_{t}^{k}$ is the
representative agent of the sub-population $\Lambda^{k}$) and the system
dynamics is governed by (5.17).

###### Assumption 2.

Suppose that the function $K_{d}(\cdot,\cdot)$ is Lipschitz in
${\mathbb{R}}^{2}$ and satisfies $c\leqslant
K_{d}(\cdot,\cdot)\leqslant\frac{1}{c}$ for some $c>0$.

The following result proves the well-posedness of the system in (5.17) and
makes a connection between the solutions of (5.17) with the system of
continuity equations. This system is deterministic and may be used for
numerical approximation.

###### Theorem 5.5.

Let $T>0$ and $\varepsilon_{k},\sigma_{k}>0$, for $k=1,\dots,P$. Suppose that
Assumption 2 holds and for all $k=\\{1,\dots,P\\}$ the kernels,
$\kappa_{d}^{k}(\cdot,\varepsilon_{k})$, satisfy Assumptions 1. Then, there
exists a unique $\\{X^{k}_{t}\\}_{k=1}^{P}\in(C^{1}[0,T])^{P}$ solving system
of DEs in (5.17). Moreover, the corresponding distributions,
$\\{\mu^{k}(t,\cdot)\\}_{k=1}^{P}$, of the sub-populations solve the system of
continuity equations

(5.19)
$\begin{cases}\mu^{k}_{t}+\Big{(}\Big{(}-\alpha_{k}x+\frac{\alpha_{k}}{\psi(x,\mu^{k})}\sum_{r=1}^{P}\lambda_{r}K_{d}(p_{kr}(t),\sigma_{k})\int_{\Omega}\kappa_{d}(x,y,\varepsilon_{k})y~{}{\rm
d}\mu^{r}(y)\Big{)}\mu^{k}\Big{)}_{x}=0\\\
\mu^{k}(0,x)=\mu^{k}_{0},\end{cases}$

in the sense of distributions.

###### Proof.

For convenience, we rewrite the system of equations in (5.19)

(5.20)
$\begin{cases}\mu^{k}_{t}+\Big{(}V(x,\tilde{\mu})\mu^{k}\Big{)}_{x}=0\\\
\mu^{k}(0,x)=\mu^{k}_{0},\end{cases}$

where $\tilde{\mu}=\\{\mu^{1},\dots,\mu^{P}\\}$ and

(5.21)
$V(x,\tilde{\mu})=\alpha_{k}\Bigg{(}-x+\frac{\sum_{r=1}^{P}\lambda_{r}K_{d}(p_{kr}(t),\sigma_{k})J^{k}(x,\mu^{r})}{\sum_{r=1}^{P}\lambda_{r}K_{d}(p_{kr}(t),\sigma_{k})\varphi_{R}^{k}(x,\mu^{r})}\Bigg{)},$

with $\varphi^{k}_{R}$, $J^{k}$ are defined by (5.10), (5.11), respectively,
for $\kappa_{d}^{k}$. To prove the existence and uniqueness of solutions to
(5.20), we following the proof of Theorem 1.1 in [44].

Note that Assumption 2 with the trigonometric inequality in Wasserstein space
implies

$\begin{split}|K_{d}(p^{1}_{kr},\sigma_{k})-K_{d}(p^{2}_{kr},\sigma_{k})|&=|K_{d}(W_{1}(\mu_{1}^{k},\mu^{r}_{1}),\sigma_{k})-K_{d}(W_{1}(\mu_{2}^{k},\mu^{r}_{2}),\sigma_{k})|\\\
&\leqslant C|W_{1}(\mu_{1}^{k},\mu^{r}_{1})-W_{1}(\mu_{2}^{k},\mu^{r}_{2})|\\\
&\leqslant
C(W_{1}(\mu_{1}^{k},\mu^{k}_{2})+W_{1}(\mu_{1}^{r},\mu^{r}_{2})).\end{split}$

