# Discrete-Time Polar Opinion Dynamics with Susceptibility

This paper considers a discrete-time opinion dynamics model in which each individual's susceptibility to being influenced by others is dependent on her current opinion. We assume that the social network has time-varying topology and that the opinions are scalars on a continuous interval. We first propose a general opinion dynamics model based on the DeGroot model, with a general function to describe the functional dependence of each individual's susceptibility on her own opinion, and show that this general model is analogous to the Friedkin-Johnsen model, which assumes a constant susceptibility for each individual. We then consider two specific functions in which the individual's susceptibility depends on the polarity of her opinion, and provide motivating social examples. First, we consider stubborn positives, who have reduced susceptibility if their opinions are at one end of the interval and increased susceptibility if their opinions are at the opposite end. A court jury is used as a motivating example. Second, we consider stubborn neutrals, who have reduced susceptibility when their opinions are in the middle of the spectrum, and our motivating examples are social networks discussing established social norms or institutionalized behavior. For each specific susceptibility model, we establish the initial and graph topology conditions in which consensus is reached, and develop necessary and sufficient conditions on the initial conditions for the final consensus value to be at either extreme of the opinion interval. Simulations are provided to show the effects of the susceptibility function when compared to the DeGroot model.

## Authors

• 103 publications
• 11 publications
• 10 publications
• 47 publications
• 22 publications
• ### Continuous-time Opinion Dynamics on Multiple Interdependent Topics

In this paper, and inspired by the recent discrete-time based works of [...
05/08/2018 ∙ by Mengbin Ye, et al. ∙ 0

• ### An Influence Network Model to Study Discrepancies in Expressed and Private Opinions

In many social situations, a discrepancy arises between an individual's ...
06/29/2018 ∙ by Mengbin Ye, et al. ∙ 0

• ### Importance of initial conditions in the polarization of complex networks

Most existing models of opinion formation use random initial conditions....
02/24/2018 ∙ by Snehal M. Shekatkar, et al. ∙ 0

• ### Evolution of Social Power in Social Networks with Dynamic Topology

The recently proposed DeGroot-Friedkin model describes the dynamical evo...
05/27/2017 ∙ by Mengbin Ye, et al. ∙ 0

• ### A Continuous Opinion Dynamic Model in Co-evolving Networks--A Novel Group Decision Approach

Opinion polarization is a ubiquitous phenomenon in opinion dynamics. In ...
05/17/2017 ∙ by Qingxing Dong, et al. ∙ 0

• ### Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities

Recently the dynamics of signed networks, where the ties among the agent...
03/20/2017 ∙ by Anton V. Proskurnikov, et al. ∙ 0

• ### Noise-synchronizability of opinion dynamics

With the analysis of noise-induced synchronization of opinion dynamics w...
10/08/2018 ∙ by Wei Su, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## I Introduction

The problem of opinion dynamics, which considers how an individual’s opinion forms and evolves through interactions with others in a social network, has been widely studied in the social sciences for decades. The classical discrete-time DeGroot model, in which each individual updates her opinion by taking a convex combination of the opinions of her neighbors at each time step, is perhaps one of the most well known models [1]. This model is closely related to discrete-time linear consensus algorithms, which have been heavily studied in multi-agent coordination literature [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Since the time the DeGroot model was proposed, numerous other models have been introduced, in both continuous- and discrete-time setting. These various models, which describe the opinion formation process in the context of different social cognitive processes, all attempt to understand the formation and evolution of opinions in social networks of all sizes, and explain observed social phenomena such as polarization or attitude extremity [12, 13, 14, 15], and subculture formation [16, 17].

There are many variants of the DeGroot model for opinion dynamics. The Altafini model, which suggests the interactions between individuals can be cooperative or antagonistic, has been studied as a discrete-time process in [18, 19, 20, 21], and the continuous-time counterpart has been considered in [22, 23, 24, 25]. It is notable because the model links the limiting opinion behavior with the structural balance of the graph representing the social network. Some other models primarily focus on linking the limiting opinion behavior with a social process. For example, the Hegselmann-Krause model shows the social cognitive process of homophily is linked to fact that opinions in the social network eventually form clusters [26, 27, 28]. It was shown in [16] that an individual’s desire to strive for uniqueness generated persistent subcultures which formed and vanished over time. On the other hand, an individual conforming to a social norm generated pluralistic ignorance [17]. Finally, some models attempt to link final opinion behavior to a combination of social processes and the underlying network structure. The Friedkin-Johnsen model [29, 30] considered individual susceptibility to influence and shows that opinions reach a persistent diversity under general graph structures. The DeGroot-Friedkin model [31] studied an individual’s ability to reflect on her impact in the opinion formation process, and showed her social power depended on the graph structure.

