# The Collective Behaviors of Self-Excitation Information Diffusion Processes for a Large Number of Individuals

Authors: Lifu Wang; Bo Shen
Journal: Complexity
 (2020)

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## Abstract

The opinion dynamics is a complex and interesting process, especially for the systems with a large number of individuals. It is usually hard to describe the evolutionary features of these systems. In some previous works, it has been shown that the self-excitation type model has superior performance in learning and predicting opinions. Following this line, we consider the self-excitation opinion model and study the collective behaviors of the self-excitation model. We propose a Mckean–Vlasov-type integrodifferential equation to describe the asymptotic behaviors of the model and show that the introduced equation, by coupling with the initial distribution, has the ability of capturing the influence of the self-excitation process, which describes the mutually exciting and recurrent nature of individuals. We also find that the steady-state distribution is a “contraction” of the initial distribution in the linear and bounded confidence (DW model) interaction cases, which is different from the results of the model with nonself-excitation interaction.


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## Body

## 1. Introduction

Expressing opinions and then influencing others are the primary forms of social behavior of people. The dynamics of public opinion has long been a major concern of social science and computational social science. There has been an increasing interest on the perception and prediction of social opinion, such as quantitative investment firms which measure investor sentiment and trade using social media Karppi and Crawford [1] and prediction of election results [2].Based on sentiment analysis with deep learning, there are many frameworks to perceive opinion on the social network. For the prediction of social opinion, there are some noticeable models proposed at the beginning of the 21st century, such as the Sznajd model [3], Hegselmann–Krause model [4, 5], Deffuant–Weisbuch model [6], and so on, which had been established to understand the co-ordinated movements of opinion as a group. However, these works held some limitations: (i) most of these models on opinion dynamics are theoretical and have not proved their effectiveness quantificationally. (ii) These models are represented by a cellular automaton, which makes it difficult to analyze the global behavior of the dynamics.More recently, some researchers began to study the models, which can provide more accurate predictions [7–9]. De et al. [9] proposed a framework of opinion dynamics named SLANT, providing accurate predictions of users’ opinions. In the early models, the opinion and the states of the next moment only depend on the states of the last moment, and thus have nothing to do with more previous ones. It means that opinion formation is seen as a Markov process. However, in the real world, this assumption may not be precise. The model given in the study by DE et al. [9] indicates that self-excitation opinion models with exponential function weight have superior performance in learning and predicting opinions.In many works [9–13], self-excitation models have shown efficient abilities to capture users’ behavior pattern. These models focus on the microbehaviors basing on cellular automata, which would not express the global dynamics of opinion evolution analytically.In this paper, we study the self-excitation opinion dynamics on the large homogeneous network. We show that when the number of the individuals is very large, the opinion distribution of this system evolves according to a nonlinear partial differential equation (PDE) of Mckean–Vlasov type. Due to the dependence on the initial value in the local rules, the individuals with different initial values will have different dynamic equations; thus, the mean-field equation deriving process is different from the cases of the previous works [14, 15]. We show that this PDE can be decomposed into two parts where the first part is the same as the Fokker–Planck model in the study by Toscani [16]. And the second part can capture the mutually exciting and recurrent nature of individual behaviors by an integral term related to the initial value. This term leads to different steady state behaviors from the Markov model, such as the model given in the study by Toscani [16]. Generally, the stationary solutions of the Toscani type model will trend to a single-point support Dirac measure when noise parameters tend to 0, but the model with additional term makes the steady state to be a “contraction” of the initial distribution.The rest of this paper is organized as follows. Section2 is for related work. In Section 3, we introduce the mathematical model of opinion dynamics we study and the main mean-field theorem in this paper. In Section 4, using this theorem, we consider the linear interaction model and the bounded confidence (DW) model and study the steady-state distribution of these models. Section 5 is the proof outline of this theorem. Section 6 is a summary. The detailed proof is in the appendix.


## 2. Related Work

### 2.1. Information Diffusion Model

The early theory of information diffusion in mathematical sociology was proposed by Sznajd in the study by Sznajd-Weron and Sznajd [3]. Sznajd’s model is designed to explain the features of opinion dynamics, in which every individual is on a lattice and have a state 0 or 1 to express the opinion. Individuals update their opinion by the opinions of their neighbors. The numerical simulations of this model are also investigated by many researchers [17–19]. On the complete graph, Slanina and Lavicka [20] show that when the number of individuals is very large, the probability densities evolve according to the partial differential equations(1)∂∂τPm,τ=∂2∂m21−m2Pm,τ,(2)∂∂τPm,τ=−∂2∂m21−m2mPm,τ,where Pm is the probability distribution of opinion. Equation (1) describes the Ochrombel simplification of the Sznajd model, and (2) does the original case. The authors showed the existence of phase transition in the original formulation and smooth behavior without transition in the model modified by Ochrombel.For the theory of continuous opinion dynamics, the Deffuant–Weisbuch model [6] and Hegselmann–Krause model [4, 5] deal with two different cases. Both these models are so-called bounded confidence models, i.e., the individuals can only be influenced by some others whose opinions are close enough to theirs. The characteristics of bounded confidence lead to the formation of communities, which coincides with the phenomenon in the real world. A very beautiful analysis of the convergence of the Deffuant–Weisbuch model is given in the study by Zhang et al. [21], which reveals the community structure of the model. Bhattacharyya [22] proves the convergence of the Hegselmann–Krause model. On the other hand, numerical simulations on complex networks [23] show that “the more heterogeneous the complex network is, the weaker the ability of polarization and consensus of the complex network will be.” Models with both continuous opinion and discrete states are also considered, such as the SHIR model [24], where the individuals have a four-state variable to represent susceptible, hesitated, infected, and removed. Opinion dynamics of the social network and the impact on the recommendation are studied in the studies by Xiong et al. [25] and Li et al. [26].Como and Fagnani [27] gave a general framework to prove that the scaling limits of the many-body continuous opinion system can be described by a measured value ordinary differential equation (ODE). In such models, every individual will change its opinion to ωX+1−ωX′, a convex combination with the neighbor with probability kX,X′. When kX,X′=10,εX−X′, it becomes the Deffuant–Weisbuch model. The key ideas of their work are the large deviation principle and some estimations based on the Lipschitz property, which are the same as the general method used to prove the chaotic propagation properties [28].Based on the Boltzmann equation with granular gas like interactions, Toscani [16] introduced a collision model, in which the communication between individuals was considered as a collision with the following form:(3)w′=1−ηPw,w∗w+ηPw,w∗w∗+ζDw,where function D and P describe the local relevance of the compromise and diffusion for a given opinion. Toscani also considered the quasi-invariant opinion limit for the Boltzmann-type equation. The main idea of that is to scale the interaction frequency, strength, and the diffusion in the integral equation, and then, the equation reduces to a Fokker–Planck type equation. Then, on the base of this model, considering the influence of the structure of the social network, Albi et al. [29] consider the kinetic opinion dynamics on the large scale networks evolving over time. The evolutions of both the network and the opinion were involved.


