## Abstract

We propose and investigate different kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider (i) opinion dynamics under the effect of opinion leadership, where each individual is characterized not only by its opinion but also by another independent variable which quantifies leadership qualities; (ii) opinion dynamics modelling political segregation in ‘The Big Sort’, a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker–Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.

## 1. Introduction

The dynamics of opinion formation have been studied with growing attention; in particular in the field of physics [1–3], in which a new research field termed *sociophysics* (going back to the pioneering work of Galam *et al.* [4]) emerged. More recently, different kinetic models to describe opinion formation have been proposed [5–12]. Such models successfully use methods from statistical mechanics to describe the behaviour of a large number of interacting individuals in a society. This leads to generalizations of the classical Boltzmann equation for gas dynamics. Then the framework from classical kinetic theory for homogeneous gases is adapted to the sociological setting by replacing molecules and their velocities by individuals and their opinion. Instead of binary collisions, one considers the process of compromise between two individuals.

The basic models typically assume a homogeneous society. To model additional sociologic effects in real societies, e.g. the influence of strong opinion leaders [7], one needs to consider *inhomogeneous* models. One solution, which arises naturally in certain situations (e.g. [13]), is to consider the time evolution of distribution functions of different, interacting species. To some extent this can be seen as the analogue to the physical problem of a mixture of gases, where the molecules of the different gases exchange momentum during collisions [14]. This leads to systems of Boltzmann-like equations for the opinion distribution functions *f*_{i}=*f*_{i}(*w*,*t*), *i*=1,…,*n*, of *n* interacting species which are of the form
*f*_{i}(*w*,*t*) because of binary interaction depending on a balance between the gain and loss of individuals with opinion *w*. The suitably chosen relaxation times *τ*_{ij} allow us to control the interaction frequencies. To model the exchange of individuals (mass) between different species, additional collision operators can be present on the right-hand side of (1.1), which are reminiscent of chemical reactions in the physical situation.

Another alternative is to study models where the distribution function depends on an additional variable as, for example, in [15–18]. This leads to *inhomogeneous Boltzmann-type equations* for the distribution function *f*=*f*(*x*,*w*,*t*), which are of the following form:
*Φ*=*Φ*(*x*,*w*) which describes the opinion flux plays a crucial role. It may not be easy to determine a suitable field from the economic or sociologic problem, in contrast with the physical situation where the law of motion yields the right choice.

In this paper, we give two examples of opinion formation problems, which can be modelled using *inhomogeneous Boltzmann-type equations*. One is concerned with opinion formation where the compromise process depends on the interacting individuals' leadership abilities. The other considers the so-called ‘Big Sort’ phenomenon, which is the clustering of individuals with similar political opinion observed in the USA. Both problems lead to an inhomogeneous Boltzmann-type equation of the form (1.2).

Our work is based on a homogeneous kinetic model for opinion formation introduced by Toscani in [5]. The idea of this kinetic model is to describe the evolution of the distribution of opinion by means of *microscopic* interactions among individuals in a society. Opinion is represented as a continuous variable *compromise process* [2,19,20], in which individuals tend to reach a compromise after exchange of opinions. The second one is *self-thinking*, where individuals change their opinion in a diffusive way, possibly influenced by exogenous information sources such as the media. Based on both Toscani [5] defines a kinetic model in which opinion is exchanged between individuals through pairwise interactions: when two individuals with pre-interaction opinion *v* and *w* meet, then their post-trade opinions *v** and *w** are given by

Herein, *compromise parameter*. The quantities *η* are independent random variables with the same distribution with mean zero and finite variance *σ*^{2}. They model *self-thinking*, which each individual performs in a random diffusive fashion through an exogenous, global access to information, e.g. through the press, television or Internet. The functions *P*(⋅) and *D*(⋅) model the local relevance of compromise and self-thinking for a given opinion. To ensure that post-interaction opinions remain in the interval *P*(⋅) and *D*(⋅); see [5] for details. In this setting, the time evolution of the distribution of opinion among individuals in a simple, homogeneous society is governed by a homogeneous Boltzmann-type equation of the form (1.1). In a suitable scaling limit, a partial differential equation of Fokker–Planck type is derived for the distribution of opinion. Similar diffusion equations are also obtained in [21] as a mean field limit the Sznajd model [3]. Mathematically, the model in [5] is related to studies of the kinetic theory of granular gases [22]. In particular, the non-local nature of the compromise process is analogous to the variable coefficient of restitution in inelastic collisions [23]. Similar models are used in the modelling of wealth and income distributions which show Pareto tails; cf. [24] and references therein.

