Dynamics Analysis for the Random Homogeneous Biased Assimilation Model

Abstract

This paper studies the evolution of opinions over random social networks subject to individual biases. An agent reviews the opinion of a randomly selected one and then updates its opinion under homogeneous biased assimilation. This study investigates the impact of biased assimilation on random opinion networks, which is different from the previous studies on fixed network structures. If the bias parameters are static, it is proven that the event in which all agents converge to extreme opinions happens almost surely. Next, the opinion polarization event is proved to be a probability one event. While if the bias parameters are dynamic, the opinion evolution is proven to depend on early finite time slots for the dynamical individual bias parameter functions independent of the biased parameter values after the time threshold. Numerical simulations further show that opinion evolution depends on early finite time slots for some nonlinear dynamical individual bias parameter functions.

1. Introduction

In our society, opinion formation among individuals and induced dynamics has been extensively studied and debated in the academic literature, including minority opinion dissemination, collective decision making, polarization and fluctuation, fashion emergence, etc. With the extensive development of network communication, such as WeChat groups, Facebook interest groups, Twitter discussion threads, etc., online interactions are becoming increasingly important in many aspects ranging from political decisions to marketing strategies [1,2]. In this setting, it is important to study the way individuals in an online social network update their attitudes.

For the traditional network topologies, the standard DeGroot model employs the discrete-time multi-agent system to simulate how public opinions may influence each other, in which an individual’s opinion toward a particular topic is often represented by a real value in the interval $[0,1]$. It was proven that as long as the underlying graph is connected, all opinions converge to a common value known as the consensus state [3]. Generalizations of this model to continuous-time dynamics and time-varying network structures have been extensively studied in the literature, e.g., [4,5,6,7]. Such convergence to consensus still holds for some deterministically switching networks, e.g., [4,5,6]. However, analyzing the social groups’ characteristics is an important way to understand the rule of the social system. Therefore, it is increasingly important to understand how macro-characteristics emerge from the micro-individual psychological and interactive effects. For example, individuals always consider the initial point of view in the process of updating [8]; according to the selective exclusion principle in social psychology, individuals in the group would choose to communicate with people with similar opinions [9]; the research objects of the stochastic DeGroot model contain stubborn agents who never modify their opinions, such as leaders and rumor disseminators [10]; agents may tend to repulse individual opinions that differ greatly from their own based on self-bias [11]. Beyond consensus, social dynamics can exhibit complex behaviors such as polarization, clustering, and opinion fluctuation [10,11,12].

Considering the wide existence of biases among individual opinion evolution, especially online comments, WeChat discussions, etc., individual biases were modeled as nonlinear weights on self-opinion and local group opinions, based on which clustering to extreme opinions in the fixed network topologies was revealed recently [13,14]. Studies from social psychology show that people are more likely to accept confirming evidence given by someone similar to themselves [15,16]. A convincing model for this biased opinion assimilation was proposed in [15] as a natural interpretation of confirmation bias. Ref. [14] provided a systemic analysis of a social opinion dynamical model with bias assimilation on fixed network topologies. However, the polarization phenomena were only shown for special fixed networks [13], and rare results on polarization are confirmed for general fixed networks [13,14]. In fact, for social systems designed expressly to facilitate collective decision making regarding complex social issues, the occurrence of polarization would not only depend on the network structures (such as the two-island network shown in [13]) but also on some random accidental factors [17,18].

Previous studies have dealt with convergence and stability analysis of such systems for some fixed network structures, and we focus on how individual opinions evolve for the random network structure. Opinion exchanges among Internet users might promote opinion consensus, polarization, and fluctuations with different psychological effects behind social interactions. This creates some new problems: Assume all internet users have the psychological effect of biased assimilation; then, how do biased individuals’ opinions evolve among the random online networks? Do the consensus phenomena always happen similarly to the fixed network topologies [14]? Through this paper, we will answer these questions partly and understand why individual-level polarization would happen, contrary to conventional wisdom, regarding the public opinions of online platforms.