### 2.2. Self-Excitation Point Process in Social Systems

Point processes type models are generally used to analyze the impact of events on the system such as CSMA/CD [30], financial data [31], and the dynamics of book sales [32]. A very succinct and effective analysis for the time series of daily views for videos on YouTube is introduced [33]. The authors showed that most of the video viewing record data could be described statistically by a Poisson process, but still, about 10% data show a point process with the instantaneous rate λt,(4)λt=Vt+∑i,tj≤tμiϕt−ti,where μi is the number of potential viewers influenced directly by person i who views the video through the social network, and Vt captures all spontaneous views that are not triggered by epidemic effects on the network.For the self-excitation social opinion models, De et al. [9] proposed an opinion model on a social network, in which every user has a latent opinion about the given topic and can post messages about the topic. The experiments on real data show that the self-excitation model, with the intensity depending on the messages sent from a neighbor in the past and the weight function having the form like e−ωt, performs much more better than the Poisson process (ω=0) model.Wang et al. [13] used point process to predicting user activities. They proposed a generic framework for point process prediction problem, and they used a mass transport equation to update the transport rate and compute the conditional mass function.


### 2.3. Mean Field Theory

Mean-field theory (MFT) is introduced to study many-body problem by using a single averaged effect to approximate the effect of all individuals. The core of MFT is to estimate the error of the mean-field approximation. If there is no interaction, all individuals are independent of each other, and then, the law of large numbers works and gives the mean-field approximation. In the general case, we still want to be able to use the law of large numbers. That is to say, if one picks a chaotic (i.i.d) initial distribution of particles, we hope that this distribution is still chaotic as time-evolving, which is the so-called propagation of chaos.The theory of propagation of chaos in the cases of Wiener noise is summed in the study by Sznitman [28]. If N particles with initial “chaotic” distribution u0⊗N satisfy the stochastic differential equation (SDE),(5)dXi=dWi+1N∑jbXi,Xjdt,the Mckean–Vlasov mean-field equation is(6)dX=dW+∫bX,yudydt,where u is the law of X. The method to show this is to use the Lipschitz character of b and Gronwall’s lemma. Mckean–Vlasov equation for SDE with Poisson jumps is in the study by Andreis et al. [15]. The authors used Burkholder–Davis–Gundy inequality for martingales. The main step in the study by Andreis et al.[15] is to use Doob–Meyer decomposition and estimate the compensated Poisson process.Using the large deviation principle [34], Arous and Guionnet [35] studied the mean-field simplification dynamics for Langevin spin glass. The framework of their proof makes use of Sanov’s theorem and Varadhan’s lemma in the path space C0,T to get the rate function for the interaction system.The main obstacle to use these methods to prove MFT is the unboundedness of coefficients in SDE. Because of Varadhan’s lemma, if we use a stop time to get the localization of SDE, we can get the so-called local large deviation principle. Dawsont and Gartner [36] gave some compactness criteria to convert the “local” result into “global” one. They also show that a Lyapunov function for the system of weakly interacting diffusions will let the compactness condition be satisfied. Puhalskii [37] set up the whole framework of local to global LDP and introduce the C-exponential tightness conditions.


## 2.1. Information Diffusion Model

The early theory of information diffusion in mathematical sociology was proposed by Sznajd in the study by Sznajd-Weron and Sznajd [3]. Sznajd’s model is designed to explain the features of opinion dynamics, in which every individual is on a lattice and have a state 0 or 1 to express the opinion. Individuals update their opinion by the opinions of their neighbors. The numerical simulations of this model are also investigated by many researchers [17–19]. On the complete graph, Slanina and Lavicka [20] show that when the number of individuals is very large, the probability densities evolve according to the partial differential equations(1)∂∂τPm,τ=∂2∂m21−m2Pm,τ,(2)∂∂τPm,τ=−∂2∂m21−m2mPm,τ,where Pm is the probability distribution of opinion. Equation (1) describes the Ochrombel simplification of the Sznajd model, and (2) does the original case. The authors showed the existence of phase transition in the original formulation and smooth behavior without transition in the model modified by Ochrombel.For the theory of continuous opinion dynamics, the Deffuant–Weisbuch model [6] and Hegselmann–Krause model [4, 5] deal with two different cases. Both these models are so-called bounded confidence models, i.e., the individuals can only be influenced by some others whose opinions are close enough to theirs. The characteristics of bounded confidence lead to the formation of communities, which coincides with the phenomenon in the real world. A very beautiful analysis of the convergence of the Deffuant–Weisbuch model is given in the study by Zhang et al. [21], which reveals the community structure of the model. Bhattacharyya [22] proves the convergence of the Hegselmann–Krause model. On the other hand, numerical simulations on complex networks [23] show that “the more heterogeneous the complex network is, the weaker the ability of polarization and consensus of the complex network will be.” Models with both continuous opinion and discrete states are also considered, such as the SHIR model [24], where the individuals have a four-state variable to represent susceptible, hesitated, infected, and removed. Opinion dynamics of the social network and the impact on the recommendation are studied in the studies by Xiong et al. [25] and Li et al. [26].Como and Fagnani [27] gave a general framework to prove that the scaling limits of the many-body continuous opinion system can be described by a measured value ordinary differential equation (ODE). In such models, every individual will change its opinion to ωX+1−ωX′, a convex combination with the neighbor with probability kX,X′. When kX,X′=10,εX−X′, it becomes the Deffuant–Weisbuch model. The key ideas of their work are the large deviation principle and some estimations based on the Lipschitz property, which are the same as the general method used to prove the chaotic propagation properties [28].Based on the Boltzmann equation with granular gas like interactions, Toscani [16] introduced a collision model, in which the communication between individuals was considered as a collision with the following form:(3)w′=1−ηPw,w∗w+ηPw,w∗w∗+ζDw,where function D and P describe the local relevance of the compromise and diffusion for a given opinion. Toscani also considered the quasi-invariant opinion limit for the Boltzmann-type equation. The main idea of that is to scale the interaction frequency, strength, and the diffusion in the integral equation, and then, the equation reduces to a Fokker–Planck type equation. Then, on the base of this model, considering the influence of the structure of the social network, Albi et al. [29] consider the kinetic opinion dynamics on the large scale networks evolving over time. The evolutions of both the network and the opinion were involved.


## 2.2. Self-Excitation Point Process in Social Systems

Point processes type models are generally used to analyze the impact of events on the system such as CSMA/CD [30], financial data [31], and the dynamics of book sales [32]. A very succinct and effective analysis for the time series of daily views for videos on YouTube is introduced [33]. The authors showed that most of the video viewing record data could be described statistically by a Poisson process, but still, about 10% data show a point process with the instantaneous rate λt,(4)λt=Vt+∑i,tj≤tμiϕt−ti,where μi is the number of potential viewers influenced directly by person i who views the video through the social network, and Vt captures all spontaneous views that are not triggered by epidemic effects on the network.For the self-excitation social opinion models, De et al. [9] proposed an opinion model on a social network, in which every user has a latent opinion about the given topic and can post messages about the topic. The experiments on real data show that the self-excitation model, with the intensity depending on the messages sent from a neighbor in the past and the weight function having the form like e−ωt, performs much more better than the Poisson process (ω=0) model.Wang et al. [13] used point process to predicting user activities. They proposed a generic framework for point process prediction problem, and they used a mass transport equation to update the transport rate and compute the conditional mass function.