The paper is organized as follows. We introduce two new, inhomogeneous models for opinion formation. In §2, we consider opinion formation dynamics which take into account the effect of opinion leadership. Each individual is not only characterized by its opinion but also by another independent variable, the assertiveness, which quantifies its leadership potential. Starting from a microscopic model for the opinion dynamics, we arrive at an inhomogeneous Boltzmann-type equation for the opinion distribution function. We show that alternatively similar dynamics can be modelled by a multi-dimensional kinetic opinion formation model. In §3, we turn to the modelling of ‘The Big Sort’, the phenomenon that US citizens increasingly prefer to live among others who share their political opinions. We propose a kinetic model of opinion formation which takes into account these preferences. Again, the time evolution of the opinion distribution is described by an inhomogeneous Boltzmann-type equation. In §4, we derive the corresponding macroscopic Fokker–Planck-type limit equations for the inhomogeneous Boltzmann-type equations in a quasi-invariant opinion limit. Details on the numerical solvers as well as results of numerical experiments are presented in §5. Section 6 concludes.

## 2. Opinion formation and the influence of opinion leadership

The prevalent literature on opinion formation has focused on election processes, referendums or public opinion tendencies. With the exception of [6,7] less attention has been paid to the important effect that opinion leaders have on the dissemination of new ideas and the diffusion of beliefs in a society. Opinion leadership is one of several sociological models trying to explain formation of opinions in a society. Certain, typical personal characteristics are supposed to characterize opinion leaders: high confidence, high self-esteem, a strong need to be unique and the ability to withstand criticism. An opinion leader is socially active, highly connected and held in high esteem by those accepting his or her opinion. Opinion leaders appear in such diverse areas as political parties and movements, advertisements for commercial products and dissemination of new technologies.

In the opinion formation model in [7], society is built of two groups—one group of opinion leaders and one of ordinary people, so-called followers. In this model, individuals from the same group can influence each other's opinions, but opinion leaders are assertive and, although able to influence followers, are unmoved by the followers' opinions. In this model, a leader always remains a leader and a follower always a follower. Hence it is not possible to describe the emergence or decline of leaders.

In the model proposed in this section, we assume that the leadership qualities (such as assertiveness, self-confidence, etc.) of each individual are characterized through an additional independent variable, which we refer to for short as *assertiveness*. In the sociological literature the term assertiveness describes a person's tendency to actively defend, pursue and speak out for his or her own values, preferences and goals. There has been a lot of research on how assertiveness is connected to leadership (e.g. [25] and references therein).

### (a) An inhomogeneous Boltzmann-type equation

Our approach is based on Toscani's model for opinion formation (1.3), but assumes that the compromise process is also influenced by the assertiveness of the interacting individuals. The assertiveness of an individual is represented by the continuous variable

The interaction for two individuals with opinion and assertiveness (*v*,*x*) and (*w*,*y*) reads:
*C*(⋅) and *D*(⋅) model the local relevance of compromise and self-thinking for a given opinion, respectively. The constant *compromise parameter* *η* are independent random variables with distribution *Θ* with finite variance *σ*^{2} and zero mean, assuming values on a set

Let us discuss the interaction described in (2.1) and its ingredients in more detail. In each such interaction, the pre-interaction opinion *v* increases (gets closer to *w*) when *v*<*w* and decreases in the opposite situation; the change of the pre-interaction opinion *w* happens in a similar way. We assume that the compromise process function *C* can be written in the following form:
*P*, *R* and *D*.