Using this, the uniform bound of the functions $K_{d}$ and $J^{k}$ from
(5.12), we get

(5.22)
$\begin{split}|K_{d}(p^{1}_{kr},\sigma_{k})J^{k}(x,\mu^{r}_{1})&-K_{d}(p^{2}_{kr},\sigma_{k})J^{k}(x,\mu^{r}_{2})|\\\
&\leqslant|K_{d}(p^{1}_{kr},\sigma_{k})||J^{k}(x,\mu^{r}_{1})-J^{k}(x,\mu^{r}_{2})|\\\
&+|J^{k}(x,\mu^{r}_{2})||K_{d}(p^{1}_{kr},\sigma_{k})-K_{d}(p^{2}_{kr},\sigma_{k})|\\\
&\leqslant
C(W_{1}(\mu_{1}^{k},\mu^{k}_{2})+W_{1}(\mu_{1}^{r},\mu^{r}_{2})),\end{split}$

for all $\tilde{\mu}_{1},\tilde{\mu}_{2}\in(\mathcal{P}({\mathbb{R}}))^{P}$.
Similarly, using (5.13) we have

(5.23)
$|K_{d}(p^{1}_{kr},\sigma_{k})\varphi_{R}^{k}(x,\mu^{r}_{1})-K_{d}(p^{2}_{kr},\sigma_{k})\varphi_{R}^{k}(x,\mu^{r}_{2})|\leqslant
C(W_{1}(\mu_{1}^{k},\mu^{k}_{2})+W_{1}(\mu_{1}^{r},\mu^{r}_{2})).$

Consequently, recalling that $K_{d},\kappa^{k}_{d}\geqslant c$ and the fact
that functions $\kappa^{k}_{d}$ are Lipschitz in $x$ from the equations in
(5.21), (5.22) and (5.23), we deduce that

(5.24) $|V(x,\tilde{\mu}_{1})-V(y,\tilde{\mu}_{2}|\leqslant
C\Big{(}|x-y|+\sum_{r=1}^{P}W_{1}(\mu^{r}_{1},\mu^{r}_{2})\Big{)},$

for all $x,y\in{\mathbb{R}}$ and
$\tilde{\mu}_{1},\tilde{\mu}_{2}\in(\mathcal{P}({\mathbb{R}}))^{P}$ .

Let $0<t_{0}\leqslant T$ and fix $\tilde{\nu}\in
L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$. Note that
$L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$ is Banach space with respect
to the distance

$d(\tilde{\nu}_{1}(t),\tilde{\nu}_{2}(t))=\sup_{t\in[0,t_{0}]}\Big{(}\sum_{r=1}^{P}W_{1}(\nu^{r}_{1}(t),\nu^{r}_{2}(t))\Big{)}.$

Now, substituting $\tilde{\nu}$ in into (5.20), we consider the following
equation

(5.25)
$\begin{cases}\mu^{k}_{t}+\Big{(}V(x,\tilde{\nu})\mu^{k}\Big{)}_{x}=0\\\
\mu^{k}(0,x)=\mu^{k}_{0}.\end{cases}$

Note that the equations of the system in (5.25) are decoupled. Furthermore,
the corresponding vector fields $v(t,x)=V(x,\tilde{\nu}(t))$ are Lipschitz in
$x$ (see (5.24)), then, by [45, Theorem 5.34] the system of equations in
(5.25) has a unique solution, $\\{\mu^{k}\\}_{k=1}^{P}\in
C([0,t_{0}],\mathcal{P}^{ac}_{c}({\mathbb{R}}))$.

