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

TL;DR

A novel opinion dynamics model is proposed to study how a discrepancy can arise in general social networks of interpersonal influence, including the establishing of conditions that ensure every individual in the network has such a discrepancy.

Abstract

In many social situations, a discrepancy arises between an individual's private and expressed opinions on a given topic. Motivated by Solomon Asch's seminal experiments on social conformity and other related socio-psychological works, we propose a novel opinion dynamics model to study how such a discrepancy can arise in general social networks of interpersonal influence. Each individual in the network has both a private and an expressed opinion: an individual's private opinion evolves under social influence from the expressed opinions of the individual's neighbours, while the individual determines his or her expressed opinion under a pressure to conform to the average expressed opinion of his or her neighbours, termed the local public opinion. General conditions on the network that guarantee exponentially fast convergence of the opinions to a limit are obtained. Further analysis of the limit yields several semi-quantitative conclusions, which have insightful social interpretations, including the establishing of conditions that ensure every individual in the network has such a discrepancy. Last, we show the generality and validity of the model by using it to explain and predict the results of Solomon Asch's seminal experiments.

Figures (10)

• Figure 1: The discussion process. Each individual $i$, at time step $t$, expresses opinion $\hat{y}_i(t)$ and learns of others' expressed opinions $\hat{y}_j(t), j\neq i$. Next, the privately held opinion $y_i(t+1)$ evolves according to Eq. (\ref{['eq:private_op']}). After this, individual $i$ then determines the new $\hat{y}_i(t+1)$ to be expressed in the next round of discussion, according to Eq. (\ref{['eq:public_op']}).
• Figure 2: Temporal evolution of opinions for 18 individuals in an influence network. The green and dotted blue lines represent the expressed and private opinions of the individuals, respectively.
• Figure 3: Temporal evolution of opinions for 18 individuals in an influence network. The green and dotted blue lines represent the expressed and private opinions of the individuals, respectively. The lack of stubbornness, $\lambda_i = 1,\forall\,i$, means that all opinions reach a consensus.
• Figure 4: Example of the Asch experiment. The individuals openly discuss their individual beliefs as to which one of $A, B, C$ has the same length as the green line. Clearly $A$ is equal in length to the green line. The test individual is the red node. The confederates (seven blue nodes) unanimously express belief in the same wrong answer, e.g. $B$.
• Figure 5: The function $f(\lambda_1)$ and $1-\lambda_1$ plotted against $\lambda_1$. The analytical calculations show that $y_1^* = f(\lambda_1)$, and thus the red line represents individual $1$'s final private belief as a function of his susceptibility to influence.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Theorem 1: Exponential Convergence
• Corollary 1: Consensus of Opinions
• Theorem 2
• Remark 3
• Corollary 2
• Corollary 3
• Lemma 1
• Lemma 2