As in [5], we assume that the ability to find a compromise is linked to the distance between opinions. The higher this distance is, the lower the possibility to find a compromise. Hence, the *localization function* *P*(⋅) is assumed to be a decreasing function of its argument. Usual choices are *P*(|*v*−*w*|)=**1**_{{|v−w|≤c}} for some constant *c*>0 and smoothed variants thereof [5].

On the other hand, we assume that the higher the assertiveness level, the lower the tendency of an individual to change their opinion. Hence, *R*(⋅) should be decreasing in its argument, too. A possible choice for *R*(⋅) may be
*k*>0. This choice is motivated by the following considerations: let *A* and *B* denote two individuals with assertiveness and opinion (*x*,*v*) and (*y*,*w*), respectively. Then the particular choice of (2.3) corresponds to the following two cases. (i) If *x*≈1 and *y*≈−1, i.e. a highly assertive individual *A* meets a weakly assertive individual *B*, then *R*(*x*−*y*)≈0 (no influence of individual *B* on *A*), but *R*(*y*−*x*)≈1, i.e. the leader *A* persuades a weakly assertive individual *B*. (ii) If both individuals have a similar assertiveness level and hence

Note that in the limit *H*(*z*) with *H* denoting the Heaviside function. In this limit, individuals are either maximally assertive in the interaction with a value of the assertiveness variable in (0,1), or minimally assertive with assertiveness in (−1,0). Effectively, one would recover in the same limit a variant of the opinion leader–follower model in [7] if the assertiveness of individuals were constant in time.

We conclude by discussing the choice of *D*(⋅). We assume that the ability to change individual opinions by self-thinking decreases as one gets closer to the extremal opinions. This reflects the fact that extremal opinions are more difficult to change. Therefore, we assume that the self-thinking function *D*(⋅) is a decreasing function of *v*^{2} with *D*(1)=0. We also need to choose the set *η* in (2.1) can assume. Clearly, it depends on the particular choice for *D*(⋅). Let us consider the upper bound at *w*=1 first. To ensure that individuals' opinions do not leave *w*=1, where we have to ensure
*D*(*v*)/(1−*v*)≤*K*_{+} it suffices to have *D*(*v*)/(1+*v*)≤*K*_{−} it suffices to have

In this setting, we are led to study the evolution of the distribution function as a function depending on the assertiveness *f*=*f*(*x*,*w*,*t*). In analogy with the classical kinetic theory of rarefied gases, we emphasize the role of the different independent variables by identifying the velocity with opinion, and the position with assertiveness. In this way, we assume at once that the variation of the distribution *f*(*x*,*w*,*t*) with respect to the opinion variable *w* depends on ‘collisions’ between agents, while the time change of distributions with respect to the assertiveness *x* depends on the transport term. Unlike in physical applications where the transport term involves the velocity field of particles, here the transport term contains an equivalent ‘opinion-velocity field’ *Φ*=*Φ*(*x*,*w*) which controls the flux depending on the independent variables *x* and *w*. This is in contrast with the physical situation of rarefied gases where the field would simply be given by *Φ*(*w*)=*w*.

The time evolution of the distribution function *f*=*f*(*x*,*w*,*t*) of individuals depending on assertiveness *inhomogeneous Boltzmann-type equation*
*Φ* is the ‘opinion-velocity field’ and *τ* is a suitable relaxation time which allows us to control the interaction frequency. The Boltzmann-like collision operator *ϕ*(*w*),
*η*.

