Convergence rate of opinion dynamics with complex interaction types

TL;DR

This paper develops a comprehensive theory using random matrix and low-rank perturbation theories to analyze how complex interaction types influence the convergence rate of opinions in social networks.

Contribution

It introduces a unified framework to quantify the impact of various interaction types on opinion convergence rates and their dependence on network parameters.

Findings

Convergence rate depends on interaction type, population size, network connectivity, and self-confidence.

Different interaction scenarios have opposite effects on convergence speed.

Optimal proportions of interaction types can be identified for faster consensus.

Abstract

The convergence rate is a crucial issue in opinion dynamics, which characterizes how quickly opinions reach a consensus and tells when the collective behavior can be formed. However, the key factors that determine the convergence rate of opinions are elusive, especially when individuals interact with complex interaction types such as friend/foe, ally/adversary, or trust/mistrust. In this paper, using random matrix theory and low-rank perturbation theory, we present a new body of theory to comprehensively study the convergence rate of opinion dynamics. First, we divide the complex interaction types into five typical scenarios: mutual trust $(+/+)$, mutual mistrust $(-/-)$, trust$/$mistrust $(+/-)$, unilateral trust $(+/0)$, and unilateral mistrust $(-/0)$. For diverse interaction types, we derive the mathematical expression of the convergence rate, and further establish the direct connection between the convergence rate and population size, the density of interactions (network connectivity), and individuals' self-confidence level. Second, taking advantage of these connections, we prove that for the $(+/+)$, $(+/-)$, $(+/0)$, and random mixture of different interaction types, the convergence rate is proportional to the population size and network connectivity, while it is inversely proportional to the individuals' self-confidence level. However, for the $(-/-)$ and $(-/0)$ scenarios, we draw the exact opposite conclusions. Third, for the $(+/+,-/-)$ and $(-/-,-/0)$ scenarios, we derive the optimal proportion of different interaction types to ensure the fast convergence of opinions. Finally, simulation examples are provided to illustrate the effectiveness and robustness of our theoretical findings.