The contribution is that we propose and study opinion dynamics over the random social network with homogeneous bias parameters. Particularly, we focus on how individual biases and randomness affect the opinion limit states. Firstly, we investigate the random bias-induced collective nonlinear network dynamics and provide conditions under which all node states converge to 0 or 1. Next, we prove that opinion polarization happens with a positive probability with homogeneous bias parameters. Finally, we prove that all node states also converge to extremal opinions even if the bias parameters are dynamic, and we show that the opinion evolution only depends on the value ranges of bias parameters in certain early time intervals. Simulations on the time interval thresholds are conducted for some dynamical bias parameters.

The remainder of the paper is organized as follows. In Section 2, we present the social network model for our study and introduce our problems of interest. Section 3 presents our main results on the fixed biased parameters. Then, Section 4 presents our main results on the dynamic biased parameters and provides some numerical simulations of periodical functions of the biased parameters. Finally, Section 5 concludes the paper with a few remarks on potential future directions.

2. Model Formulation

Our opinion formulation process unfolds over the random social network represented by random weighted directed graphs $G\left(t\right)=(V,E(t),W)$. Time is slotted at $t=0,1,2,\dots$. $V=\{1,2,\dots ,n\}$.

At each time t, each node $i\in V$ randomly selects one node $r_i\left(t\right)\in V$ as its neighbor from the network node set V independent of other nodes’ selections. This results in a random set of neighbors (clusters), which are denoted by $\left\{r_i\left(t\right)\right\}$, for $i\in V$ and $t=0,1,\dots$. Note that $(i,j)\in E\left(t\right)$ if and only if the agent selected by agent i is $r_i\left(t\right)=j$, representing the other node j that influences i.

For $W={\left[w_{ij}\right]}_{n\times n}$, $0<w_{ij}<1$ represents the influence weight between two nodes i and j. Without loss of generality, the node i’s self-confidence is represented by $w_{ii}=1$.

Each node i holds an opinion $x_i\left(t\right)\in [0,1]$ at time t. Let $b_i$ be a positive number associated with node i as a bias parameter. The evolution of the $x_i\left(t\right)$, $i\in V$ is described by:

$$\begin{array}{c}\hfill x_i(t+1)=\frac{x_i\left(t\right)+w_{i,r_i\left(t\right)}{\left(x_i\left(t\right)\right)}^{b_i}{x}_{r_i\left(t\right)}\left(t\right)}{1+w_{i,r_i\left(t\right)}\left[{\left(x_i\left(t\right)\right)}^{b_i}{x}_{r_i\left(t\right)}\left(t\right)+{(1-x_i\left(t\right))}^{b_i}(1-x_{r_i\left(t\right)}\left(t\right))\right]}\end{array}$$

(1)

This model reflects the social psychology phenomenon named biased assimilation. For any node i, $w_{i,r_i\left(t\right)}$ is the inspired influence weight that the node i is influenced by the node $r_i\left(t\right)\in V$. For the right side of the model (1), the factor ${\left(x_i\left(t\right)\right)}^{b_i}$ weighting $x_{r_i\left(t\right)}\left(t\right)$ means the biased manner degree on its previous “relevant disconfirming empirical” opinion $x_i\left(t\right)$, while the factor ${(1-x_i\left(t\right))}^{b_i}$ weighting $1-x_{r_i\left(t\right)}\left(t\right)$ means the biased manner degree on its previous “relevant disconfirming empirical” opinion $1-x_i\left(t\right)$.

Note that ${(1-x_i\left(t\right))}^{b_i}(1-x_{r_i\left(t\right)}\left(t\right))\ge 0$ if $x_i\left(t\right)\in [0,1]$; thus, the denominator is not smaller than the numerator. Thus, $x_i\left(0\right)\in [0,1]$ for all $i\in V$ guarantees that $x_i\left(t\right)\in [0,1]$ for all $t\ge 0$ and $i\in V$. In addition, 0 and 1 represent the extreme opinion of opposing or supposing on the given topic, respectively. Based on the results of [14,15], opinion evolution depends on whether all $b_i>1$ or all $b_i\in (0,1)$, except the network structure and other parameter constraints. Therefore, it is necessary to induce and classify the bias parameters as follows.

Definition 1. 

If all opinion bias parameter $b_i>1$, $i\in V$ or all opinion bias parameter $b_i\in (0,1)$, $i\in V$, then the biased assimilation model (1) is homogeneous. Correspondingly, if $b_i>1$ for $i\in V_1$ where $V_1$ is nonempty, $b_i\in (0,1)$ for $i\in V_2$, ($V_1\cup V_2)\subset V$, then the biased assimilation model (1) is heterogeneous.