Choosing *ϕ*(*w*)=1 as the test function in (2.5) and denoting the mass by
*C*≡1 and *D*≡0), *v**+*w**=*v*+*w*, or, more generally, conservation of opinion in the mean in each interaction in (2.1) (e.g. choosing *C*≡1), 〈*v**+*w**〉=*v*+*w*, then the first moment, the mean opinion, is also conserved. This can be seen by choosing *ϕ*(*w*)=*w* in (2.5). Denoting the mean opinion by

We still have to specify the ‘opinion-velocity field’ *Φ*(*x*,*w*). A possible choice can be
*G*=*G*(*x*,*w*,*t*) models the increase or decrease in the assertiveness level, while the prefactor (1−*x*^{2})^{α} ensures that the assertiveness level stays inside the domain

This leads to an *inhomogeneous Boltzmann-type equation* of the following form:

### (b) A multi-dimensional Boltzmann-type equation

Alternatively, we can consider that both change of opinion and change of the assertiveness level happen through binary collisions. In this case, we have the following interaction rules:
*P*, *R* and *D* play the same role as in the previous section and are assumed to fulfil the assumptions introduced earlier. The constant parameters *η* and *μ* are independent random variables with distribution *Θ* with finite variances

To specify

In this setting, we are led to study the evolution of the distribution function as a function depending on the assertiveness *f*=*f*(*x*,*w*,*t*). Different from the previous section, we assume that the variation of the distribution *f*(*x*,*w*,*t*) with respect to both variables, assertiveness *x* and opinion *w*, depends on collisions between individuals. The time evolution of the distribution function *f*=*f*(*x*,*w*,*t*) of individuals depending on assertiveness *multi-dimensional homogeneous Boltzmann-type equation*,

where *τ* is a suitable relaxation time which allows us to control the interaction frequency. The Boltzmann-like collision operator *ϕ*(*x*,*w*), where 〈⋅〉 denotes the operation of the mean with respect to the random quantities *η* and *μ*.

## 3. Political segregation: ‘The Big Sort’

In this section, we are interested in modelling the clustering of individuals who share similar political opinions, a process that has been observed in the USA over recent decades.

In 2008, journalist Bill Bishop achieved popularity as the author of the book *The Big Sort: Why the Clustering of Like-Minded America is Tearing us Apart* [27]. Bishop's thesis is that US citizens increasingly choose to live among politically like-minded neighbours. Based on county-level election results of US presidential elections in the past 30 years, he observed a doubling of so-called ‘landslide counties’, that is, counties in which either candidate won or lost by 20 percentage points or more. Such a segregation of political supporters may result in making political debates more bitter and hamper the political decision-making by consensus.

Bishop's findings were discussed and acclaimed in many newspapers and magazines, and former president Bill Clinton urged audiences to read the book. On the other hand, his claims also met opposition from political sociologists [28] who argued that political segregation is only a by-product of the correlation of political opinions with other sociologic (cultural background, race, etc.) and economic factors which drive citizens' residential preferences. One could consider including such additional factors in a generalized kinetic model, and potentially even couple the opinion dynamics with a kinetic model for wealth distribution [15,24].

‘The Big Sort’ lends itself as an ideal subject to study inhomogeneous kinetic models for opinion formation. In this context, we are faced with spatial inhomogeneity rather than inhomogeneity in assertiveness as in §2. In the following, we propose a kinetic model for opinion formation when the individuals are driven towards others with a similar political opinion.

### (a) An inhomogeneous Boltzmann-type equation

We study the evolution of the distribution function of political opinion as a function depending on three independent variables, the continuous political opinion variable *w*∈[−1,1], the position *f*=*f*(*x*,*w*,*t*). In analogy with the classical kinetic theory of rarefied gases, we assume that the variation of the distribution *f*(*x*,*w*,*t*) with respect to the political opinion variable *w* depends on collisions between individuals, while the time change of distributions with respect to the position *x* depends on the transport term, which contains the ‘opinion-velocity field’ *Φ*=*Φ*(*x*,*w*). The specific choice of *Φ* is discussed further below.

The exchange of opinion is modelled by binary collisions in the operator *x*=(*x*_{1},*x*_{2}) and *y*=(*y*_{1},*y*_{2}) denote the positions of two individuals with opinion *v* and *w*, respectively. Then the interaction rule reads
*r*>0. (ii) The functions *P* and *D* and the random quantities *η* satisfy the same assumptions as in §2.