## 2.3. Mean Field Theory

Mean-field theory (MFT) is introduced to study many-body problem by using a single averaged effect to approximate the effect of all individuals. The core of MFT is to estimate the error of the mean-field approximation. If there is no interaction, all individuals are independent of each other, and then, the law of large numbers works and gives the mean-field approximation. In the general case, we still want to be able to use the law of large numbers. That is to say, if one picks a chaotic (i.i.d) initial distribution of particles, we hope that this distribution is still chaotic as time-evolving, which is the so-called propagation of chaos.The theory of propagation of chaos in the cases of Wiener noise is summed in the study by Sznitman [28]. If N particles with initial “chaotic” distribution u0⊗N satisfy the stochastic differential equation (SDE),(5)dXi=dWi+1N∑jbXi,Xjdt,the Mckean–Vlasov mean-field equation is(6)dX=dW+∫bX,yudydt,where u is the law of X. The method to show this is to use the Lipschitz character of b and Gronwall’s lemma. Mckean–Vlasov equation for SDE with Poisson jumps is in the study by Andreis et al. [15]. The authors used Burkholder–Davis–Gundy inequality for martingales. The main step in the study by Andreis et al.[15] is to use Doob–Meyer decomposition and estimate the compensated Poisson process.Using the large deviation principle [34], Arous and Guionnet [35] studied the mean-field simplification dynamics for Langevin spin glass. The framework of their proof makes use of Sanov’s theorem and Varadhan’s lemma in the path space C0,T to get the rate function for the interaction system.The main obstacle to use these methods to prove MFT is the unboundedness of coefficients in SDE. Because of Varadhan’s lemma, if we use a stop time to get the localization of SDE, we can get the so-called local large deviation principle. Dawsont and Gartner [36] gave some compactness criteria to convert the “local” result into “global” one. They also show that a Lyapunov function for the system of weakly interacting diffusions will let the compactness condition be satisfied. Puhalskii [37] set up the whole framework of local to global LDP and introduce the C-exponential tightness conditions.


## 3. Mathematical Model

### 3.1. Self-Excitation Opinion Model

In this paper, we consider a simplified version of the model [9] and study the large number limit of the self-excitation opinion model with the following form:There areN individuals with opinion xi, which is a function of the time t, such that(7)xit=xi0+∑jhxj,xikt∗dNjt,where hxj,hi is a interaction function, ∗ is the convolution operator, kt=e−ωt is the triggering kernel, and Nt is a Poisson process. This model has been previously studied [11, 12].The original model proposed in the study by De et al. [9] has the following form:(8)xit=bi+∑jαi,j∑tj∈Hjtmjkt−tj=bi+∑jαi,jmjkt∗dNjt,where mj is the message sent by user j, αij is the interaction intensity between i and j, bi=xi0 is the initial opinion of user i, Hjt is the history of events up to time t for user j, and kt is a triggering kernel. The values of message m come from a sentiment distribution related to the opinion x and Emixi=xi.As the models have been previously studied [11, 12], we only consider the opinion part, the simplified model (7). In the case that hxi,xj=hxj, where xj is the opinion of the individual j, De et al. and Wang et al. [9, 11] showed the validity of the model in the study by DE et al. [9] on the real world data. And when hxj,xi=xj−xi, the form of this model will degenerate into the form of the DW model [6] and Toscani model [16] when kt=1.In studies by De et al. and Wang et al. [9, 11], the self-excitation opinion model with exponential function type weight was shown to be very effective in the opinion dynamics; therefore, we set kt=e−ωt, where ω is a constant. Due to the fact that differential of the convolution of two functions df∗g=f0g+g∗df, where there is a jump part in g, and dkt=−ωktdt. We can use a stochastic differential equation (SDE) to describe xt [11].(9)dXi=ωbi−Xitdt+∑jαhXj,XidNj.It is also necessary to addσdW into the model to represent noise, where W is the Wiener noise with W0=0.(10)dXi=ωXi0−Xitdt+∑jαhXj,XidNj+σdWi.The above two equations are given in the study by Wang et al. [11]. Because of the dependence on the path of the history, it is apparently not a Markov process. We consider the limit that the population size tends to infinity. Let XN=X1,X2,…,XN be the opinion variable of the N-particles system. X satisfies(11)dXi=ωbi−Xitdt+1N∑j≠iNαhXj,XidNj+σdWi,where bi=Xi0. In the case that ω=0 and hx,y=x−y10,εx−y, this is a Deffuant–Weisbuch-like model.If we only consider the caseω=0 and replace the Poisson Noise dN with dt and consider the form(12)dXi=dWi+1N∑jhXi,Xjdt.The mean-field theory of this equation is well studied [28]. The Mckean–Vlasov mean-field equation is(13)dX=dW+∫bX,yudydt.In this paper, we first use Doob–Meyer decomposition fordN. Then, we estimate the compensated Poisson process by Burkholder–Davis–Gundy inequality for martingales. This method helps us to reduce our equation to the form in (12). As for the dependence on the initial values, we regard the initial as a random media. Using the methods in the study by Pra and Hollander [38]. The Mckean–Vlasov equation can be obtained using Sanov’s theorem and Varadhan’s lemma in the path space C0,T. In this paper, we use a direct estimation to prove the convergence to the Mckean–Vlasov equation.


### 3.2. Main Results

The main result in this paper is the following theorem:Theorem 1.
WhenN tends to infinity, there is a limit process X for XN in equation (11), with SDE:(14)dX=∫dPyhy,Xλdt+ωb−Xdt+σdW,where P is the law of X, and b=X0 is a random variable. And given the initial distribution μ, the law of X is P=∫Pbdμb, where Pb is the solution of the Fokker–Planck equation associated with the Mckean–Vlasov process:(15)∂∂tPb=−∂∂xβb,PPb+12σ2∂2∂x2Pb,Pb0=δb,βb,P=∫μdb′∫Pb′dyαλhy,x+ωb−x,where δb is a Dirac measure with support on b.
From this theorem, we can study the asymptotic behavior of the self-excitation opinion model. In Section4, we will study the steady state of this system by using Fokker–Planck equation and show the effects of self-excitation, which makes it impossible to give a perfect consensus, such that it can avoid Abelson’s diversity puzzle [39].
In order to prove this theorem, we need some assumptions. Stemming from the physical meaning, we can assume that the support of initial distributionμb is bounded, such as the uniform distribution on [0, 1]. Also, for the proof of Mckean–Vlasov limit, we need the function hx to have good enough properties to make(16)EhXjb′,Xib<∞,(17)∫x2Pdx<∞,where P is the solution of the Mckean–Vlasov equation.
Ifh is bounded, the first condition is satisfied trivially. In the appendix, we will give other examples of h that can make these conditions be satisfied. Also, we assume that hx,y is Lipschitz for both of the variables. However, this is not the necessary condition [14].(18)hx1,y1−hx2,y2<Lx1−x2+Ky1−y2.
Most of these conditions can be relaxed, but we hope to prove our results without excessive technical details on tightness; therefore, these restrictions are imposed.


