Convergence Properties of the Heterogeneous Deffuant-Weisbuch Model\thanksreffootnoteinfo

# Convergence Properties of the Heterogeneous Deffuant-Weisbuch Model\thanksreffootnoteinfo

## Abstract

The Deffuant-Weisbuch (DW) model is a bounded-confidence opinion dynamics model that has attracted much recent interest. Despite its simplicity and appeal, the DW model has proved technically hard to analyze and its most basic convergence properties, easy to observe numerically, are only conjectures.

This paper solves the convergence problem for the heterogeneous DW model. We establish that, for any positive confidence bounds and initial values, the opinion of each agent will converge to a limit value almost surely. Additionally, we show that the limiting opinions of any two agents either are the same or have a distance larger than the confidence bounds of the two agents. Moreover, we provide some sufficient conditions for the heterogeneous DW model to reach consensus. Finally, we show the mean-square convergence rate of the heterogeneous DW model is exponential.

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Convergence of Heterogeneous DW Model

1

footnoteinfo]This work was supported by the National Key Basic Research Program of China (973 program) under grant 2016YFB0800404, the National Natural Science Foundation of China under grants 11688101, and the Leading Research Projects of Chinese Academy of Sciences under grant QYZDJ-SSW-JSC003. Additionally, this material is based upon work supported by, or in part by, the U. S. Army Research Laboratory and the U. S. Army Research Office under grant numbers W911NF-15-1-0577.

Chen]Ge Chen, Su1]Wei Su, Mei]Wenjun Mei, Bullo]Francesco Bullo,

pinion dynamics, consensus, Deffuant model, gossip model, bounded confidence model

## 1 Introduction

The field of opinion dynamics studies the dynamical processes regarding the formation, diffusion, and evolution of public opinion about certain events and object of interest in social systems. The study of opinion dynamics can be traced back to the two-step communication flow model in (Katz and Lazarsfeld, 1955) and the social power and averaging model in (French Jr., 1956). The model by French Jr. (1956) was then elaborated by Harary (1959) and rediscovered by DeGroot (1974). Other notable developments include the model by Friedkin and Johnsen (1990) with attachment to initial opinions, a general influence network theory (Friedkin, 1998), social impact theory (Latané, 1981), and dynamic social impact theory (Latané, 1996). A comprehensive review of opinion dynamics models is given in the two tutorials Proskurnikov and Tempo (2017, 2018) and the textbook Bullo (2018).

In recent years, significant attention has focused on so-called bounded confidence (BC) models of opinion dynamics. In these models one individual is willing to accord influence to another only if their pair-wise opinion difference is below a threshold (i.e., the confidence bound). (Deffuant et al., 2000) propose their now well-known BC model called the Deffuant-Weisbuch (DW) model or Deffuant model. In this model a pair of individuals is selected randomly at each discrete time step and each individual updates its opinion if the other individual’s opinion lies within its confidence bound. A second well-known BC model is the Hegselmann-Krause (HK) model (Hegselmann and Krause, 2002), where all individuals update their opinions synchronously by averaging the opinions of individuals within their confidence bounds.

As reported in (Lorenz, 2007, 2010), simulation results for the DW model have revealed numerous interesting phenomena such as consensus, polarization and fragmentation. However, the DW model is in general hard to analyze due to the nonlinear state-dependent inter-agent topology. Current analysis results focus on the homogeneous case in which all the agents have the same confidence bound. The convergence of the homogeneous DW model has been proved in (Lorenz, 2005) and its convergence rate is established in (Zhang and Chen, 2015). Some research has considered also modified DW models. For example, (Como and Fagnani, 2011) consider a generalized DW model with an interaction kernel and investigate its scaling limits when the number of agents grows to infinity; (Zhang and Hong, 2013) generalize the DW model by assuming that each agent can choose multiple neighbors to exchange opinion at each time step. Despite all this progress, the analysis of the heterogeneous DW model is still incomplete in that its convergence properties are yet to be established.