If there exist at least two agents $i,j$ such that $b_i\in (0,1)$ and $b_j>1$, then we say the biased assimilation model (1) is heterogeneous.

Denote $R\left(t\right)=(r_1\left(t\right),r_2\left(t\right),\dots ,r_n\left(t\right))$ as the selection vector for any $t\ge 0$. We impose the following assumptions for the selection rule and bias parameters.

Assumption 1. 

$\left\{r_i\left(t\right)\right\}$ are independent with each other for any $i\in V$ and $t\ge 0$, and $r_i\left(t\right)$ is any discrete distribution on $\{1,2,\dots ,n\}$ where $r_i\left(t\right)=j$ is a positive probability event for any $i,j\in V$ and $t\in \mathbb{N}$.

Assumption 2. 

The bias assimilation parameter $\left\{b_i\right\}$ satisfies $b_i>0$ for any $i\in V$.

3. Fixed Bias Parameters

In this section, we investigate the opinion limit analysis for the model (1) where the bias parameters are fixed and homogeneous for $t\ge 0$.

3.1. Probability Space

Let $\left\{R\right(t\left)\right\}$ be a non-repetitive selection vector set, that is, any $R\left(t\right)$ for $t\ge 0$ is a permutations sort of $(1,2,\dots ,n)$. Thus, for a given $t\ge 0$, there is n different vectors for $R\left(t\right)$. Denote $\mathsf{\Omega }$ as the set composed by all n different permutations sort of $(1,2,\dots ,n)$. By Assumption 1, we can construct a probability space $(V,\mathcal{F},P)$. Furthermore, because the selections are independent among different $t\ge 0$, thus, the probability space reflecting any opinion selection trajectories is independent among $t\in \mathbb{N}$.

3.2. Some Lemmas

In this subsection, we introduce some lemmas for the model (1).

Lemma 1. 

For the probability measure P, $\underset{t\to \infty }{lim}{x}_i\left(t\right)=x_i^{*}$ a.s. is equivalent to $x_i\left(t\right)$, which almost uniformly converges to $x_i^{*}$. That is to say, $\forall \epsilon >0$, $\underset{t\to \infty }{lim}P\left({\cup }_{k=t}^{+\infty }\left(\right|x_i\left(t\right)-x_i^{*}|\ge \epsilon )\right)=0$ for any $i\in V$.

Lemma 2. 

If $x_i\left(0\right)\in (0,\frac{1}{2})$ for all $i\in V$, then $x_i\left(t\right)\in (0,\frac{1}{2})$ for any $t\in \mathbb{N}$. Similarly, if $x_i\left(0\right)\in (\frac{1}{2},1)$ for all $i\in V$, then $x_i\left(t\right)\in (\frac{1}{2},1)$ for any $t\in \mathbb{N}$.

Proof of Lemma 2 is listed in Appendix A. Lemma 2 explains some realistic social phenomena. If all people own negative opinions (<0.5), then all of them will always keep the negative opinions. A similar phenomenon appears when they all have positive opinions (>0.5). Different from the results in [14], ${max}_{i\in V}\left\{x_i\left(t\right)\right\}$ is not monotonic due to the randomness selection of $\left\{r_i\left(t\right)\right\}$. Therefore, some of the following main results would be different from the ones on the fixed network topologies. To analyze the nonlinearity of the model (1), we provide the following lemma.

Lemma 3. 

For the function $f(x,y)=\frac{x+w{x}^{b}y}{1+w(x^{b}y+{(1-x)}^b(1-y))}$ where $w>0$, $b>0$ and $x,y\in [0,1]$, we have:

(i) 

$f(0,y)\equiv 0$ for any $y\in [0,1]$;

(ii) 

$f(1,y)\equiv 1$ for any $y\in [0,1]$;

(iii) 

$f(x,0)=\frac{x}{1+w{(1-x)}^b}$ and it is a lower convex function;

(iv) 

$f(x,1)=\frac{x+w{x}^b}{1+w{x}^b}$ and it is a upper convex function;

(v) 

$f(1-x,x)+f(x,1-x)=f(x,y)+f(1-x,1-y)=1$.

This lemma can be easily obtained and the proof is omitted.