## 3.1. Self-Excitation Opinion Model

In this paper, we consider a simplified version of the model [9] and study the large number limit of the self-excitation opinion model with the following form:There areN individuals with opinion xi, which is a function of the time t, such that(7)xit=xi0+∑jhxj,xikt∗dNjt,where hxj,hi is a interaction function, ∗ is the convolution operator, kt=e−ωt is the triggering kernel, and Nt is a Poisson process. This model has been previously studied [11, 12].The original model proposed in the study by De et al. [9] has the following form:(8)xit=bi+∑jαi,j∑tj∈Hjtmjkt−tj=bi+∑jαi,jmjkt∗dNjt,where mj is the message sent by user j, αij is the interaction intensity between i and j, bi=xi0 is the initial opinion of user i, Hjt is the history of events up to time t for user j, and kt is a triggering kernel. The values of message m come from a sentiment distribution related to the opinion x and Emixi=xi.As the models have been previously studied [11, 12], we only consider the opinion part, the simplified model (7). In the case that hxi,xj=hxj, where xj is the opinion of the individual j, De et al. and Wang et al. [9, 11] showed the validity of the model in the study by DE et al. [9] on the real world data. And when hxj,xi=xj−xi, the form of this model will degenerate into the form of the DW model [6] and Toscani model [16] when kt=1.In studies by De et al. and Wang et al. [9, 11], the self-excitation opinion model with exponential function type weight was shown to be very effective in the opinion dynamics; therefore, we set kt=e−ωt, where ω is a constant. Due to the fact that differential of the convolution of two functions df∗g=f0g+g∗df, where there is a jump part in g, and dkt=−ωktdt. We can use a stochastic differential equation (SDE) to describe xt [11].(9)dXi=ωbi−Xitdt+∑jαhXj,XidNj.It is also necessary to addσdW into the model to represent noise, where W is the Wiener noise with W0=0.(10)dXi=ωXi0−Xitdt+∑jαhXj,XidNj+σdWi.The above two equations are given in the study by Wang et al. [11]. Because of the dependence on the path of the history, it is apparently not a Markov process. We consider the limit that the population size tends to infinity. Let XN=X1,X2,…,XN be the opinion variable of the N-particles system. X satisfies(11)dXi=ωbi−Xitdt+1N∑j≠iNαhXj,XidNj+σdWi,where bi=Xi0. In the case that ω=0 and hx,y=x−y10,εx−y, this is a Deffuant–Weisbuch-like model.If we only consider the caseω=0 and replace the Poisson Noise dN with dt and consider the form(12)dXi=dWi+1N∑jhXi,Xjdt.The mean-field theory of this equation is well studied [28]. The Mckean–Vlasov mean-field equation is(13)dX=dW+∫bX,yudydt.In this paper, we first use Doob–Meyer decomposition fordN. Then, we estimate the compensated Poisson process by Burkholder–Davis–Gundy inequality for martingales. This method helps us to reduce our equation to the form in (12). As for the dependence on the initial values, we regard the initial as a random media. Using the methods in the study by Pra and Hollander [38]. The Mckean–Vlasov equation can be obtained using Sanov’s theorem and Varadhan’s lemma in the path space C0,T. In this paper, we use a direct estimation to prove the convergence to the Mckean–Vlasov equation.


## 3.2. Main Results

The main result in this paper is the following theorem:Theorem 1.
WhenN tends to infinity, there is a limit process X for XN in equation (11), with SDE:(14)dX=∫dPyhy,Xλdt+ωb−Xdt+σdW,where P is the law of X, and b=X0 is a random variable. And given the initial distribution μ, the law of X is P=∫Pbdμb, where Pb is the solution of the Fokker–Planck equation associated with the Mckean–Vlasov process:(15)∂∂tPb=−∂∂xβb,PPb+12σ2∂2∂x2Pb,Pb0=δb,βb,P=∫μdb′∫Pb′dyαλhy,x+ωb−x,where δb is a Dirac measure with support on b.
From this theorem, we can study the asymptotic behavior of the self-excitation opinion model. In Section4, we will study the steady state of this system by using Fokker–Planck equation and show the effects of self-excitation, which makes it impossible to give a perfect consensus, such that it can avoid Abelson’s diversity puzzle [39].
In order to prove this theorem, we need some assumptions. Stemming from the physical meaning, we can assume that the support of initial distributionμb is bounded, such as the uniform distribution on [0, 1]. Also, for the proof of Mckean–Vlasov limit, we need the function hx to have good enough properties to make(16)EhXjb′,Xib<∞,(17)∫x2Pdx<∞,where P is the solution of the Mckean–Vlasov equation.
Ifh is bounded, the first condition is satisfied trivially. In the appendix, we will give other examples of h that can make these conditions be satisfied. Also, we assume that hx,y is Lipschitz for both of the variables. However, this is not the necessary condition [14].(18)hx1,y1−hx2,y2<Lx1−x2+Ky1−y2.
Most of these conditions can be relaxed, but we hope to prove our results without excessive technical details on tightness; therefore, these restrictions are imposed.


## 4. The Properties of Mckean–Vlasov Dynamics

### 4.1. Mckean–Vlasov Equation

For the equation(19)∂∂tPb=−∂∂xβb,PPb+12σ2∂2∂x2Pb,Pb0=δb,βb,P=∫μdb′∫Pb′dyλhy,x+ωb−x,we consider a simple case(20)λ=1,σ=1,hy,x=αy−x.Then, we can see that(21)βb,P=αm−x+ωb−x,where m=∫xdPx, the mean value of x.Sinceh is an odd function, it is easy to see that(22)dEXtdt=Eα∫PdYY−X+ωb−Xt=0+ωEX0−Xt.So,EXt−X0=Ce−ωt since EX0−X0=0, C=0, and m=EXt=EX0.Then, we have the Fokker–Planck equation associated with the Mckean–Vlasov process:(23)∂∂tPb=−∂∂xαm−x+ωb−xPb+12∂2∂x2Pb.Integral overμb(24)∂∂tP=−∂∂xαm−xP+ω∫dμbb−xPb+12∂2∂x2P.Another example is the bounded confidence type model [4, 6] where every agent interacts only within a certain level of confidence:(25)hy,x=y−xky−x,where ky−x is a continuous function, and if y−x<Δ1,ky−x=1, and if y−x>Δ2,ky−x=0. Since, in this case, hx is bounded and Lipschitz, our assumptions are satisfied.Note that for the case thathy,x=αy, which is used in the SLANT system [9],(26)∂∂tPb=−∂∂xαmt+ωb−xPb+12σ2∂2∂x2Pb,where mt=∫xdPx,t. Since(27)dEXtdt=Eα∫PdYY+ωb−Xt=α−ωEX+ωEX0.As a sanity check, this formula matches (4) in the study by De et al. [9]. Solving this equation, we get(28)mt+ωα−ωm0=Ceα−ωt,C=αα−ω.