A key aspect of the DeGroot model is the interpersonal influence, which describes the amount of influence each individual’s neighbors have in determining that individual’s new opinion. Some of the results consider arbitrary, time-varying interpersonal influence, e.g. [6, 19, 24]. However, many of the aforementioned models consider influence determined by a social process, e.g. homophily [28], social distancing [16, 17], conformity [17], desire for uniqueness [16], biased assimilation [32], or reflected self-appraisal [31]. Because the social process is often dependent on the states, i.e. opinions (which change with time), then necessarily the interpersonal influences are state-dependent, and thus time-varying. In a recent paper [33], a continuous-time model has been proposed for fixed social network topology which considers, separately, three different cognitive processes to drive the influence change. In [33], the term “polar opinion dynamics” relates to the fact that the level of influence is dependent on how extreme, i.e. polar, an individual’s opinion is.

In this paper, we study a discrete-time opinion dynamics model where an individual’s susceptibility to influence is dependent on her current opinion, and allow the social network topology to vary over time. We first propose a general model, show it can be considered as a generalization of both the DeGroot model and the Friedkin-Johnsen model [29, 30], and establish some general properties of the model. We then investigate discrete-time versions of two of the three cognitive process introduced in [33], bearing in mind that discrete-time models are more appropriate to describe opinion dynamics, at least from the point that individuals change their minds from time to time, instead of continuously. In addition, we provide social examples from existing literature to motivate these susceptibility functions. For each function, we provide sufficient conditions on the graph topology and initial opinions for the social network to reach a consensus. Importantly, we establish necessary and sufficient conditions for the social network to hold opinions at either extremes of the opinion interval, whereas [33] provided only sufficient conditions, or no conditions at all for extremity of the final consensus in the continuous-time model. Lastly, it turns out that while some of the limiting behaviors are similar to the continuous-time model, in other cases, the limiting behaviors are not the same.

The remainder of this paper is organized as follows. Some notations and preliminaries are introduced in Section I-A. In Section II, the discrete-time polar opinion dynamics model with susceptibility is introduced. The main results of the paper are presented in Section III, which are illustrated and compared with the DeGroot model via simulations in Section IV. The paper ends with some concluding remarks in Section V.

### I-a Preliminaries

For any positive integer , we use to denote the index set

. We view vectors as column vectors and write

to denote the transpose of a vector . For a vector , we use to denote the th entry of . For any matrix , we use to denote its th entry. A nonnegative

matrix is called a stochastic matrix if its row sums are all equal to

. We use and to denote the vectors whose entries all equal 0 and 1, respectively, and

to denote the identity matrix, while the dimensions of the vectors and matrices are to be understood from the context. For any real number

, we use to denote the absolute value of . For any two real vectors , we write if for all , if and , and if for all . For any two sets and , we use to denote the set of elements in but not in . The graph of an matrix with real-valued entries is an -vertex directed graph defined so that is an arc from vertex to vertex in the graph whenever the th entry of is nonzero. We will use the terms “individual” and “agent” interchangeably.

## Ii The General Model

In this section, we propose a general model for describing opinion dynamics where each individual’s susceptibility to being influenced by others is affected by some social process, and give some results on the trajectories of the opinions. In the next section, we shall propose two specific models to describe two different variants of a social process.

Consider a social network of agents, labeled through , discussing opinions on a given topic.111 The purpose of labeling of the agents is only for convenience. We do not require a global labeling of the agents in the network. We only assume that each agent can identify her own neighbors. Each agent can only learn, and be influenced by, the opinions of certain other agents called the neighbors of agent . Neighbor relationships among the agents are described by a directed graph , called the neighbor graph, which may change over time. Agent is a neighbor of agent at time whenever is an arc in . Thus, the directions of arcs indicate the directions of information flow (specifically opinion flow). For convenience, we assume that each agent is always a neighbor of herself. Thus, has self-arcs at all vertices for all time . Each agent has control over a real-valued quantity , called agent ’s opinion.