We still need to specify the ‘opinion-velocity field’ *Φ*(*x*,*w*). As a first approach it helps to think of *Φ*(*x*,*w*) modelling the movement of individuals with respect to a given initial configuration: we consider *Ω*_{i} which constitute a disjoint cover of *w*>0) or ‘blue/Democratic’ (*w*<0). In our numerical experiments presented later we will typically restrict ourselves to bounded domains

We assume that supporters of a party are attracted to counties which are controlled by the party they support. We model this effect by defining *Φ*(*x*,*w*) as a potential that drives the dynamics and is given by a superposition of (signed) Gaussians *C*(*x*) centred around *σ*_{i}. We also assume that stronger supporters, i.e. individuals with more extreme opinion values, are able to retain their positions. A possible way to take these effects into account is to choose

The time evolution of the distribution function *f*=*f*(*x*,*w*,*t*) of individuals with political opinion *inhomogeneous Boltzmann-type equation* for the distribution function *f*=*f*(*x*,*w*,*t*) which is of the form
*t* by computing
*C*(*x*) in (3.2) accordingly in time.

## 4. Fokker–Planck limits

### (a) Fokker–Planck limit for the inhomogeneous Boltzmann equation

In general, equations such as (3.3) (and (2.7)) are rather difficult to treat and it is usual in kinetic theory to study certain asymptotics, which frequently lead to simplified models of Fokker–Planck type. To this end, we study by formal asymptotics the quasi-invariant opinion limit (*γ*,*σ*_{η}→0 while keeping

Let us introduce some notation. First, consider test functions *δ*>0. We use the usual Hölder norms
*A*, we define by *p*th moment. In the following all our probability densities belong to *Θ* is obtained from a random variable *Y* with zero mean and unit variance. We then obtain
*E*[|*Y* |^{p}] is finite. The weak form of (3.3) is given by
*γ*≪1 the transformation *t* and *x*)
*ϕ* up to second order around *w* of the right-hand side of (4.3) leads to
*κ*∈[0,1] and
*γ*,*σ*_{η}→0 while keeping

We first show that the remainder term *R*(*γ*,*σ*_{η}) vanishes in this limit, as in [5]. Note first that, as *f*(*s*):=|*s*|^{2+δ} and the fact that *η* has variance *R*(*γ*,*σ*_{η}) vanishes in the limit *γ*,*σ*_{η}→0 while keeping *x*. After integration by parts we obtain the right-hand side of (the weak form of) the Fokker–Planck equation
*w* (which result from the integration by parts).

### (b) Fokker–Planck limit for the multi-dimensional model

We follow a similar approach to that in the previous section, now in the two-dimensional setting of the opinion formation model (2.8) (see also [29]). Our aim is to study by formal asymptotics the quasi-invariant opinion limit, where *γ*,*δ*,*σ*_{η},*σ*_{μ}→0 while keeping *c*_{1}=*δ*/*γ* and *c*_{2}=*σ*_{μ}/*σ*_{η}.

We consider test functions *δ*>0. As above we use the usual Hölder norms and denote by *A* with finite *p*th moment. In the following all our probability densities belong to *Θ* is obtained from a random variable *Y* with zero mean and unit variance. We then obtain
*E*[|*Y* |^{p}] is finite. The weak form of (2.10) is given by
*γ*≪1 the transformation *f*(*x*,*w*,0)=*g*(*x*,*w*,0) and the evolution of the scaled density *t*)
*ϕ* up to second order around (*x*,*w*) of the right-hand side of (4.7). Recalling that the random quantities *η*, *μ* have mean zero, variance *σ*_{η} and *σ*_{μ}, respectively, and are uncorrelated, we can follow along the lines of the computations of the previous subsection to obtain
*R*(*γ*,*σ*_{η},*σ*_{μ}) is the remainder term. Our aim is to consider the formal limit *γ*,*δ*,*σ*_{η},*σ*_{μ}→0 while keeping *c*_{1}=*δ*/*γ* and *c*_{2}=*σ*_{μ}/*σ*_{η}. The remainder term *R*(*γ*,*σ*_{η},*σ*_{μ}) depends on the higher moments of the (uncorrelated) random quantities and can be shown to vanish in this limit, as in the previous subsection (we omit the details). In the same limit, the term on the right-hand side of (4.7) then converges to

with *x*. After integration by parts we obtain the right-hand side of (the weak form of) the Fokker–Planck equation

## 5. Numerical experiments

In this section, we illustrate the behaviour of the different kinetic models and the limiting Fokker–Planck-type equations with various simulations. We first discuss the numerical discretization of the different Boltzmann-type equations and the corresponding Fokker–Planck-type equations, and then present results of numerical experiments.