### 4.2. Discussion

Como and Fagnani [27] showed that when the population size tends to infinity, the limit behavior of the Deffuant–Weisbuch model can be described by a measure-valued ODE. And also, Toscani introduced a kinetic model of opinion formation [16]. In this model, the opinion is changed by the binary interaction of collision, and the dynamics of the opinion distribution is modeled by Boltzmann-type integrodifferential equation.In these models, the dynamics of opinion distribution is depicted as follows:(29)ddtϕ,μt=ϕ,Hμt,where μt is the measure of opinion distribution, and ϕ is an arbitrary test function. The right side of (29) has the following form [27]:(30)ϕ,Hμt=∫∫ϕ1−ωx+ωy−ϕxkx,ydμxdμy,and in Boltzmann case [16],(31)ϕ,Hμt=∫∫ϕw′+ϕw∗′−ϕw−ϕw∗dμwdμw∗,w′=1−ηPw,w∗w+ηPw,w∗w∗,w∗′=1−ηPw∗,ww+ηPw∗,ww.Intuitively, both models can be derived from the stochastic differential equation with Poisson jump interactions by the method mentioned in the study by Méléard [14]. Events that interact with other particles are regarded as a Poisson process.For the system withhy,x=y−x,(32)∂∂tP=−∂∂xαm−xP+ω∫dμbb−xPb+12∂2∂x2P,in the case that ω=0, which is the nonself-excitation case, it degenerates into the general opinion dynamics model, or equally, the quasi-invariant opinion limit for Toscani’s Boltzmann equation [16]:(33)∂∂tP=λ2∂2∂x2Dx2P−∂∂xPxm−xP.Theω term is a modification to consider the influence of the self-excitation process.In this paper, we only consider the homogeneous network. However, using the method in the study by Pra and Hollander and Albi et al. [38, 40], we can study the large system of agents interacting through a network with a given distribution of the number of connections by the following way:(1)
Consider a random mediac∈0,1,2,…,cmax, which is a discrete variable describing the number of connections. c should be a random variable to represent the complex network. For example, c obeys power law distribution.(2)
Construct a new interaction equation as in the study by Albi et al. [40].(34)dXi=ωbi−Xitdt+∑jαhXj,Xi,cdNj.Then, it is the same as Theorem1 in the study by Pra and Hollander [38] such that in this random media, we can also prove a mean-field theorem. And there is a distribution Pc,x,t about the variable c, opinion x, and time t. Using a similar argument as Albi et al. [40], we have obtained the stationary solutions.


### 4.3. Steady-State Distribution of the Linear Case

Considering a simple case of initial distributionP0=1/2δ−10+1/2δ10. We will use this simple form to explain how the initial distribution is coupled to the equation. Then, the equation will become(35)∂∂tP−10=−∂∂xαx¯−xP−10+ω−10−xP−10+12∂2∂x2P−10,∂∂tP10=−∂∂xαx¯−xP10+ω10−xP10+12∂2∂x2P10.The overall distributionP is equal to 1/2P−10+1/2P10.The evolution process of this distribution is shown in in Figure1(a) . And Figures 1(b) and 1(c) show the differences between two kinds of steady states.Figure 1
Simulation results. (a) Evolution of the opinion density, withs = 0 : 02; a = 0 : 02; w = 0 : 01. (b) Steady state for the dynamics with parameters s = 0 : 02; a = 0 : 02; w = 0. (c) Solution profiles at time t = 200, with parameters s = 0 : 02; a = 0 : 02; w = 0 : 01.
(a)(b)(c)As shown in Figure1(b), for the initial distribution P0, the steady state distribution is normal. In fact, this is true for any initial distribution, and we have the following theorem.Theorem 2.
In the case thatω=0, for all type of initial distribution P0, we have the steady-state distribution P∞=1/Zeαx−m/σ2, where Z is the normalization constant.
This is because the form of PDE forP has the form(36)∂∂tP=−∂∂xαm−xP+σ212∂2∂x2P.
The steady-state equation(37)−∂∂xαm−xP+σ212∂2∂x2P=0,and this steady state is asymptotically stable, such that when t⟶∞, the solution of the equation will trend to the steady state. It is easy to see that the steady state is a normal distribution as shown in Figure 1(b).
From Theorem2, this ω=0 model cannot describe the actual situation since, in this model, the final state of the opinion distribution has no community structures but a perfect consensus (in fact, this is true for any monotonic function h, since the steady state is the Gibbs distribution of a potential energy Vx, such that ∇V=h), which is so-called Abelson’s diversity puzzle [39], a persistent research puzzle in the social sciences. Generally, this puzzle is solved by the bounded-confidence mode, such as HK [4] and DW [6] models. It is proved in the study by Como and Fagnani [27] for a large number of individuals case and by Zhang et al. [21] for the general case that the steady state of the DW model has the form ∑iCiδxi, where xi is the opinion of the community i, and xi−xj>R if i≠j, where R is the bounded confidence distance.
However, in the caseω>0, the equation has the steady state as in Figure 1(c) when the initial distribution has the form 1/2δ−10+δ10. In the general case, we haveTheorem 3.
In the case thatω>0, given an initial distribution μx, we have the steady-state distribution Px,∞=1/Z∫μbe−α+ωx−αm+ωb/σ2, where Z is the normalization constant.
LetP=∫dμbPb, and Pb is the solution of Mckean–Vlasov equation,(38)∂∂tPb=−∂∂xαm−x+ωb−xPb+12σ2∂2∂x2Pb.
The steady state ofPb is easy to calculate:(39)Pbx,∞=1Ze−α+ωx−αm+ωb/σ2,where Z is the normalization constant, so we have P∞=∫dμbPb∞.
Considering the limit behaviorσ⟶0, the steady state of Mckean–Vlasov equation when hy,x=y−x has the form(40)Px,∞=∫dμbδx−αm+ωbα+ω.
So that even in the case that the variance of the noise trends to 0, this equation will not converge to a single Dirac measure, which is different from theω=0 case. Intuitively, the term ωb−x makes the opinion try not to deviate from the initial position too far, which can be considered as soft-bounded confidence.
In the case thathy,x=αy, which is used in the SLANT system in the study by De et al. [9], supposing ω>α, the steady state will have the similar form since the equation of which is(41)−∂∂xαm+ωb−xPb+12σ2∂2∂x2Pb=0,where m=m∞=α/ω−αm0 (this is from (25)), the final mean value of x. We have P∞=∫dμbδx−αm+ωb/ω.