In the time-varying DeGroot model222The original DeGroot model was proposed for a fixed graph [1]. Subsequent results expanded this to time-varying graphs, e.g. [3, 6]., each agent updates her opinion at each discrete time by setting

 xi(t+1)=∑j∈Ni(t)wij(t)xj(t),i∈[n], (1)

where denotes the set of neighbors of agent at time including herself, and are positive influence weights satisfying for all and time . We assume that the weights change in a manner which is entirely independent of . We rewrite the above model as

 xi(t+1) = xi(t)+⎛⎝∑j∈Ni(t)wij(t)xj(t)−xi(t)⎞⎠ = xi(t)+∑j∈Ni(t)wij(t)(xj(t)−xi(t)),

with the second equality obtained by noting that , and define

 ui(t)=∑j∈Ni(t)wij(t)(xj(t)−xi(t)).

Then, represents the influence of agent ’s neighbors, which generates a change in the opinion of agent , i.e. can be viewed as the control input of agent at time . We now suppose that agent may not fully accept the influence of her neighbors, and her openness to influence, or susceptibility, is captured by the real-valued function . We make the following assumption on :

###### Assumption 1

The susceptibility function takes on values in .

We consider the following model for opinion dynamics with susceptibility:

 xi(t+1)=xi(t)+fi(xi(t))∑j∈Ni(t)wij(t)(xj(t)−xi(t)). (2)

If at time , agent fully accepts her neighbors’ influence at time , and the model reduces to the DeGroot model. In the case when , agent will ignore her neighbors and not change her opinion at time ; in such a case, the agent is sometimes called stubborn [34, 29]. It is worth emphasizing that an agent’s susceptibility function depends on its current opinion, i.e. state. This is consistent with the many works discussed in the introduction, which considers social cognitive processes which are dependent on the individual’s opinion and, in some instances, the opinions of her neighbor.

There is also another interpretation of the model. Inspired by the Friedkin-Johnsen model [29, 30, 35], we assume that each agent updates her opinion as a convex combination of her own opinion and the weighted average of her neighbors’ opinions;333 In the Friedkin-Johnsen model, of the second term on the right of (3) is replaced by . specifically,

 xi(t+1)=λi∑j∈Ni(t)wij(t)xj(t)+(1−λi)xi(t), (3)

where the constant is agent ’s susceptibility or openness to being influenced by her neighbors’ opinions. Let us replace with the state-dependent susceptibility function . It follows that

 xi(t+1) =fi(xi(t))∑j∈Ni(t)wij(t)xj(t)+(1−fi(xi(t)))xi(t) (4) =xi(t)−fi(xi(t))⎛⎝xi(t)−∑j∈Ni(t)wij(t)xj(t)⎞⎠ =xi(t)−fi(xi(t))⎛⎝∑j∈Ni(t)wij(t)(xi(t)−xj(t))⎞⎠ =xi(t)+fi(xi(t))ui(t),

which is the same as (2).

###### Remark 1

We note here that there are two types of time-dependency in the influence term . Firstly, we have assumed that the influence weights may be time-varying but state-independent. This may occur in situations where an individual decides to at time , stop sharing her opinion, stop listening to certain neighbors, start listening to other neighbors, or adjust weight magnitudes (perhaps by becoming more persuasive) etc. This differs from the continuous-time work in [33], which considered static influence weights. The second is time-dependency arising from the fact that the susceptibility of individual , , is state-dependent. In the Friedkin-Johnsen model, susceptibility was assumed to be constant. While some papers have studied state-dependent susceptibility in discrete-time models, they provide only simulations, and have not provided rigorous analysis or considered time-varying influence weights [17].

In this paper, we assume that all the initial opinions , , lie in the interval , where and represent the extreme positive and negative opinions, respectively. Such a scaling is typical in opinion dynamics problems where may represent individual ’s attitude towards an idea, e.g. the legalization of recreational marijuana, with maximally supportive and maximally opposing. The following lemma shows that is an invariant set of each agent’s opinion dynamics given by (2).

###### Lemma 1

Suppose that each agent follows the update rule (2) and that for all . Then, for all and time .