It is worth remarking that the analysis of the HK model is also similarly restricted to the homogeneous case; the convergence of the heterogeneous HK model is only conjectured in our previous work (MirTabatabaei and Bullo, 2012) and has since been established in (Chazelle and Wang, 2017) only for the special case that the confidence bound of each agent is either or . In general, numerous conjectures remain open for heterogeneous bounded-confidence models.

This paper establishes the convergence properties of the heterogeneous DW model. We show that, for any positive confidence bounds and initial opinions, the opinion of each agent converges almost surely to a limiting value. Additionally we prove that the limiting values of any two agents’ opinions are either identical or have a distance larger than the confidence bounds of the two agents. Moreover, we show that a sufficient, and in some cases also necessary, condition for almost sure consensus; the intuitive condition is expressed as a function of the largest confidence bound in the group. Finally, we show the exponential convergence of the mean square error.

The paper is organized as follows. Section 2 introduces the heterogeneous DW model and our almost surely convergence results. Section 3 contains the proofs of our convergence results. Section 4 contains the analysis of the convergence rate and, finally, Section 5 concludes the paper.

## 2 The heterogeneous DW model and our main convergence results

Following (Lorenz, 2007), this paper considers the following basic DW model. In a group of agents, we assume each agent has a real-valued opinion at each discrete time . We let assume, without loss of generality, that . We let denote the confidence bound of the agent and we assume, without loss of generality,

 r1≥r2≥⋯≥rn>0.

We let denote the indicator function, i.e., we let if the property holds true and otherwise. At each time , a pair is independently and uniformly selected from the set of all pairs . Subsequently, the opinions of the agents and are updated according to

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩xit(t+1)=xit(t)+12\mathbbm1{|xjt(t)−xit(t)|≤rit}(xjt(t)−xit(t)),xjt(t+1)=xjt(t)+12\mathbbm1{|xjt(t)−xit(t)|≤rjt}(xit(t)−xjt(t)), (1)

whereas the other agents’ opinions remain unchanged:

 xk(t+1)=xk(t),for k∈{1,…,n}∖{it,jt}. (2)

If , the DW model is called homogeneous, otherwise heterogeneous.

Previous works (Lorenz, 2005) show that the homogeneous DW model (1)-(2) always converges to a limit opinion profile. Simulations reported in (Lorenz, 2007) show that this property holds also for the heterogeneous case; but a proof for this statement is lacking. We note that the original DW model (Deffuant et al., 2000) contains a weighting factor instead of factor in our protocol (1)-(2). Simulations in (Deffuant et al., 2000; Weisbuch et al., 2002) show that the parameter affects only the convergence time and so previous works (Lorenz, 2007, 2010) simplified the model by setting . Following these previous works, also this paper adopts the simplification.

Before stating our convergence results, we need to define the probability space of the DW model. If the initial state is a deterministic vector, we let be the sample space, be the Borel -algebra of , and be the probability measure on . Then the probability space of the DW model is written as . If the initial state is a random vector, we let be the sample space and, similarly to the case of deterministic initial state, the probability space is defined by .

{thm}

(Almost sure convergence of heterogeneous DW model) Consider the heterogeneous DW model (1)-(2) with positive confidence bounds. For any initial state , there exists a random vector satisfying

1. or for all , and

2. converges to almost surely (a.s.) as , that is,

 P(ω∈Ω:limt→∞x(t)(ω)=x∗(ω))=1.

The proof of Theorem 2 is postponed to Section 3. Fig. 1 displays the simulation results for a heterogeneous DW model (1)-(2) with agents and with confidence bounds equal to respectively. Consistently with Theorem 2, Fig. 1 shows that the individual opinions converge to two distinct limit values and that the distance between the two values is larger than .

Theorem 2 leads to two corollaries on convergence to consensus. By consensus we mean that all agents’ opinions converge to the same value.

{cor}

(Almost sure consensus for large confidence bound) Consider the heterogeneous DW model (1)-(2) with positive confidence bounds. If the largest confidence bound satisfies , then for any initial state the system reaches consensus a.s.