3.3. Results on Homogeneous Bias Parameters

In this subsection, we study the limits of the model (1) where Assumptions 1 and 2 are satisfied. Here, all bias parameters are homogeneous. Specially, if $b_i=0$ for any $i\in V$, then $x_i(t+1)=\frac{1}{1+w_{ij}}{x}_i\left(t\right)+\frac{w_{ij}}{1+w_{ij}}{x}_j\left(t\right)$. By the standard DeGroot model, all agent opinions reach a consensus. If $b_i=1$ for any $i\in V$, we obtain the following lemma.

Lemma 4. 

For the model (1),

$$P\{\underset{t\to \infty }{lim}{x}_i\left(t\right)=1\}=1$$

for any $i\in V$, if $b_i=1$ and $x_i\left(0\right)\in (\frac{1}{2},1]$ for any $i\in V$;

$$P\{\underset{t\to \infty }{lim}{x}_i\left(t\right)=0\}=1$$

for any $i\in V$, if $b_i=1$ and $x_i\left(0\right)\in [0,\frac{1}{2})$ for any $i\in V$.

The proof of Lemma 4 is shown in Appendix B. In the following, we generalize the results of Lemma 4 to any homogeneous bias parameters $b_i>0$.

Theorem 1. 

For the model (1), if $\left\{b_i\right\}$ are homogeneous, then

(i) 

$P\{\underset{t\to \infty }{lim}{x}_i\left(t\right)=1\}=1$ for any $i\in V$, if $x_i\left(0\right)\in (\frac{1}{2},1]$ for any $i\in V$;

(ii) 

$P\{\underset{t\to \infty }{lim}{x}_i\left(t\right)=0\}=1$ for any $i\in V$, if $x_i\left(0\right)\in [0,\frac{1}{2})$ for any $i\in V$.

The proof of Theorem 1 is shown in Appendix C. Although the dynamics of model (1) is different from the one in [14], the result of Theorem 1 is similar to Theorem 4 of [14]. Therefore, we can weaken the condition of Theorem 1 and distinguish the results of different network structures, and we provide the following theorem.

Theorem 2. 

For the homogeneous parameters $\left\{b_i\right\}$, if $x_i\left(t\right)\in (\frac{1}{2},1)$ infinitely often (i.o.) for certain $i\in V$, then $\underset{t\to \infty }{lim}{x}_i\left(t\right)=1$ a.s. Similarly, if $x_i\left(t\right)\in (0,\frac{1}{2})$ i.o. for certain $i\in V$, then $\underset{t\to \infty }{lim}{x}_i\left(t\right)=0$ a.s.

The proof of Theorem 2 is shown in Appendix D. Theorem 2 illustrates that the social group that always owns negative opinions (<$0.5$) will finally reach the extremely negative attitude (0) after sufficient communication. Similarly, the social group that always owns positive opinions (>$0.5$) will finally reach an extremely positive attitude (1). These phenomena can usually be found in online interest groups.

Based on Theorem 2, we can show that opinion polarization happens with a positive probability, which is much different from the results on fixed social networks [13,14]. Denote

$$E_{polarization}=\{\exists V_1,V_2\subset V,V_1\cup V_2=Vs.t.:\underset{t\to \infty }{lim}{x}_i\left(t\right)=0,i\in V_1;\underset{t\to \infty }{lim}{x}_i\left(t\right)=1,i\in V_2\}.$$

Furthermore, we denote

$$E_{con}=\{\underset{t\to \infty }{lim}{x}_i\left(t\right)=0\mathrm{or}1,\forall i\in V\}.$$

According to Theorem 1, $E_{polarization}$ is a zero probability event if $x_i\left(0\right)\in [0,\frac{1}{2})$ or $x_i\left(0\right)\in (\frac{1}{2},1]$. Generally, we obtain the following theorem.

Theorem 3. 

For the model (1), $P\left\{E_{con}\right\}=1$ and $P\left\{E_{polarization}\right\}>0$ if $\left\{b_i\right\}$ are homogeneous.

The proof of Theorem 3 is shown in Appendix E. The result of Theorem 3 is different from anyone in the fixed network topologies [13,14]. The proof of Theorem 3 shows that opinion polarization depends on opinion selection sequence $\{r_i\left(t\right),i\in V,t\in \mathbb{N}\}$, not only the initial opinions $\left\{x_i\left(0\right)\right\}$, which shows that some accidental factors could also affect the opinion evolution for the model (1).