Proof: From (4) and the assumption that , each agent ’s updated value is a convex combination of and ; i.e. a convex combination of the current opinions of her neighbors. Using induction, it is easy to see that if for all , it follows that for all and time .

This shows that individual ’s opinion will remain bounded from above and below, and ensures that an individual’s opinion cannot become increasingly extreme in either the positive or negative direction. More can be said about the most extreme opinions in the social network. Specifically, the most negative and positive opinions will never become more negative and more positive, respectively.

###### Lemma 2

Suppose that each agent follows the update rule (2). Then, is nondecreasing and is nonincreasing as increases.

Proof: From (2), it follows that

 xi(t+1) = xi(t)+fi(xi(t))⎛⎝∑j∈Ni(t)wij(t)xj(t)−xi(t)⎞⎠ = (1−fi(xi(t)))xi(t)+fi(xi(t))∑j∈Ni(t)wij(t)xj(t)

The above set of equations can be combined into state form. Toward this end, let be the vector in whose th entry equals , be the diagonal matrix whose th diagonal entry equals with , and be the matrix whose th entry equals . Then, it follows that

 x(t+1) = (I−F(x(t)))x(t)+F(x(t))W(t)x(t) (5) = S(x(t),t)x(t),

where

 S(x(t),t)=I−F(x(t))+F(x(t))W(t).

It is worth noting that is a function of as is so. Thus, (5) is a nonlinear system. From Lemma 1 and Assumption 2, is a nonnegative matrix for all time . Since is a stochastic matrix, it follows that

 S(x(t),t)1 = (I−F(x(t))+F(x(t))W(t))1 = 1−F(x(t))1+F(x(t))1 = 1,

which implies that is a stochastic matrix for all time . Thus, each is a convex combination of all , , which implies that is nondecreasing and is nonincreasing as increases.

We also impose the following set of assumptions on the weights throughout the rest of the paper.

###### Assumption 2

For all and , there hold if and otherwise. There exists a positive number such that, for all and , if , then . For all and , there holds .

Such a set of assumptions implies that is a stochastic matrix for all time , and was widely used in the DeGroot (and consensus) studies [11]. For the (time-varying) DeGroot model (1), we have a standard result, which we state after first defining some connectivity conditions for time-varying graphs.

A directed graph is strongly connected if there is a directed path between each pair of its distinct vertices. We say that a finite sequence of directed graphs with the same vertex set is jointly strongly connected if the union444 The union of a finite sequence of directed graphs with the same vertex set is a directed graph with the same vertex set and the arc set which is the union of the arc sets of all directed graphs in the sequence. of the directed graphs in this sequence is strongly connected. We say that an infinite sequence of directed graphs with the same vertex set is repeatedly jointly strongly connected if there exist positive integers and for which each finite sequence , , is jointly strongly connected. Repeatedly jointly strongly connected graphs are equivalent to so-called “-connected” graphs in the consensus literature [36] whose definition is in a slightly different form.

###### Proposition 1

(Theorem 2 in [11]) Suppose that Assumption 2 holds. If the sequence of neighbor graphs is repeatedly jointly strongly connected555 The result still holds if is repeatedly jointly rooted [10]., then all , , in (1) will reach a consensus exponentially fast as for all initial conditions.

It is worth noting that for each time , neighbor graph is the same as the graph of weight matrix .

We will use this result in our analysis of the model (2).

## Iii Susceptibility Functions

In this section, we will consider two specific susceptibility functions, and study the behavior of the corresponding models. We give motivating examples from sociology for the susceptibility functions taking on these specific forms.

### Iii-a Stubborn Positives

We begin with the case where , for all . An agent which has this susceptibility function is called a stubborn positive.

#### Motivation

Note here that for stubborn positive agents, her susceptibility decreases as , and increases as . In other words, the closer the agent is to a “positive opinion” (respectively a “negative opinion”), the more stubborn or unwilling (respectively more open or susceptible) she is to changing her opinion. Our motivating example is a jury panel. The paper [37] conducted extensive surveys of criminal juries after trials were complete. A clear pattern was observed: a juror was more likely to be extremely stubborn when believing the defendant should be acquitted, than when believing the defendant should be convicted. In our context, a juror with is maximally supportive of acquitting the defendant, while a juror with is maximally opposing acquittal (and thus supportive of convicting)666We note here that this is the opinion of the juror, as opposed to the final action taken by the juror. An individual may privately take one opinion and express another due to the social circumstances [38].. It was suggested that this asymmetric stubbornness arose from the fact that a false conviction carried an enormous amount of consequence for defendants in criminal cases, e.g. a prison sentence. In summary, scenarios which involve social networks with stubborn positives can arise in discussions where the outcome for one result has drastically different severity of consequences compared to the opposite result.