{cor}

(Almost sure consensus if and only if large confidence bound) Consider the heterogeneous DW model (1)-(2) with positive confidence bounds. Assume that the initial state is randomly distributed in and that its joint probability density has a lower bound , that is, for any real numbers , , with ,

 P(n⋂i=1{xi(0)∈[ai,bi]})≥ρ\textupminn∏i=1(bi−ai). (3)

Then the heterogeneous DW model reaches consensus almost surely if and only if .

Corollary 2 provides a sufficient and necessary condition for almost sure consensus when the initial opinions are randomly distributed. However, for settings when almost sure consensus is not guaranteed, the probability of achieving consensus is unknown. In the remainder of this section, we provide simulation results for the consensus probability of the heterogeneous DW model. Let and suppose that agent has a maximal confidence bound whose value is chosen over the set . We approximate the consensus probability via the Monte Carlo method. We run 1000 samples for each value of . In each sample, we assume the initial opinions are independently and uniformly distributed on , while the confidence bounds of agents are independently and uniformly distributed on . Fig. 2 shows the estimated consensus probability of the heterogeneous DW model (1)-(2) as a function of the maximal confidence bound .

## 3 Proof of convergence results

The proof of Theorem 2 requires multiple steps. We adopt the method of “transforming the analysis of a stochastic system into the design of control algorithms” first proposed by (Chen, 2017). This method requires the construction of a new system called as DW-control system to help with the analysis of the DW model.

### 3.1 DW-control system and connection to DW model

Consider the DW protocol (1)-(2) where, at each time , the pair is not selected randomly but instead treated as a control input. In other words, assume that is chosen from the set arbitrarily as a control signal. We call such a control system the DW-control system.

Given , we say is reached at time if and is reached in the time interval if there exists such that .

{defn}

Let be two state sets. Under the DW-control system, is said to be (uniformly) finite-time reachable from if there exists a duration such that for any , we can find a sequence of pairs for opinion update which guarantees is reached in the time interval .

Based on these definitions we can get the following result.

{lem}

(Connection between DW model and DW-control system) Let be a set of states. Assume is finite-time reachable from under the DW-control system. Then, under the DW protocol, for any initial state , there exist constants and such that

 P(τ≥t)≤a⌊t/T⌋,    ∀t≥1,

where is the time when is firstly reached.

{pf}

First according to the rule of the DW protocol (1)-(2) we get for all . Also, since is reached in finite time from under the DW-control system, by Definition 3.1 there exist an integer such that for any , we can find a sequence of pairs which guarantees is reached in . From this and the definition of the DW-control system, for any and , there exists a sequence of pairs such that is reached in . Thus, under the DW protocol, for any and we have

 P({S is reached in [t,t+t∗]}|x(t)) ≥P(t+t∗−1⋂s=t{(is,js)=(i′s,j′s)}|x(t)) =P((it,jt)=(i′t,j′t)|x(t)) ⋅P(t+t∗−1⋂s=t+1{(is,js)=(i′s,j′s)}|x(t),(it,jt)=(i′t,j′t)) =⋯=P((it,jt)=(i′t,j′t)|x(t)) ⋅P({(it+1,jt+1)=(i′t+1,j′t+1)}|x(t),(it,jt)=(i′t,j′t)) ⋯P({(it+t∗−1,jt+t∗−1)=(i′t+t∗−1,j′t+t∗−1)} |x(t),(is,js)=(i′s,j′s),s∈[t,t+t∗−2]), (4)

where all the equations use Bayes’ Theorem. Because is uniformly and independently selected from the set , for any we get

 P{(is,js)=(i′s,j′s)|x(s)}=1|N|=2n(n−1), (5)

where denotes the cardinality of the set . Substituting (5) into (3.1) yields

 P({S is reached in [t,t+t∗]}|x(t))≥2t∗nt∗(n−1)t∗. (6)

Set to be the event that is reached in , and let be the complement set of . For any integer and , Bayes’ Theorem again and equation (6) imply

 P({S is not reached in [0,(t∗+1)M−1]}|x(0)) =P(M−1⋂m=0Ecm(t∗+1)∣∣x(0)) =P(Ec0|x(0))M−1∏m=1P(Ecm(t∗+1)∣∣x(0),⋂0≤m′

Let . For any integer and , by (3.1) we have

 P(τ ≥(t∗+1)M|x(0)) =P({S is not reached in [0,(t∗+1)M−1]}|x(0)) ≤aM. (8)

Let . From (8) we can get

According to Lemma 3.1, to prove the convergence of the DW model, we only need to design control algorithms for DW-control system such that a convergence set is reached. Before the design of such control algorithms we introduce some useful notions.