Theorem 3 can explain the social phenomena on online social networks. If there is a group of people who process their information in a biased manner, then opinion polarization happens with a positive probability. For the fixed networks, Ref. [13] shows that opinion polarization happens on the two-island network with strict parameter conditions. Theorem 3 extends it into the case of random selection rules.

According to Theorem 3, we obtain that all opinions will converge to 0 or 1, a.s., for different initial values and biased parameter $b_i$. Figure 1 shows how opinions $\left\{x_i\left(t\right)\right\}$ change for different initial values and selection processes when the agent number $n=20$, the termination time is $T=200$ and the bias parameters $b_i=2.2$ for any $i\in V$.

4. Dynamic Biased Parameters

In this section, we will analyze the generalized form of the model (1) where the biased assimilation parameters $\left\{b_i\right\}$ are functions of time t, that is,

$$\begin{array}{c}\hfill x_i(t+1)=\frac{x_i\left(t\right)+w_{i,r_i\left(t\right)}{\left(x_i\left(t\right)\right)}^{b_i\left(t\right)}{x}_{r_i\left(t\right)}\left(t\right)}{1+w_{i,r_i\left(t\right)}\left[{\left(x_i\left(t\right)\right)}^{b_i\left(t\right)}{x}_{r_i\left(t\right)}\left(t\right)+{(1-x_i\left(t\right))}^{b_i\left(t\right)}(1-x_{r_i\left(t\right)}\left(t\right))\right]}.\end{array}$$

(2)

4.1. Results on Dynamic Bias Parameters

For the function $b_i\left(t\right)$, we assume that $b_i\left(t\right)>0$ for any $i\in V$ and $t\ge 0$. In this section, we call $\left\{b_i\left(t\right)\right\}$ homogeneous if $b_i\left(t\right)>1$ for any $i\in V$ and $t\in \mathbb{N}$ simultaneously, or $b_i\left(t\right)\in (0,1)$ for any $i\in V$ and $t\in \mathbb{N}$ simultaneously. Obviously, if $b_i\left(t\right)\in (0,1)$ for all $i=1,2,\dots ,n$ and $t\ge 0$, then opinions will almost surely converge to 1. While if $b_i\left(t\right)>1$ for all $i=1,2,\dots ,n$ and $t\ge 0$, then opinions will almost surely converge to 0. Generally, we obtain the following lemma.

Lemma 5. 

For the model (2), $P\left\{E_{con}\right\}=1$ and $P\left\{E_{polarization}\right\}>0$ if $\left\{b_i\left(t\right)\right\}$ are homogeneous.

Note that in the proof of Theorem 3, the analysis on $b_i$ only depends on the current period t. Thus, the proof of Theorem 3 can be naturally extended to the case of Lemma 5; thus, the proof of Lemma 5 is omitted. Different from the previous results, we propose that agent opinion evolution on the model (2) only depends on the early time slots.

Theorem 4. 

For the model (2), there exists a finite time threshold $T_1^{*}>0$, if $b_i\left(t\right)>1$, $x_i\left(0\right)\in (0,\frac{1}{2})$ for any $i\in V$ and $t\in (0,T_1^{*})$, then

$$\underset{t\to \infty }{lim}{x}_i\left(t\right)=0 a.s.\mathit{for}\mathit{any}i\in V;$$

Similarly, there exists a finite time threshold $T_2^{*}>0$, if $b_i\left(t\right)\in (0,1)$, $x_i\left(0\right)\in (\frac{1}{2},1)$ for any $i\in V$ and $t\in (0,T_2^{*})$, then

$$\underset{t\to \infty }{lim}{x}_i\left(t\right)=1 a.s.\mathit{for}\mathit{any}i\in V.$$

This theorem shows that there always exists a finite time threshold $T^{*}$, such that the opinion evolution only depends on the time interval $[0,T^{*}]$. However, it is difficult to provide a mathematical expression of $T^{*}$. In the following, we demonstrate the threshold $T^{*}$ for some periodic functions and monotone functions.