### 4.4. Steady-State Distribution of Bounded Confidence Models

In the case thatω=0 and hy,x=y−xky−x, the steady-state distribution is studied [27]. We can follow their methods to study the weak solution of our equation.Theorem 4.
Letω=0, hy,x=y−xky−x, and kz=0 if z>R, otherwise kz>0. We have Px,∞⟶∑iCδx−xii when σ⟶0, where xi−xj>R if i≠j.Proof.
Considering equation (19) with ω=0, let Qt=∫μbx2Pbx=∫x2Pdx. We have(42)ddtQt=−∫μdbx2∂∂x∫Pdyy−xky−x+ωb−xPbdx+12∫σ2x2∂2∂x2Pdx=∫2xy−xky−xPdyPdx12∫σ2x2∂2∂x2Pdx=a−∫y−x2ky−xPdyPdx+σ2∫Pdx,where ky−x is a bounded confidence kernel. In (a) of (42), we use the symmetry of Px and Py and the integration by parts.
Considering the steady state andσ⟶0 limit, we have d/dtQt=0, yet if in the support of P∞, there are two points x and y such that x−y<R and x≠y, and we have −∫y−x2ky−xPyPx<0, which is a contradiction, so that our claim follows. □
Supposingω>0, it is hard to obtain a steady state directly. However, we can use Qt function to give an estimation. We can show that in the bounded confidence case, the steady-state solution is still a “contraction” of initial distribution.(43)ddtQt=−∫μbx2∂∂x∫Pdyy−xky−xPbdx+12∫σ2x2∂2∂x2Pdx=∫2xy−xky−xPdyPdx+∫2ωxb−xμdbPbdx+12∫σ2x2∂2∂x2Pdx=−∫y−x2ky−xPdyPdx+∫2ωxb−xμdbPbdx+σ2∫Pdx.
Since−∫y−x2ky−xPdyPdx≤0 and σ⟶0, we have(44)∫xb−xμdbPbdx≥0,such that in the sense of mean value, x<b=x0. And the degree of offset is depending on the value of ∫y−x2ky−xPdyPdx. We can see that ∫y−x2ky−xPdyPdx will increase as the radius R of the kernel ky−x grows. When R⟶∞ and ky−x=1, the model will degenerate into the linear case in Theorem 3.


## 4.1. Mckean–Vlasov Equation

For the equation(19)∂∂tPb=−∂∂xβb,PPb+12σ2∂2∂x2Pb,Pb0=δb,βb,P=∫μdb′∫Pb′dyλhy,x+ωb−x,we consider a simple case(20)λ=1,σ=1,hy,x=αy−x.Then, we can see that(21)βb,P=αm−x+ωb−x,where m=∫xdPx, the mean value of x.Sinceh is an odd function, it is easy to see that(22)dEXtdt=Eα∫PdYY−X+ωb−Xt=0+ωEX0−Xt.So,EXt−X0=Ce−ωt since EX0−X0=0, C=0, and m=EXt=EX0.Then, we have the Fokker–Planck equation associated with the Mckean–Vlasov process:(23)∂∂tPb=−∂∂xαm−x+ωb−xPb+12∂2∂x2Pb.Integral overμb(24)∂∂tP=−∂∂xαm−xP+ω∫dμbb−xPb+12∂2∂x2P.Another example is the bounded confidence type model [4, 6] where every agent interacts only within a certain level of confidence:(25)hy,x=y−xky−x,where ky−x is a continuous function, and if y−x<Δ1,ky−x=1, and if y−x>Δ2,ky−x=0. Since, in this case, hx is bounded and Lipschitz, our assumptions are satisfied.Note that for the case thathy,x=αy, which is used in the SLANT system [9],(26)∂∂tPb=−∂∂xαmt+ωb−xPb+12σ2∂2∂x2Pb,where mt=∫xdPx,t. Since(27)dEXtdt=Eα∫PdYY+ωb−Xt=α−ωEX+ωEX0.As a sanity check, this formula matches (4) in the study by De et al. [9]. Solving this equation, we get(28)mt+ωα−ωm0=Ceα−ωt,C=αα−ω.


## 4.2. Discussion

Como and Fagnani [27] showed that when the population size tends to infinity, the limit behavior of the Deffuant–Weisbuch model can be described by a measure-valued ODE. And also, Toscani introduced a kinetic model of opinion formation [16]. In this model, the opinion is changed by the binary interaction of collision, and the dynamics of the opinion distribution is modeled by Boltzmann-type integrodifferential equation.In these models, the dynamics of opinion distribution is depicted as follows:(29)ddtϕ,μt=ϕ,Hμt,where μt is the measure of opinion distribution, and ϕ is an arbitrary test function. The right side of (29) has the following form [27]:(30)ϕ,Hμt=∫∫ϕ1−ωx+ωy−ϕxkx,ydμxdμy,and in Boltzmann case [16],(31)ϕ,Hμt=∫∫ϕw′+ϕw∗′−ϕw−ϕw∗dμwdμw∗,w′=1−ηPw,w∗w+ηPw,w∗w∗,w∗′=1−ηPw∗,ww+ηPw∗,ww.Intuitively, both models can be derived from the stochastic differential equation with Poisson jump interactions by the method mentioned in the study by Méléard [14]. Events that interact with other particles are regarded as a Poisson process.For the system withhy,x=y−x,(32)∂∂tP=−∂∂xαm−xP+ω∫dμbb−xPb+12∂2∂x2P,in the case that ω=0, which is the nonself-excitation case, it degenerates into the general opinion dynamics model, or equally, the quasi-invariant opinion limit for Toscani’s Boltzmann equation [16]:(33)∂∂tP=λ2∂2∂x2Dx2P−∂∂xPxm−xP.Theω term is a modification to consider the influence of the self-excitation process.In this paper, we only consider the homogeneous network. However, using the method in the study by Pra and Hollander and Albi et al. [38, 40], we can study the large system of agents interacting through a network with a given distribution of the number of connections by the following way:(1)
Consider a random mediac∈0,1,2,…,cmax, which is a discrete variable describing the number of connections. c should be a random variable to represent the complex network. For example, c obeys power law distribution.(2)
Construct a new interaction equation as in the study by Albi et al. [40].(34)dXi=ωbi−Xitdt+∑jαhXj,Xi,cdNj.Then, it is the same as Theorem1 in the study by Pra and Hollander [38] such that in this random media, we can also prove a mean-field theorem. And there is a distribution Pc,x,t about the variable c, opinion x, and time t. Using a similar argument as Albi et al. [40], we have obtained the stationary solutions.


