Opinion Dynamics: A Comprehensive Overview

TL;DR

The paper provides a unified framework for two broad classes of opinion-diffusion models (discrete and continuous), synthesizing models from computer science, mathematics, sociology, and physics under common notation. It surveys convergence properties, viral marketing formulations, and how user heterogeneity shapes diffusion, highlighting key results on consensus versus polarization and convergence times. It collects algorithmic and probabilistic findings across diffusion models (LT, IC, SIR/SIS/SI, Ising, Voter, Sznajd, GK/HK/DW, etc.) and discusses practical implications for campaigns, interventions, and online environments. It also points to future directions, including neural message passing as a bridge to graph neural networks and cross-disciplinary collaboration to advance theoretical and applied understanding of opinion dynamics.

Abstract

Opinion dynamics, the evolution of individuals through social interactions, is an important area of research with applications ranging from politics to marketing. Due to its interdisciplinary relevance, studies of opinion dynamics remain fragmented across computer science, mathematics, the social sciences, and physics, and often lack shared frameworks. This survey bridges these gaps by reviewing well-known models of opinion dynamics within a unified framework and categorizing them into distinct classes based on their properties. Furthermore, the key findings on these models are covered in three parts: convergence properties, viral marketing, and user characteristics. We first analyze the final configuration (consensus vs polarized) and convergence time for each model. We then review the main algorithmic, complexity, and combinatorial results in the context of viral marketing. Finally, we explore how node characteristics, such as stubbornness, activeness, or neutrality, shape diffusion outcomes. By unifying terminology, methods, and challenges across disciplines, this paper aims to foster cross-disciplinary collaboration and accelerate progress in understanding and harnessing opinion dynamics.

Figures (13)

• Figure 1: Voter Model: Initially, node $i$ is $+1$ (blue). Node $i+1$ will flip to $+1$ anyway as it adopts node $i$'s state. Node $i+3$ will be $+1$ with probability $w_{i,i+3}$.
• Figure 2: Majority Model: Each node updates its opinion based on the majority of its in-neighbors. Here, node $i+1$ changes from $-1$ (red) to $+1$ (blue).
• Figure 3: Random Majority Model: Each node updates based on the majority of its in-neighbors, with ties resolved randomly. Here, $i+1$ updates to blue ($+1$) from $i$ regardless. Two possible outcomes (with equal probability) are shown: $i+3$ remains red ($-1$) or changes to blue ($+1$ ).
• Figure 4: Sznajd Model: Node $i$ disagrees with neighbor $i+1$, causing $i-1$ to flip its opinion. Node $i+2$ already disagrees with $i+1$ and remains unchanged.
• Figure 5: PUSH Model: Node $i$ is initially blue ($+1$) and randomly pushes its opinion to one of its out-neighbors, node $i+1$ and $i+3$. Two possible outcomes (with equal probability) are shown after one round.

Theorems & Definitions (25)

• Definition : Weighted Graph
• Definition : Adjacency Matrix
• Definition : Undirected Graph
• Definition : In-Neighborhood and Out-Neighborhood
• Definition : In-Degree and Out-Degree
• Definition : Degree Matrix
• Definition : Unweighted or Simple Graph
• Definition 1: Opinion Space
• Definition 2: Update Function
• Definition 3: Synchronous/Asynchronous Update