Next, we prove that $\operatorname{supp}(\mu^{k}(t))\subseteq\Omega$ for all
$t\in[0,t_{0}]$. Consider the corresponding flow for (5.25)

(5.26) $\begin{cases}\frac{{\rm d}X^{k}_{t}}{{\rm
d}t}=-\frac{\alpha_{k}}{\psi(X_{t}^{k},\nu^{k})}\Bigg{(}\sum_{r=1}^{P}\lambda_{r}K_{d}(q_{kr}(t),\sigma_{k})\int_{\Omega}\kappa_{d}(X_{t}^{k},y,\varepsilon_{k})\Big{(}X_{t}^{k}-y\Big{)}~{}{\rm
d}\nu^{r}(y)\Bigg{)},\\\ X^{k}_{0}\sim\mu^{k}_{0},\end{cases}$

where $q_{kr}(t)=W_{1}(\nu^{k}(t),\nu^{r}(t))$ and $\psi$ is defined by
(5.18). Recalling that
$\operatorname{supp}(\mu^{r}(0))\subseteq\Omega=[-1,1]$, for all
$r=1,\dots,P$, and using the continuity of the solution to (5.26), we denote
by $0\leqslant t_{e}\leqslant t_{0}$ the smallest time for which there exits
$k\in\\{1,\dots,P\\}$ such that $X^{k}_{t_{e}}=1$. The proof is completed if
such $t_{e}$ does not exist or $t_{e}=t_{0}$. Now, suppose that $0\leqslant
t_{e}<t_{0}$, hence, $X^{k}_{t_{e}}=1$. Consequently, recalling that
$\nu^{r}(t_{e})$ is the push forward of $\nu^{r}(0)$ with respect
$X^{k}_{t_{e}}$, we have
$\operatorname{supp}(\nu^{r}(t_{e}))\subseteq\Omega=[-1,1]$, for all
$r=\\{1,\dots,P\\}$, which yields

$\int_{\Omega}\kappa_{d}(X_{t_{e}}^{k},y,\varepsilon_{k})\Big{(}X_{t_{e}}^{k}-y\Big{)}~{}{\rm
d}\nu^{r}(y,t_{e})=\int_{\Omega}\kappa_{d}(1,y,\varepsilon_{k})\Big{(}1-y\Big{)}~{}{\rm
d}\nu^{r}(y,t_{e})\geqslant 0.$

Therefore, from (5.26) it follows that $\dot{X}^{k}_{t_{e}}\leqslant 0$, which
implies that there exists $\Delta t$ such that $X^{k}_{t}\leqslant 1$, for all
$t\in[0,t_{e}+\Delta t]$. Repeating the same arguments finite time, we
conclude that $X^{k}_{t}\leqslant 1$, for all $k=1,\dots,P$ and
$t\in[0,t_{0}]$. Similarly, we prove that $X^{k}_{t}\geqslant-1$, for all
$k=1,\dots,P$ and $t\in[0,t_{0}]$. Thus, taking into account that the solution
to (5.25), $\mu^{k}(t)$, is the push forward of $\mu_{0}^{k}$ through
$X^{k}_{t}$ and $X^{k}_{t}\in\Omega$, we deduce that
$\operatorname{supp}(\mu^{k}(t))\subseteq\Omega$ for $k=1,\dots,P$, hence,
$\mu\in L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$.

Let $S:L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})\to
L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$ is defined as follows: for
all $\tilde{\nu}\in L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$,
$S(\tilde{\nu})=\tilde{\mu}$, where $\tilde{\mu}\in
L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$ is the unique solution to
(5.25).

Next, we chose $t_{0}$ such that the map, $S$, becomes a contraction. Because
in (5.25) the state variable $x$ belongs to $\Omega$ and $V$ satisfies the
Lipschitz condition in (5.24) the Proposition 4.2 in [44] holds true in our
setting as well. Let $\tilde{\nu}_{1},\tilde{\nu}_{2}\in
L^{\infty}([0,t_{0}],(\mathcal{P}(\Omega))^{P})$,
$S(\tilde{\nu}_{i})=\tilde{\mu}_{i}$, $i=1,2$. Noticing that $\tilde{\mu}_{1}$
and $\tilde{\mu}_{2}$ have the same initial conditions from [44, Proposition
4.2], we get