#### Analysis

In this case, from (2), each agent updates her opinion by setting

 xi(t+1) = xi(t)+12(1−xi(t))∑j∈Ni(t)wij(t)(xj(t)−xi(t)) = xi(t)+12(1−xi(t))⎛⎝∑j∈Ni(t)wij(t)xj(t)−xi(t)⎞⎠ = 12(1+xi(t))xi(t)+12(1−xi(t))∑j∈Ni(t)wij(t)xj(t).

It follows that

 x(t+1) = 12(I+X(t))x(t)+12(I−X(t))W(t)x(t) = (12(I+X(t))+12(I−X(t))W(t))x(t),

where is the diagonal matrix whose th diagonal entry equals .

The following theorem characterizes the limiting behavior of system (LABEL:positive1).

###### Theorem 1

Suppose that Assumption 2 holds and that the sequence of neighbor graphs is repeatedly jointly strongly connected. If for all , then all , , in (LABEL:positive) will reach a consensus exponentially fast at some value in the interval ; moreover, in this case, the consensus value equals if, and only if, for all . If for at least one , then all , , in (LABEL:positive) will reach a consensus at value .

Proof: From (LABEL:positive1), where

 S(x(t),t)=12(I+X(t))+12(I−X(t))W(t). (8)

From the proof of Lemma 2, is a stochastic matrix for all time .

First suppose that for all . There must exist a positive number such that for all . It follows from Lemma 2 that for all and time . From (8), we obtain the following two inequalities for , the entries of . For each diagonal entry,

 sii(t) = 12(1+xi(t))+12(1−xi(t))wii(t) ≥ 12(1−xi(t))wii(t) ≥ 12αβ,

where the last inequality makes use of Assumption 2 and the fact that we assumed every node in has a self-loop, for all . For each off-diagonal entry,

 sij(t)=12(1−xi(t))wij(t).

Thus, is nonzero if and only if is nonzero (because ), which implies that the graph of has the same edge and vertex set (but with different edge weights) as the graph of , as well as neighbor graph . Moreover, it can be seen that when , it must hold that . From Proposition 5, all , , will reach a consensus exponentially fast. Since for all and time , the consensus value must lie in .

Next we show that all , , in (LABEL:positive) will reach a consensus at value if and only if for all . Suppose that, to the contrary, there exists at least one such that . From (LABEL:positive), since for all , there holds

 xi(t+1) ≥ 12(1+xi(t))xi(t)−12(1−xi(t)).

Suppose that . Then, it follows that

 xi(t+1)>−12(1+xi(t))−12(1−xi(t))=−1,

which implies that if , then for all . Let denote the set of agents whose opinions are greater than at time . From the hypothesis and preceding discussion, is nonempty for all time . Let be the set of agents whose opinions equal at time . If is empty, i.e., for all , then from Lemma 2, the system cannot reach a consensus at . Suppose that is nonempty, i.e., there exists at least one agent whose initial opinion is . Since the sequence of neighbor graphs is repeatedly jointly strongly connected, there must exist a finite time and an agent such that it has a neighbor , i.e., and with . From (LABEL:positive), it can be seen that . Using the same arguments, there exists a finite time such that for all , which contradicts the hypothesis that all , , will reach a consensus at . Therefore, the consensus at will be reached if and only if for all .

Now we consider the case when for at least one . Consider the Lyapunov function

 V(x(t))=1−mini∈[n]xi(t).