While the multi-dimensional Boltzmann-type equation (2.10) can be solved by a classical kinetic Monte Carlo method, the Monte Carlo simulations for the inhomogeneous Boltzmann-type equations (1.2) are more involved. On the macroscopic level the high dimensionality poses a significant challenge—hence we propose a time splitting as well as a finite-element discretization with mass lumping in space.

### (a) Monte Carlo simulations for the multi-dimensional Boltzmann equation

We perform a series of kinetic Monte Carlo simulations for the Boltzmann-type models presented in §2a,b. In this kind of simulation, known as direct simulation Monte Carlo (DSMC) or Bird's scheme, pairs of individuals are randomly and non-exclusively selected for binary collisions, and exchange opinion (and assertiveness in the multi-dimensional model from §2b), according to the relevant interaction rule.

In each simulation we consider *N*=5000 individuals, which are uniformly distributed in *t*=0. One time step in our simulation corresponds to *N* interactions. The average steady-state opinion distribution *f*=*f*(*x*,*w*,*t*) is calculated using *M*=10 realizations. To compute a good approximation of the steady state, each realization is carried out for *n*=2×10^{6} time steps, then the particle distribution is averaged over 5000 time steps. The random variables are chosen such that *η*_{i} and *μ*_{i} assume only values ±*ν*=±0.02 with equal probability. We assume that the diffusion has the form *D*(*w*)=(1−*w*^{2})^{α} to ensure that the opinion *w* remains inside the interval *α* and *τ* are set to *α*=2 and *τ*=1 if not mentioned otherwise.

### (b) Monte Carlo simulations for the inhomogeneous Boltzmann equation

The Monte Carlo simulations for the inhomogeneous Boltzmann-type equations (1.2) are more involved. In equation (1.2), the advection term is of conservative form, hence the transport step cannot be translated directly to the particle simulation. Pareschi & Seaid [30] as well as Herty *et al.*[31] propose a Monte Carlo method that is based on a relaxation approximation of conservation laws. It corresponds to a semilinear system with linear characteristic variables. We briefly review the underlying idea for the conservative transport operator in (1.2) in the following.

Let us consider the linear conservation law in one spatial dimension for the function *ρ*=*ρ*(*x*,*t*) with a given flux function *ρ*(*x*,0)=*ρ*_{0}(*x*) and *v*(*x*,0)=*b*(*x*,0)*ρ*_{0}(*x*) and *v* approaches the solution at local equilibrium *v*=*φ*(*u*), if *p* and *q* with *ρ*=*p*+*q* and *p*≥0, *q*≥0, *p*/*ρ*+*q*/*ρ*=1 and ∂_{t}*ρ*=0 in the relaxation step, we can calculate the solution explicitly and obtain
*t*^{n+1}=(*n*+1)Δ*t* reads
*e*^{−Δt/ε}. Let *p*^{n}=*p*(*x*,*t*^{n}), *q*^{n}=*q*(*x*,*t*^{n}) and *ρ*^{n}=*p*^{n}+*q*^{n}. As 0≤λ≤1 we can define the probability density
*P*^{n}(*ξ*)≤1 and *E*^{n}(*ξ*)=*b*(*x*,*t*^{n})*ρ*^{n} if *E*^{n}(*ξ*)=−*b*(*x*,*t*^{n})*ρ*^{n} if *b*(*x*,*t*^{n})*ρ*^{n} in the relaxation step. For more details, see [30].

### (c) Fokker–Planck simulations

The discretization of the Fokker–Planck-type equation (4.5) is based on a time-splitting algorithm. The splitting strategy allows us to consider the interactions in the opinion variable, i.e. the right-hand side of equation (4.5), and the transport step in space separately.