(5.27)
$\sup_{t\in[0,t_{0}]}\Big{(}\sum_{r=1}^{P}W_{1}(\nu^{r}_{1}(t),\nu^{r}_{2}(t))\Big{)}\leqslant
Ct_{0}e^{Ct_{0}}\sup_{t\in[0,t_{0}]}\Big{(}\sum_{r=1}^{P}W_{1}(\mu^{r}(t),\mu^{r}_{2}(t))\Big{)}.$

Choosing $t_{0}$ such that $Ct_{0}e^{Ct_{0}}<1$, we obtain that the map $S$ is
a contraction. Therefore, by Banach’s fixed point Theorem, we deduce that
there exists a unique solution to (5.20) in $[0,t_{0}]$. Finally, note that
the constant $C$ in (5.27) only depends on problem data, hence, iterating the
arguments, we obtain the existence and uniqueness of solutions to (5.20) on
the whole interval $[0,T]$.

Because the coefficients of the system in (5.17) are Lipschitz by [44, Theorem
2.7], the existence and uniqueness of solutions of problems in (5.17) and
(5.25). ∎

Note that in the dynamics in (5.17), the distributions, therefore, population-
wise weights, are updated by the system of continuity equations, (5.8). Next,
we present an example that demonstrates a scenario where the isolated and
integrated models lead to different behaviors due to the small number of
agents.

###### Example 5.6.

Isolated and Integrated Models. Let we are given two populations whose
distributions are drowned in the first plot of the second row of Figure 8. As
one can see the actual distributions of these two populations are quite
different, and their Wasserstein distance is quite big. However, taking a
small number of agents from each population (taking a small number of samples
from the corresponding distributions), it may happen that all considered
agents’ opinions are close to each other’s; that is, their empirical
distinctions are close to each other, see the first plot of the first row of
Figure 8. Hence, in the case of the isolated model, (5.3), these two
populations converge, the first row of Figure 8. In contrast, the integrated
model, (5.17), leads to polarization due to the big difference of the actual
distributions, the second row of Figure 8.

Fig. 8. Isolated and Integrated Population: empirical initial opinion profiles
$\mu_{0}^{p1_{emp}}$ and $\mu_{0}^{p2_{emp}}$ are sampled from uniform
distribution, $N=200,\,\ \alpha=0.5,\,\ \varepsilon=0.5,\,\
\kappa_{d}=uniform\ kernel$, see equation in (2.9).

##  6\. Multi-Identity Multi-Population Model

In this section, we introduce a multi-identity multi-population model. Unlike
the models in the previous section, here, a single agent may have multiple
identities (e.g., gender, race, political orientation), and the influence of
these identities on opinion formation may vary depending on the focal issue.

In the preceding sections, the whole population is assumed to be divided into
unique, mutually exclusive sub-populations. So, the agents are assumed to be
part of a uniquely given sub-population, and their opinions are affected by
how these sub-populations interact (based on the distribution-driven
dynamics). In reality, people are part of many different groups: gender,
ethnic group, age, political orientation, etc. However, in general, most of
the group affiliations are neglectable for a given focal issue. For example,
in the case of Russia-Ukraine war, the nationality of people and their
political orientations play crucial role, and the effect of other group
affiliations may be neglected. Combining identity-invariant and time-varying
models, we introduce an opinion dynamics model, which considers multi-group
affiliation as well.

In our setting, a group affiliation corresponds to a single partition of the
population, and multi-group affiliations corresponds to several partitions of
the population. Particularly, let $I$ be the set of agents. Consider $(M+1)$
different partitions of $I$, each of the partition corresponds to different
group affiliations of agents, and let $\mathcal{M}$ is be the set of all
partitions. We denote $\Lambda_{r}^{k}$ the $k$-th sub-population of the
$r$-th partition and by $\Lambda_{r}(i)$, we denote the sub-population of
agent $i$ for $r$-th partition. For $r=0,1,\dots,M$, we have
$\Lambda_{r}^{k}\cap\Lambda_{r}^{j}=\emptyset$ for $k\neq j$ and
$\cup_{k=1}^{P_{r}}\Lambda_{r}^{k}=I$, where $P_{r}$ is the size of the
partition or the number of sub-populations with respect to $r$ division.