From (LABEL:positive), if , then , which implies that if , then for all time . Thus, if for all at some time , then and . Suppose that there exists at least one agent such that at a specific time . In this case, and thus . From Lemma 2, there holds for all and . Let be the set of agents whose opinions are the smallest at time , i.e., for each . Since for at least one , it follows that is nonempty for all . Since the sequence of neighbor graphs is repeatedly jointly strongly connected, there must exist a finite time and an agent such that it has a neighbor , i.e., with (else the agents of would induce a disconnected subgraph, which contradicts the repeatedly jointly strongly connected nature of the neighbor graphs). From (LABEL:positive), it can be seen that . Using the same arguments, there must exist a finite time such that , which implies that . Therefore, will converge to for all .

Theorem 1 implies that system (LABEL:positive1) will reach a consensus for any initial condition. Necessary and sufficient conditions for the two extreme opinions are also given. Specifically, the consensus will be reached at if and only if , and at if and only if does not hold.

###### Remark 2

The discrete-time model with stubborn positives has the same limiting behavior as the continuous-time model considered in [33]. We consider the general time-varying case whereas only the time-invariant case was studied in [33]. Moreover, we establish necessary and sufficient conditions for the two extreme opinions, i.e., and , whereas only consensus to was studied in [33].

###### Remark 3

We now compare the discrete-time model with stubborn positives (LABEL:positive) with the original DeGroot model (1) for the case in which the neighbor graph does not change over time and is a strongly connected graph . Thus, the corresponding weight matrix is also time-invariant. Since is strongly connected, is irreducible. It is well known that in this case, all , , in the DeGroot model will reach a consensus at value , where

is the unique left eigenvector of

associated with eigenvalue

which satisfies ; moreover, . Therefore, as long as there exists at least one agent for which , all the agents will not reach a consensus at value . This is a significant difference from the model (LABEL:positive) in which as long as at least one agent has initial opinion at , all the agents’ opinions will converge to .

### Iii-B Stubborn Neutrals

Now we consider the case where each individual has susceptibility function . We call such an individual a stubborn neutral.

#### Motivation

Observe that for stubborn neutral agents, her susceptibility to being influenced decreases as , and increases as . This means that the closer the individual’s opinion is to “neutral”, i.e. , the more stubborn she becomes. In networks with stubborn neutrals, we consider the neutral opinion as an established, socially normative opinion. For an illustrative example, suppose that the topic was on the level of environmental regulations, e.g. for nuclear power. Then represents individual favoring the maintaining of the status quo, represents favoring increasing regulation, and represents favoring of decreasing regulation. Some literature showed that pressures existed on individuals in a social network to conform with the group norm [39, 38], with deviants being punished [40] or receiving additional pressure to conform [41]. Merei showed in [42] that established traditions heavily influenced the behavior of individuals despite a strong leader attempting to influence change. In the context of our paper, tradition is and the leader is an individual with , and with large, for any individual who listens to the leader. Lastly, institutionalization has been linked to the persistence of cultures and resistance to changing the status quo [43]. In summary, stubborn neutrals may occur in social networks where individuals are reluctant to change from the established norm because of associated risks, or due to institutionalization, or because of pressure to conform.

#### Analysis

In this case, from (2), each agent updates her opinion by setting

 xi(t+1) (9) = xi(t)+xi(t)2∑j∈Ni(t)wij(t)(xj(t)−xi(t)) = xi(t)+xi(t)2⎛⎝∑j∈Ni(t)wij(t)xj(t)−xi(t)⎞⎠ = xi(t)−xi(t)3+xi(t)2∑j∈Ni(t)wij(t)xj(t).

Then, it follows that

 x(t+1) = (I−X(t)2+X(t)2W(t))x(t). (10)

The following theorem characterizes some limiting behavior of system (10).

###### Theorem 2

Suppose that Assumption 2 holds and that the sequence of neighbor graphs is repeatedly jointly strongly connected. If for all , then all , , in (9) will reach a consensus exponentially fast at some value in the interval ; moreover, in this case, the consensus value equals if, and only if, for all . If for all , then all , , in (9) will reach a consensus exponentially fast at some value in the interval ; moreover, in this case, the consensus value equals if, and only if, for all .

Proof: From (10), where

 S(x(t),t)=I−X(t)2+X(t)2W(t). (11)

From the proof of Lemma 2, is a stochastic matrix for all time .