### 3.2 Maximal-confidence clusters and properties

Recall that we assume . For any opinion state , let be the set of the agents that can connect to agent directly or indirectly with the confidence bound , i.e., if and only if or there exists some agents such that . From this definition we have .

Set . If is not empty, we let and define to be the set of the agents that can connect to agent directly or indirectly with the confidence bound . Set . If is not empty, we let and define to be the set of the agents that can connect to agent directly or indirectly with the confidence bound . Repeat this process until there exists an integer such that . We call the sets maximal-confidence (MC) clusters. Note that MC clusters are quite different from connected components in graph theory.

To illustrate the definition of MC clusters we give an example, visualized Fig. 3: Assume that and that the agents are labeled by . We suppose . With the confidence bound the agent can connect to agents and , and the agent can connect to agent ; however agent cannot connect to agent . Thus, the first MC cluster is . The remaining agents are and . With the confidence the agent can connect to agent , and the agent can connect to agent , so the second MC cluster is .

The following lemma describes the distance between MC clusters. {lem} (Distance between maximal-confidence clusters) For any opinion state and two different MC clusters and , let be the maximal confidence bound of all agents in and . Then, the opinion values of agents in are all bigger or smaller than those in , i.e.,

 xk−xl>rijmax    ∀k∈Ci(x),l∈Cj(x),

or

 xl−xk>rijmax    ∀k∈Ci(x),l∈Cj(x).
{pf}

Without loss of generality we assume that . Let and denote the minimal and maximal opinion values of all agents in respectively. For any , if , by the definition of the MC cluster we have , which is contradictory with . Thus, for any , we get

 xl

or

 xl>ximax+rijmax≥xk+rijmax,  ∀k∈Ci(x). (10)

Since is also a MC cluster, there is no agent in whose opinion value is located in the interval . Thus, either (9) or (10) holds for all .

Under the DW protocol (1)-(2), the MC clusters have the convex property as follows.

{lem}

(Convexity of maximal-confidence clusters) Consider the DW protocol (1)-(2) with arbitrary initial state and update pairs . For any and any MC cluster , the opinion values of all agents in will always stay in the interval at the time , i.e.,

 ximin(t)≤xj(s)≤ximax(t),    ∀j∈Ci(x(t)),s≥t,

where and denote the minimal and maximal opinion values of all agents in respectively.

{pf}

Assume that at time all MC clusters are . By Lemma 3.2 we can order these clusters as

 Cj1≺Cj2≺⋯≺CjK,

and get, for ,

 minl∈Cjk+1xl(t)−maxl∈Cjkxl(t)>rk,k+1, (11)

where means that at time the opinion values of the agents in are all less than those in , and .

By the DW protocol (1)-(2), if the update pair belongs to different MC clusters then from (11) we have and ; if belongs to a same MC cluster then and will stay in the interval . Thus, for ,

 minl∈Cjk+1xl(t+1)−maxl∈Cjkxl(t+1)>rk,k+1.

Repeating this process yields our result.

With the definition and properties of MC clusters we can design control algorithms and complete final proof of our results in the following subsection.

### 3.3 Design of control algorithms and final proofs

For any opinion state and any MC cluster , we say that is a complete cluster if any agent in can interact with others with the minimal confidence bound of , i.e.,

 maxj,k∈Ci(x)|xj−xk|≤minj∈Ci(x)rj.
{lem}

Let and be arbitrarily given. Let be an arbitrary MC cluster, in which the agents’ maximal and minimal confidence bounds are and respectively. Assume

 maxM,m∈Ci(x(t))[xM(t)−xm(t)]>rimin. (12)

Then, under the DW-control system, there is a sequence of agent pairs with

 t∗≤(|Ci(x(t))|−1)2(1+⌈log2⌈rimax/rimin⌉⌉)

for opinion update, such that one of the following two results holds:

1. the agents in split into different MC clusters at time ; and

2. we have

 maxM,m∈Ci(x(t))[xM(t+t∗)−xm(t+t∗)]≤maxM,m∈Ci(x(t))[xM(t)−xm(t)]−rimin/4.