$\begin{cases}\frac{{\rm d}X^{i}_{t}}{{\rm
d}t}=-\alpha_{i}X^{i}_{t}+\frac{\alpha_{i}}{Z_{M}}\sum_{j=1}^{N}K(\mathcal{M})\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i})X^{j}_{t}\\\
X^{i}_{0}\sim\bar{\mu}^{i}_{0},\end{cases}$

where

$K(\mathcal{M})=\sum_{r=0}^{M}\gamma_{r}Q^{r}(W_{p}(\mu^{\Lambda_{r}(i)}_{t},\mu^{\Lambda_{r}(j)}_{t}),\sigma^{r}_{i}),\,\
Z_{M}=\sum_{j=1}^{N}K(\mathcal{M})\kappa_{d}(X^{i}_{t},X^{j}_{t},\varepsilon_{i}),$

with $Q_{r}$ is being either identity invariant kernel, see Section 3, or
time-variant kernel, see Section 4; that is,

$Q_{r}=\beta_{r}(t)K^{r}_{d}(W_{p}(\mu^{\Lambda_{r}(i)}_{0},\mu^{\Lambda_{r}(j)}_{0}),\sigma^{r}_{i})\,\
\text{or}\,\
Q_{r}=K^{r}_{d}(W_{p}(\mu^{\Lambda_{r}(i)}_{t},\mu^{\Lambda_{r}(j)}_{t}),\sigma_{i}).$

In this model, we assume that for the given focal issue, the group
affiliations that have the most effect on opinion formation have bigger
coefficients $\gamma$.

###### Example 6.1.

Group-Identity Dominance on Opinion Evolution. In this example, we consider
multi-identity multi-population model with to identity invariant kernels.

Consider a population of agents that have multi-group affiliations (see Figure
9). The first group affiliation gives two sub-groups that have opinions
distributed neutrally but with different variations, while the second
affiliation corresponds to ideology partitioning (i.e, whether or not an
agent’s initial opinion is larger than $0$). The group identities coming from
different group affiliations affects agents’ interactions differently. Here,
we’re using the parameter $\gamma$ to quantify the influence from different
group identities: higher value of $\gamma$ implies the dominance of the second
group affiliation, and vice versa.

In the first column of Figure 10, it shows the overall flow of opinions
without distinguishing between group identities. The second and third columns
highlight the opinion evolution with different partitioning.

With increasing $\gamma$, the first group affiliations begin to lose their
influence, depicted in the second column, we observe the emergence of opinion
clusters with mixing colors (orange and blue). In contrast, the third column
of Figure 10 reflects a population where ideological divisions are more
pronounced. When $\gamma$ is high, indicating the dominance of the second
group affiliation, agents’ opinions polarize significantly around the
ideology’s stance. Here, opinions are sharply divided, with little overlap
between the groups, illustrating the impact of strong ideological group
identities on social consensus and cohesion.

Fig. 9. Opinion Distribution with Different Partitions.

(a) $\beta=0.1$

(b) $\beta=0.5$

(c) $\beta=0.9$

Fig. 10. Opinion Trajectories of Different Partitions with $\gamma=0.1,\ 0.5,\
0.9$.

##  7\. Case Study

In this section, we present several numerical experiments using the models
introduced in the preceding sections. Each of the examples captures certain
social phenomenon.