## 4.3. Steady-State Distribution of the Linear Case

Considering a simple case of initial distributionP0=1/2δ−10+1/2δ10. We will use this simple form to explain how the initial distribution is coupled to the equation. Then, the equation will become(35)∂∂tP−10=−∂∂xαx¯−xP−10+ω−10−xP−10+12∂2∂x2P−10,∂∂tP10=−∂∂xαx¯−xP10+ω10−xP10+12∂2∂x2P10.The overall distributionP is equal to 1/2P−10+1/2P10.The evolution process of this distribution is shown in in Figure1(a) . And Figures 1(b) and 1(c) show the differences between two kinds of steady states.Figure 1
Simulation results. (a) Evolution of the opinion density, withs = 0 : 02; a = 0 : 02; w = 0 : 01. (b) Steady state for the dynamics with parameters s = 0 : 02; a = 0 : 02; w = 0. (c) Solution profiles at time t = 200, with parameters s = 0 : 02; a = 0 : 02; w = 0 : 01.
(a)(b)(c)As shown in Figure1(b), for the initial distribution P0, the steady state distribution is normal. In fact, this is true for any initial distribution, and we have the following theorem.Theorem 2.
In the case thatω=0, for all type of initial distribution P0, we have the steady-state distribution P∞=1/Zeαx−m/σ2, where Z is the normalization constant.
This is because the form of PDE forP has the form(36)∂∂tP=−∂∂xαm−xP+σ212∂2∂x2P.
The steady-state equation(37)−∂∂xαm−xP+σ212∂2∂x2P=0,and this steady state is asymptotically stable, such that when t⟶∞, the solution of the equation will trend to the steady state. It is easy to see that the steady state is a normal distribution as shown in Figure 1(b).
From Theorem2, this ω=0 model cannot describe the actual situation since, in this model, the final state of the opinion distribution has no community structures but a perfect consensus (in fact, this is true for any monotonic function h, since the steady state is the Gibbs distribution of a potential energy Vx, such that ∇V=h), which is so-called Abelson’s diversity puzzle [39], a persistent research puzzle in the social sciences. Generally, this puzzle is solved by the bounded-confidence mode, such as HK [4] and DW [6] models. It is proved in the study by Como and Fagnani [27] for a large number of individuals case and by Zhang et al. [21] for the general case that the steady state of the DW model has the form ∑iCiδxi, where xi is the opinion of the community i, and xi−xj>R if i≠j, where R is the bounded confidence distance.
However, in the caseω>0, the equation has the steady state as in Figure 1(c) when the initial distribution has the form 1/2δ−10+δ10. In the general case, we haveTheorem 3.
In the case thatω>0, given an initial distribution μx, we have the steady-state distribution Px,∞=1/Z∫μbe−α+ωx−αm+ωb/σ2, where Z is the normalization constant.
LetP=∫dμbPb, and Pb is the solution of Mckean–Vlasov equation,(38)∂∂tPb=−∂∂xαm−x+ωb−xPb+12σ2∂2∂x2Pb.
The steady state ofPb is easy to calculate:(39)Pbx,∞=1Ze−α+ωx−αm+ωb/σ2,where Z is the normalization constant, so we have P∞=∫dμbPb∞.
Considering the limit behaviorσ⟶0, the steady state of Mckean–Vlasov equation when hy,x=y−x has the form(40)Px,∞=∫dμbδx−αm+ωbα+ω.
So that even in the case that the variance of the noise trends to 0, this equation will not converge to a single Dirac measure, which is different from theω=0 case. Intuitively, the term ωb−x makes the opinion try not to deviate from the initial position too far, which can be considered as soft-bounded confidence.
In the case thathy,x=αy, which is used in the SLANT system in the study by De et al. [9], supposing ω>α, the steady state will have the similar form since the equation of which is(41)−∂∂xαm+ωb−xPb+12σ2∂2∂x2Pb=0,where m=m∞=α/ω−αm0 (this is from (25)), the final mean value of x. We have P∞=∫dμbδx−αm+ωb/ω.


## 4.4. Steady-State Distribution of Bounded Confidence Models

In the case thatω=0 and hy,x=y−xky−x, the steady-state distribution is studied [27]. We can follow their methods to study the weak solution of our equation.Theorem 4.
Letω=0, hy,x=y−xky−x, and kz=0 if z>R, otherwise kz>0. We have Px,∞⟶∑iCδx−xii when σ⟶0, where xi−xj>R if i≠j.Proof.
Considering equation (19) with ω=0, let Qt=∫μbx2Pbx=∫x2Pdx. We have(42)ddtQt=−∫μdbx2∂∂x∫Pdyy−xky−x+ωb−xPbdx+12∫σ2x2∂2∂x2Pdx=∫2xy−xky−xPdyPdx12∫σ2x2∂2∂x2Pdx=a−∫y−x2ky−xPdyPdx+σ2∫Pdx,where ky−x is a bounded confidence kernel. In (a) of (42), we use the symmetry of Px and Py and the integration by parts.
Considering the steady state andσ⟶0 limit, we have d/dtQt=0, yet if in the support of P∞, there are two points x and y such that x−y<R and x≠y, and we have −∫y−x2ky−xPyPx<0, which is a contradiction, so that our claim follows. □
Supposingω>0, it is hard to obtain a steady state directly. However, we can use Qt function to give an estimation. We can show that in the bounded confidence case, the steady-state solution is still a “contraction” of initial distribution.(43)ddtQt=−∫μbx2∂∂x∫Pdyy−xky−xPbdx+12∫σ2x2∂2∂x2Pdx=∫2xy−xky−xPdyPdx+∫2ωxb−xμdbPbdx+12∫σ2x2∂2∂x2Pdx=−∫y−x2ky−xPdyPdx+∫2ωxb−xμdbPbdx+σ2∫Pdx.
Since−∫y−x2ky−xPdyPdx≤0 and σ⟶0, we have(44)∫xb−xμdbPbdx≥0,such that in the sense of mean value, x<b=x0. And the degree of offset is depending on the value of ∫y−x2ky−xPdyPdx. We can see that ∫y−x2ky−xPdyPdx will increase as the radius R of the kernel ky−x grows. When R⟶∞ and ky−x=1, the model will degenerate into the linear case in Theorem 3.


## 5. The Outline of the Proof

### 5.1. SDE Related to Initial Values

Since the process we study is not a Markov process, we need to consider SDE with the form(45)dX=FX0,X,tdt+dW.It has no Markov generator, but the non-Markovian of it is “not so bad”. The law ofX can be solved by the following method. For a given X0, we can solve the law by etAP0, where A is the generator of the Fokker–Planck equation, and P0=δx0 is the initial Dirac distribution. Then, the solution of the SDE has the form(46)∫dP0X0etAX0δX0,where AX0 is the generator with given X0, δX0 is the Dirac measure with the support on X0, and P0 is the initial distribution of X0.The essence of this method is to regardX0 as an additional random variable. When SDE has the above form, we need the double layer empirical measure, which will regain the symmetry.We will consider the SDEs(47)dXi=ωbi−Xitdt+∑j≠iαhXj,XidNj+σdWi,where bi is a stochastic variable such that Xi0=bi; then, we will prove the large number law for the double layer empirical measure 1/N∑iδXi,bi, which will give the Mckean–Vlasov limit. This is equal to consider SDEs(48)dXib=ωb−Xibtdt+∑j≠i∫dμb′αhXjb′,XibdNj+σdWi,where Xb is the stochastic process with given b, and EX=E∫dμbXb.


### 5.2. Intermediate Process

Andreis et al. [15] proved that the Mckean–Vlasov limit of the equation(49)dXiN=FXiNdt+σXiNdWi+1N∑j≠iNhXjN,XiNdNj,has the form(50)dX=FX+∫dPYhY,Xλdt+σXdW,where P is the law of X.We use the same way as Andreis et al.[15] to reduce the jump-SDE to the “averaging dynamics.” Their method is to decompose the jump terms into a martingale and a continuous part. Then, the martingale can be easily estimated by Burkholder–Davis–Gundy inequality.We will show that there is an intermediate processYi with the following SDE:(51)dYib=b−Yidt+σdWi+∫dμb′1N∑j=1NαhYjb′,Yibλdt,such that(52)∫dμb1N∑j=1NEsupXib−Yib≤CN.By comparing these two process directly, we can notice that(53)∫μbEsuprXib−Yib≤∫μbF+Θ,where F=ω∫0tXibs−Yibsds, and Θ=Esupr1/N∑j∫0rdsαhXj,XidNjs−αhYj,Yiλds.Using Doob–Meyer decomposition,(54)dN=dN˜+λdt,where N˜ is a martingale.Intuitively, the supremum of a martingale would not be very large. This can be proved by Burkholder–Davis–Gundy inequality and our assumption onh. Then, we can obtain the approximation (52). See the second part of the appendix for a detailed proof of (52).