Let Δ*t* denote the size of each time step and *t*^{k}=*k*Δ*t*. Then the splitting scheme consists of a transport step *S*^{1}(*g*,Δ*t*) for a small time interval Δ*t*,
*S*^{2}(*g*,Δ*t*),
*t*=*t*^{k+1} is given by
*g** at the discrete time steps *t*^{k}=*k*Δ*t* and *p*_{i},*i*=1,…(#points), in space as well as opinion. Then the discrete transport step (5.4) in space has the form
** M**=〈

*p*

_{i},

*p*

_{j}〉

_{ij}corresponds to the mass matrix for element-wise linear basis functions and

**=〈**

*T**Φ*(

*x*)∇

*p*

_{i},

*p*

_{j}〉

_{ij}denotes the matrix corresponding to the convective field

*Φ*of the form (2.6). In the interaction step, we approximate the mass matrix by the corresponding lumped mass matrix

**=〈**

*C**K*(

*x*,

*w*)

*p*

_{i},∇

*p*

_{j}〉 and Laplacian

**=〈**

*L**D*(

*w*)∇

*p*

_{i}, ∇

*p*

_{j}〉. Here, the vector

*K*corresponds to the discrete convolution operator

### (d) Numerical results: opinion formation and opinion leadership

We present numerical results for the kinetic models for opinion formation including the assertiveness of individuals from §2. We consider the inhomogeneous Boltzmann-type model of §2a and the multi-dimensional Boltzmann-type equation of §2b which are discretized using the Monte Carlo methods presented in the previous sections.

First, we would like to illustrate the influence of the interaction radius *r*. We assume *w* that the functions in both models which relate to the increase or decrease of the assertiveness (see (2.6) and (2.9)) define a constant increase in the individual assertiveness level, i.e.
*k*=10. We start each Monte Carlo simulation with a uniformly distributed number of individuals within (*v*,*x*)∈(−1,1)×(−1,1). We expect the formation of one or several peaks at the highest assertiveness level due to the constant increase caused by the functions *G* and *δ*=*γ*=0.25 and two different radii, *r*=0.5 and *r*=1. For *r*=1, we observe the formation of a single peak located at the highest assertiveness level, *w*=1, in figure 1 in the inhomogeneous and multi-dimensional model. In the case of the smaller interaction radius, *r*=0.5, two peaks form at the highest assertiveness level. These results are in accordance with numerical simulations of the original Toscani model (1.3).

Next we focus on the influence of the functions *G* and *r*.

Our last example illustrates the rich behaviour of the proposed models. We assume that leadership qualities are directly related to the opinion of an individual. Individuals with a popular, ‘mainstream’ opinion, i.e. *w*=±0.5, gain confidence, while extreme opinions are not promoting leadership. To model this we set
*k*=10. Here, the second factor models the assumption that individuals change their opinion more if they have a low assertiveness level.

At time *t*=0, we equally distribute the individuals within (*w*,*x*)∈[−0.25,0.25]×[−0.75,−0.25]—hence we assume that initially no extreme opinions exist and no leaders are present. The interaction radius is set to *r*=1. Figure 3 illustrates the very interesting behaviour in this case. In the multi-dimensional simulation, we observe the formation of a single peak at a low assertiveness level, while individuals with a higher assertiveness level group around the ‘mainstream’ opinion *w*=±0.5. In the case of the inhomogeneous model, the potential *Φ* initiates an increase in the assertiveness level for all individuals. Therefore, we do not observe the formation of a single peak at a low assertiveness level in this case.

This example illustrates the emergence of complex macroscopic phenomena from individual interactions in the multi-dimensional model as well as the limitations of the inhomogeneous approach. In the first case, the formation of either a single or two peaks at a low or high assertiveness level is initiated by the different individual dynamics. As the multi-dimensional model allows for more general interactions than the inhomogeneous approach it triggers complex behaviour naturally.