First suppose that for all . There must exist a positive number such that for all . It follows from Lemma 2 that for all and time . From (11), we obtain the following two inequalities for , the entries of . For each diagonal entry,

 sii(t) = 1−xi(t)2+xi(t)2wii(t) ≥ xi(t)2wii(t) ≥ α2β,

where the last inequality makes use of Assumption 2 and the fact that we assumed every node in has a self-loop, for all . For each off-diagonal entry,

 sij(t)=xi(t)2wij(t).

Thus, is nonzero if and only if is nonzero, which implies that the graph of is the same as the graph of , as well as neighbor graph . Moreover, it can be seen that when , it must hold that . From Proposition 5, all , , will reach a consensus exponentially fast. Since for all and time , the consensus value must lie in .

Next we show that all , , in (9) will reach a consensus at value if and only if for all . Suppose that, to the contrary, there exists at least one such that . From (9), since for all implies , there holds

 xi(t+1) ≤ xi(t)−xi(t)3+xi(t)2.

It can be verified that increases as increases when and assumes the value at . Suppose that . Then, it follows that , which implies that if , then for all . Let denote the set of agents whose opinions are less than at time . From the hypothesis and preceding discussion, is nonempty for all time . Let be the set of agents whose opinions equal at time . If is empty, i.e., for all , then from Lemma 2, the system cannot reach a consensus at . Suppose that is nonempty, i.e., there exists at least one agent whose initial opinion is . Since the sequence of neighbor graphs is repeatedly jointly strongly connected, there must exist a finite time and an agent such that it has a neighbor , i.e., and with . From (9), and with replaced by , it can be seen that . Using the same arguments, there exists a finite time such that for all , which contradicts the hypothesis that all , , will reach a consensus at . Therefore, the consensus at will be reached if and only if for all .

Now suppose that for all . There must exist a positive number such that (i.e., ) for all . It follows from Lemma 2 that (i.e., ) for all and time . From (11), we obtain the following two inequalities for , the entries of . For each diagonal entry,

 sii(t) = 1−xi(t)2+xi(t)2wii(t) ≥ xi(t)2wii(t) ≥ α2β,

where the last inequality makes use of Assumption 2. For each off-diagonal entry,

 sij(t)=xi(t)2wij(t).

Thus, is nonzero if and only if is nonzero, which implies that the graph of is the same as the graph of , as well as neighbor graph . Moreover, it can be seen that when , it must hold that . From Proposition 5, all , , will reach a consensus exponentially fast. Since for all and time , the consensus value must lie in .

Next we show that all , , in (9) will reach a consensus at value if and only if for all . Suppose that, to the contrary, there exists at least one such that . From (9), since for all implies , there holds

 xi(t+1) ≥ xi(t)−xi(t)3−xi(t)2.

It can be verified that increases as increases when . Suppose that . Then, it follows that , which implies that if , then for all . Let denote the set of agents whose opinions are greater than at time . From the hypothesis and preceding discussion, is nonempty for all time . Let be the set of agents whose opinions equal at time . If is empty, i.e., for all , then from Lemma 2, the system cannot reach a consensus at . Suppose that is nonempty, i.e., there exists at least one agent whose initial opinion is . Since the sequence of neighbor graphs is repeatedly jointly strongly connected, there must exist a finite time and an agent such that it has a neighbor , i.e., and with . From (9), and with replaced by , it can be seen that . Using the same arguments, there exists a finite time such that for all , which contradicts the hypothesis that all , , will reach a consensus at . Therefore, the consensus at will be reached if and only if for all .

###### Remark 4

It should be noted that Theorem 2 does not consider the case where there exist such that and . It has been shown in [33] that for this case, the continuous-time model has . However, this is not always the case for the discrete-time model with stubborn neutrals, as we will show shortly, since in discrete-time, the opinion of an agent can jump from to , but this is not possible in the continuous-time case. Although we will only characterize partial limiting behavior of the discrete-time model for this case (see Theorem 3), we consider the general time-varying graph case whereas only the time-invariant graph case was studied in [33]. Moreover, we establish necessary and sufficient conditions for the two extreme opinions, i.e., and , which were not provided in [33].

The following example shows that the discrete-time model with stubborn neutrals (9) has different limiting behaviors from the continuous-time model considered in [33] (cf. Theorem 5 in [33]).

###### Example 1

Suppose that there are 4 agents labeled 1 through 4. The neighbor graph is a complete graph and all the weights equal . Suppose that the initial opinions are and for . Thus,

 W=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1414