4.2. Simulations on Dynamic Bias Parameters

In this subsection, simulations that explore the threshold $T^{*}$ of the model (2) are presented based on MATLAB software. Set agent number $n=20$ and the termination time $T=200$. The initial opinions are equally distributed on the interval $[0,1]$ and the simulation number is 200. We use ${\widehat{T}}_1^{*}={min}_{t\in \mathbb{N}}\{t:x_i\left(t\right)\in (0,\frac{1}{2})\forall i\in V\}$ (or ${\widehat{T}}_2^{*}={min}_{t\in \mathbb{N}}\{t:x_i\left(t\right)\in (\frac{1}{2},1)\forall i\in V\}$) to substitute for $T_1^{*}$ (or $T_2^{*}$) of Theorem 4. Here, the estimated threshold $T^{*}$ is a weighted combination of ${\widehat{T}}_1^{*}$ and ${\widehat{T}}_1^{*}$ where the weight parameters are the frequencies of the events that opinions converge to 0 or 1, respectively. Specially, we set all $b_i\left(t\right)$ to be the same for $i\in V$.

(1) Figure 2 shows that the average estimated threshold $T^{*}$ oscillatory decreases as h changes from 0 to 1.1, where the biased assimilation functions are $b_i\left(t\right)=0.8mod(x,10+30h)/(10+30h)+0.8$ and $b_i\left(t\right)=0.8exp(-0.1hx)+0.4$, respectively.

In fact, as h increases from 0 to 1, the probabilities of opinions converge to 1 and decrease to 0. By Theorem 4, there exists a time threshold $T_1^{*}>0$, such that ${lim}_{t\to \infty }{x}_i\left(t\right)=0 \mathrm{a}.\mathrm{s}.$ for any $i\in V$ if $b_i\left(t\right)>1$, $x_i\left(0\right)\in (0,\frac{1}{2})$ for any $i\in V$ and $t\in (0,T_1^{*})$.

The following figures (Figure 3) show how the first range where $b_i\left(t\right)>1$ enlarges when h increases from 0 to 1. According to Lemma 5, this is corresponding to Figure 2 where the frequency of opinions converging to 0 increases.

Similarly, Figure 4 shows that the frequency of opinions converging to 1 also oscillatory decreases as h changes from 0 to 1, where the biased assimilation function $b_i\left(t\right)=0.8sin\left(\frac{t}{4+10h}\right)+0.8$ and $b_i\left(t\right)=0.8cos\left(\frac{t}{4+10h}\right)+0.8$. The analysis is similar to Figure 2.

(2) Probability of consensus for exponential functions of $b_i\left(t\right)$: In this part, the probabilities of opinion consensus to 0 or 1 for the model (1) are demonstrated.

If $b_i\left(t\right)=0.8e^{-0.1t}+0.4+h$ where h changes from 0 to 1, then $\{b_i\left(t\right)>1\}=[0,10ln\left(\frac{8}{6-10h}\right)]$. When h increases from 0 to 1, $10ln\frac{8}{6-10h}$ increases. Thus, the range $[0,T_2^{*}]$ where $b_i\left(t\right)>1$ enlarges, and the frequency of opinions converging to 0 increases. This is corresponding to Figure 5. The similar analysis can be obtained for $b_i\left(t\right)=0.8e^{-0.1ht}+0.4$.

(3) Probability of consensus for index periodic functions of $b_i\left(t\right)$: In this part, the probabilities of opinion consensus to 0 or 1 for the model (1) are demonstrated.

According to Figure 6, when h changes from 0 to 1, the first ranges where $0<0.8mod(x,10+30h)/(10+30h)+0.8<1$ and $0.8mod(5+15h+x,10+30h)/(10+30h)+0.8>1$ increase. Figure 7 reflects the probability where opinions converging to 1 oscillatory increase and oscillatory decrease, respectively.

5. Conclusions

We have systematically analyzed online social opinion dynamics subject to individually biased assimilation. With initial opinions being independently and identically distributed, at each time step, peers review the selected opinions of a randomly selected clique with biased assimilation. The contributions are that a series of results on the asymptotic behaviors of the social opinions at a system level were provided, focusing on polarization and consensus. The results show that convergence happens almost surely and polarization happens with a positive probability. Future works include studying the limit states if $\{b_i,i\in V\}$ are heterogeneous, extending the results to general network structures, and validating the established opinion formations with real-world social network data.