### (e) Numerical results: ‘The Big Sort’ in Arizona

In our final example, we present numerical simulations of the corresponding Fokker–Planck equation (4.5), which illustrate ‘The Big Sort’ by considering the state of Arizona. Arizona is a state in the southwest of the USA with 15 electoral counties. In recent years, the Republican Party dominated Arizona's politics; see, for example, the outcome of the presidential elections from the years 1992 to 2004 in figure 4. The colours red and blue correspond to Republicans and Democrats, respectively. The colour intensity reflects the election outcome in per cent, i.e. dark blue corresponds to Democrats 60–70%, medium blue to Democrats 50–60% and light blue to Democrats 40–50%. Similar colour codes are used for the Republicans. The election results illustrate the clustering trend as the election results per county become more and more pronounced over the years.

We solve the Fokker–Planck equation (4.5) on a bounded physical domain *t*=2×10^{−4}.

We choose an initial distribution which is proportional to the election result in 1992,
*f*_{D,1992} and *f*_{R,1992} correspond to the distribution of Democrats and Republicans estimated from the election results in 1992. We approximate the distributions by assigning different constants to the respective percentages in the electoral vote,
*Φ*, given by (3.2), attracts individuals to the counties controlled by the party they support. We assume that the potential *C*=*C*(*x*) is directly related to the electoral results of the year 1996, and satisfies the following boundary value problem:
*C*, and the right-hand side corresponds to the election results in the year 1996 (using the same constants as in (5.6) with the ± sign corresponding to the Republicans and Democrats, respectively). We choose Neumann boundary conditions to ensure that individuals stay inside the physical domain. The calculated potential *C* is shown in figure 5. The simulation parameters are set to λ=0.1, *τ*=0.25 and the interaction radius to *r*=5.

Figure 6 shows the spatial distribution of the marginals (3.4), which correspond to Democrats and Republicans, respectively, at time *t*=0.5. We observe the formation of two larger clusters of democratic voters in the south and the northeast of Arizona. The Republicans move towards the northwest as well as the southeast. This simulation reproduces the patterns of the electoral results from 2004 fairly well, except for the second county from the right in the northeast (Navajo). However, given the electoral data from 1992 and 1996 only it is not possible to initiate such opinion dynamics. This would require more in-depth information such as the demographic distribution within the counties or the incorporation of several sets of electoral results. We leave such extensions for future research.

## 6. Conclusion

We proposed and examined different inhomogeneous kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable. Examples included opinion dynamics under the effect of opinion leadership and opinion dynamics modelling political segregation. Starting from microscopic opinion consensus dynamics we derived Boltzmann-type equations for the opinion distribution. In a quasi-invariant opinion limit they can be approximated by macroscopic Fokker–Planck-type equations. We presented numerical experiments to illustrate the models' rich behaviour. Using presidential election results in the state of Arizona, we showed an example modelling the process of political segregation in ‘The Big Sort’—the process of clustering of individuals who share similar political opinions.

The numerical simulations illustrate the potential and restrictions of the different approaches. The inhomogeneous model allows us to embed external knowledge (such as previous election results) in the opinion formation process. At the same time, the one-dimensional opinion formation process limits the complexity of the macroscopic dynamics. The multi-dimensional approach allows for much more flexibility in the individual interactions, which initiates the emergence of more complex phenomena. Depending on the application considered either approach has its merits; the combination of both models will be an interest of future research.

## Author' contributions

Both authors contributed equally to this publication.

## Competing interests

The authors have no competing interests.

## Funding

M.-T.W. acknowledges financial support from the Austrian Academy of Sciences (ÖAW) via the New Frontiers grant no. NFG0001. B.D. is supported by the Leverhulme Trust research project grant ‘Novel discretisations for higher-order nonlinear PDE’ (RPG-2015-69).

## Acknowledgements

The authors are grateful to Professor Richard Tsai (UT Austin) for suggesting ‘The Big Sort’ as an example for an inhomogeneous opinion formation process. The authors thank the two anonymous reviewers for their helpful suggestions.

- Received May 27, 2015.
- Accepted September 3, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.