### 5.3. Mckean–Vlasov Process

The intermediate process has the form like(55)dXi=1N∑jfXi,Xjdt+gXi,ωidt+dW,where ωi is a random variable.The asymptotic behavior of the double layer empirical measure, with the case thatf and g are bounded, has been studied [38]. They use Varadhan’s lemma to reduce the system to the case without interaction. Since f and g are not bounded, we use the method in the study by Sznitman [28] to compare the two SDEs directly. Using the Lipschitz condition, we can control EsuprXr−X¯r by the first Wasserstein distance ρ.The first Wasserstein distance is defined by(56)W1μ1,μ2=inf∫Rd×Rdx−yηdx,dy:η∈Hμ1,μ2,where Hμ1,μ2 is the set of all probability measures on Rd×Rd with marginals μ1 and μ2.By Kantorovich–Rubinstein duality theorem, it has a dual representation:(57)supfL≤1∫fxdμ1−μ2x,where fL is the minimal Lipschitz constant for f.Using this representation, since we have assumed that the coefficients are Lipschitz, we can see that(58)EsuprXr−X¯r≤L∫0TEρμXNt,μtdt,where μXN is the empirical measure, and μt is the solution of Mckean–Vlasov equation.ForEρμXNt,μt, we have(59)EρμXNt,μt≤EρμXNt,μX¯Nt+EρμX¯Nt,μtN,where μX¯N is the empirical measure for N independent Mckean–Vlasov processes.Then, we can derive our result by considering the Wasserstein distance between the solution of Mckean–Vlasov equation and empirical measure. Using Theorem1 in the study by Zanella and Guillin [41], if the two order moment is finite,(60)EX¯2<∞,when the size tends to infinity and the Wasserstein distance trends to zero, so our claim follows. The detailed proof is given in the appendix.


## 5.1. SDE Related to Initial Values

Since the process we study is not a Markov process, we need to consider SDE with the form(45)dX=FX0,X,tdt+dW.It has no Markov generator, but the non-Markovian of it is “not so bad”. The law ofX can be solved by the following method. For a given X0, we can solve the law by etAP0, where A is the generator of the Fokker–Planck equation, and P0=δx0 is the initial Dirac distribution. Then, the solution of the SDE has the form(46)∫dP0X0etAX0δX0,where AX0 is the generator with given X0, δX0 is the Dirac measure with the support on X0, and P0 is the initial distribution of X0.The essence of this method is to regardX0 as an additional random variable. When SDE has the above form, we need the double layer empirical measure, which will regain the symmetry.We will consider the SDEs(47)dXi=ωbi−Xitdt+∑j≠iαhXj,XidNj+σdWi,where bi is a stochastic variable such that Xi0=bi; then, we will prove the large number law for the double layer empirical measure 1/N∑iδXi,bi, which will give the Mckean–Vlasov limit. This is equal to consider SDEs(48)dXib=ωb−Xibtdt+∑j≠i∫dμb′αhXjb′,XibdNj+σdWi,where Xb is the stochastic process with given b, and EX=E∫dμbXb.


## 5.2. Intermediate Process

Andreis et al. [15] proved that the Mckean–Vlasov limit of the equation(49)dXiN=FXiNdt+σXiNdWi+1N∑j≠iNhXjN,XiNdNj,has the form(50)dX=FX+∫dPYhY,Xλdt+σXdW,where P is the law of X.We use the same way as Andreis et al.[15] to reduce the jump-SDE to the “averaging dynamics.” Their method is to decompose the jump terms into a martingale and a continuous part. Then, the martingale can be easily estimated by Burkholder–Davis–Gundy inequality.We will show that there is an intermediate processYi with the following SDE:(51)dYib=b−Yidt+σdWi+∫dμb′1N∑j=1NαhYjb′,Yibλdt,such that(52)∫dμb1N∑j=1NEsupXib−Yib≤CN.By comparing these two process directly, we can notice that(53)∫μbEsuprXib−Yib≤∫μbF+Θ,where F=ω∫0tXibs−Yibsds, and Θ=Esupr1/N∑j∫0rdsαhXj,XidNjs−αhYj,Yiλds.Using Doob–Meyer decomposition,(54)dN=dN˜+λdt,where N˜ is a martingale.Intuitively, the supremum of a martingale would not be very large. This can be proved by Burkholder–Davis–Gundy inequality and our assumption onh. Then, we can obtain the approximation (52). See the second part of the appendix for a detailed proof of (52).


## 5.3. Mckean–Vlasov Process

The intermediate process has the form like(55)dXi=1N∑jfXi,Xjdt+gXi,ωidt+dW,where ωi is a random variable.The asymptotic behavior of the double layer empirical measure, with the case thatf and g are bounded, has been studied [38]. They use Varadhan’s lemma to reduce the system to the case without interaction. Since f and g are not bounded, we use the method in the study by Sznitman [28] to compare the two SDEs directly. Using the Lipschitz condition, we can control EsuprXr−X¯r by the first Wasserstein distance ρ.The first Wasserstein distance is defined by(56)W1μ1,μ2=inf∫Rd×Rdx−yηdx,dy:η∈Hμ1,μ2,where Hμ1,μ2 is the set of all probability measures on Rd×Rd with marginals μ1 and μ2.By Kantorovich–Rubinstein duality theorem, it has a dual representation:(57)supfL≤1∫fxdμ1−μ2x,where fL is the minimal Lipschitz constant for f.Using this representation, since we have assumed that the coefficients are Lipschitz, we can see that(58)EsuprXr−X¯r≤L∫0TEρμXNt,μtdt,where μXN is the empirical measure, and μt is the solution of Mckean–Vlasov equation.ForEρμXNt,μt, we have(59)EρμXNt,μt≤EρμXNt,μX¯Nt+EρμX¯Nt,μtN,where μX¯N is the empirical measure for N independent Mckean–Vlasov processes.Then, we can derive our result by considering the Wasserstein distance between the solution of Mckean–Vlasov equation and empirical measure. Using Theorem1 in the study by Zanella and Guillin [41], if the two order moment is finite,(60)EX¯2<∞,when the size tends to infinity and the Wasserstein distance trends to zero, so our claim follows. The detailed proof is given in the appendix.


## 6. Conclusion

In this paper, we studied the asymptotic behaviors of the exponential function weight of the self-excitation interaction systemXN on the large homogeneous network when N, the number of the individuals, trends to infinity. We proved that there is a Mckean–Vlasov process X¯, such that limN⟶+∞EXN−X¯ is zero, whose opinion distribution evolve according to a Mckean–Vlasov type integrodifferential equation, which couples with the initial distribution. The steady state of this equation is also studied. We investigated the steady state distribution of linear and bounded confidence models and showed that the steady distribution of these models is a “contraction” of initial distribution. By virtue of the self-excitation interaction, even if we do not consider bounded confidence, it will give not a perfect consensus, but a distribution depending on the initial distribution, such as this model, can avoid Abelson’s diversity puzzle.



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*Source: 1861583-2020-08-24.xml*