The proof of Lemma 3.3 is quite complicated. We put it in Appendix A.

{lem}

Let and be arbitrarily given. Let be an arbitrary MC cluster. Assume

 maxM,m∈Ci(x(t))[xM(t)−xm(t)]≤minm∈Ci(x(t))rm. (13)

Then, under the DW-control system, there is a sequence of agent pairs with for opinion update, such that

 maxM,m∈Ci(x(t))[xM(t+t∗)−xm(t+t∗)]≤23maxM,m∈Ci(x(t))[xM(t)−xm(t)].
{pf}

The proof of this lemma is similar for the cases . To simplify the exposition we consider only the case when . We set

 ximin(0) =minm∈Ci(x(0))xm(0), ximax(0) =maxm∈Ci(x(0))xm(0).

Let . Set

 B––(s):={m∈Ci(x(0)):xm(s)

and

 ¯¯¯¯B(s):={m∈Ci(x(0)):xm(s)>ximax(0)−d(0)/3}.

Take and . Without loss of generality we assume .

Label the elements in as , and the elements in as . For , we choose as the agent pair for opinion update at time , then by the protocol (1)-(2) and (13) we get

 xik(I)=xjk(I)=xik(k+1)=xik(k)+xjk(k)2=xik(0)+xjk(0)2∈(ximin(0)+ximax(0)−d(0)32,      ximin(0)+d(0)3+ximax(0)2)=(ximin(0)+d(0)3,ximax(0)−d(0)3),

which proves . Combining this equality with Lemma 3.2 we obtain

 maxM,m∈Ci(x(t))[xM(I)−xm(I)]≤ximax(0)−ximin(0)−d(0)3=2d(0)3.

Finally, because , our result follows.

For any opinion state , let denote the maximal diameter among all the MC clusters , i.e.,

 D(x)=max1≤i≤Kmaxj,k∈Ci(x)|xj−xk|. (14)
{lem}

Consider the DW-control system. Let and be the maximal and minimal confidence bounds of all agents. Then for any initial state and constant , there exists a sequence of agent pairs with

 t∗≤(n−1)2(1+⌈log2⌈rmaxrmin⌉⌉)⌈4(1−rmin)rmin⌉    +n2⌈−logεlog(3/2)⌉.

for opinion update such that . {pf} Assume there are MC clusters ,
, at time . Using Lemma 3.3 repeatedly there exists a sequence of agent pairs for opinion update such that

 maxM,m∈Ci(x(T1))[xM(T1)−xm(T1)]≤minm∈Ci(x(T1))rm

for all . Since for any positive integers , by Lemma 3.3 it can be computed that

 T1≤(n−1)2    ×(1+⌈log2⌈rmaxrmin⌉⌉)⌈4(1−rmin)rmin⌉.

Further, using Lemma 3.3 repeatedly there exists a sequence of agent pairs , for opinion update such that

 D(x(T1+T2))≤ε.

Lemma 3.3 now implies .

{pf*}

Proof of Theorem 2 For any constant , let be the state set defined by

 Sε:={x∈[0,1]n:D(x)≤ε},

where is the maximal diameter of all MC clusters defined by (14). By Lemma 3.3, is finite-time reachable from under the DW-control system. Let be the time when is firstly reached under the DW protocol (1)-(2). By Lemma 3.1, for any . By the convexity of MC clusters (Lemma 3.2) we have for all . Let we can get

 P(ω∈Ω:limt→+∞D(x(t))(ω)=0)=1.

From this and Lemma 3.2 we have that a.s. converges to a random vector . By Lemma 3.2 we obtain or for any .

{pf*}

Proof of Corollary 2 By Theorem 2 we have a.s. converges to a limit point which satisfies either or for all . Because , we have for all , which indicates is a consensus state.

{pf*}

Proof of Corollary