Consensus Measurement in Complex Space: Statistical Modeling and Empirical Validation on Social Network Data

Measurement of public opinion and consensus faces unprecedented challenges in the digital age, where attitudes are increasingly expressed as complex, multidimensional constructs rather than simple binary positions. Traditional analytical approaches based on sentiment analysis or unidimensional variance measures prove inadequate for capturing the nuanced relationships between different dimensions of opinion in these complex spaces. This study proposes a novel statistical framework for measuring consensus based on principles from multivariate analysis of variance (MANOVA) and distance metrics in multidimensional Euclidean space. The methodology applies this model to a comprehensive dataset of Twitter discourse concerning COVID-19 vaccination, utilizing both topic modeling and transformer-based embeddings to represent opinions in high-dimensional spaces. Our results demonstrate the model's effectiveness in quantifying consensus on a continuous scale from 0 to 1, revealing that online discussions frequently maintain multipolar opinion structures rather than trending toward either complete consensus or polarization. Furthermore, we identify strong correlations between network properties—particularly community structure and density—and internal consensus levels within subcommunities. The proposed Consensus Index provides a powerful quantitative tool for analyzing opinion dynamics in complex spaces, with significant applications in computational social science, social media marketing, and polarization research.