###  7.1. Decaying Effects

The phenomenon of opinion dynamics with decaying effects relates to
individuals gradually reverting to their original social identities after
active involvement in discussions. The study of such effects has significant
implications in modelling public opinion on break out event (e.g., a
significant political development, natural disaster, or technological
innovation). Given that such events often generate an immediate, intense focus
of public attention, it often induces a transient structural pattern in public
opinion, manifesting as either consensus or polarization. However, as the
fervor surrounding the event decays, these transient patterns tend to evolve
towards more ”stable” configurations that are more closely aligned with
individuals’ original social identities (see for example [46]). We capture
such decaying effects by adding a time-decaying factor on the population
kernel:

$K(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0},\sigma_{i})=\begin{cases}\exp(-\Gamma(t)W_{p}(\mu^{\Lambda(i)}_{0},\mu^{\Lambda(j)}_{0})),\quad&W_{p}(\mu_{\mathcal{X}^{i}_{t}},\mu_{\mathcal{X}^{j}_{t}})\leqslant\sigma_{i}\\\
0,\quad&\text{otherwise}.\end{cases}$

where $\Gamma(t)$ is a monotonically increasing function of time $t$ with
$\Gamma(0)>0$.

In particular, the population consists of three sub-groups: two extreme ($p1$
and $p2$) and one neutral ($p3$) sub-populations (see Figure 11). The decaying
effect is imposed on the neutral population to model its ”reaction” to a
break-out event. As shown in Figure 11, a continuous active interaction
between neutral and extreme populations lead to a break-down of the neutral
population: it loses its neutrality identity and arrives at a polarization
configuration at steady state. On the other hand, when the decaying effect is
imposed, a transient polarization pattern (from time step 0 to 8) is replaced
by a new consensus configuration, indicating that the group reverting back to
its original neutrality identity.

Fig. 11. Decaying Effect: for $p_{1}$ and $p_{2}$, initial opinion profiles
$\mu_{0}^{p1}$ and $\mu_{0}^{p2}$ are sampled from truncated Gaussian
distribution and $p_{3}$ is sampled from uniform distribution,
$N=400,\,\alpha=0.5,\,\varepsilon=0.5,\,\kappa_{d}=uniform\ kernel$, see
equation in (2.9).

###  7.2. Reducing polarization

Here, we reproduce the social experiment conducted in [33] by using identity
invariant model (see Section 4).

In [33] conducted social experiment about wealth distribution in US. The
results show that political orientations of participants plays crucial rule of
the opinion formation and leads to polarization. The experiment demonstrated
that interactions between participants with opposing views led to reduced
feelings of closeness, thereby diminishing their impact on each other. In
contrast, the interactions among participants with consistent views increased
closeness, therefore, the influence. The study suggests that the influence of
individuals from different political parties on each other is relatively
negligible compared to interactions within the same political party. However,
the authors observed that introducing participants from different parties to
each other in non-political matters increases the interaction between them,
hence, reducing the polarization.

We consider two populations, denoted as $p1$ (orange) and $p2$ (blue), which
observe each other but do not interact due to a large group bias, as
illustrated in the first row of Figure 12. To expose each other’s opinions, we
introduce the third population, $p3$ (green). There is no population bias
between populations $p1$ and $p3$, as well as between populations $p2$ and
$p3$. The representatives of population $p3$ are stubborn, meaning they
influence representatives of other populations but remain unchanged
themselves. In other words, this population serves as an information
transferrer. So, we make populations $p1$ and $p2$ take into account each
other’s opinions through the population $p3$. As shown in the second row of
Figure 12, the information explosion reduces the polarization and may lead to
the consensus. However, with a strong bias between the two populations,
achieving consensus becomes unrealistic, meaning that while polarization is
reduced, the populations remain polarized. We simulate this behavior by
considering the threshold below kernel (see Section 4), see the third row of
Figure 12. Specifically, when populations $p1$ and $p2$ come too close to each
other, the group bias is activated, causing them to stop interacting with
population $p3$. Hence, they stay polarized.

Fig. 12. Polarization Reduction.

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