Multipolar social systems: Measuring polarization beyond dichotomous contexts

TL;DR

We present a multidimensional framework for measuring polarization in multipolar social systems by placing $n$ poles at the vertices of a regular $n-1$-simplex and inferring multidimensional opinions on a directed interaction network via a Friedkin–Johnsen–style model. Polarization is quantified with a covariance-based metric, using the total variation $TV = \mathrm{tr}(\mathrm{Cov}[\vec{Y},\vec{Y}])$ (normalized to $1$ at $u=1$) and PCA to identify pole-constraint via explained variance, revealing the latent ideological structure. The method is validated on real Twitter data from quadripolar 2015 and pentapolar 2019 Spanish elections, uncovering left–right axes and secondary dimensions, and showing that ideology alone does not fully explain polarization. The framework supports de-escalation, disinformation assessment, and adaptive party-system analysis by providing flexible, data-driven insight into high-dimensional opinion landscapes.

Abstract

Social polarization is a growing concern worldwide, as it strains social relations, erodes trust in institutions, and thus threatens democratic societies. Academic efforts to understand this phenomenon have traditionally approached it from a one-dimensional perspective, focusing on bipolar or dichotomous systems. However, political conflicts often involve not only two, but multiple potentially dissenting factions. The most representative examples are multi-party democracies, where the multilateral tensions among different parties often lead to gridlock and uncertainty. Despite the prevalence of these multipolar systems, there is still a lack of suitable analytical tools to study their intricate polarization patterns. In this work, we develop an analytical framework consisting of an inherently multipolar model for unbiased ideological spaces, a method to infer multidimensional opinions from interaction networks, and novel multidimensional polarization metrics that quantify several aspects of ideological polarization and bring new insights into the analysis of high-dimensional opinion distributions. Crucially, our multidimensional framework does not assume the underlying ideological structure, such as conservative vs progressive, liberal vs authoritarian, etc. Instead, it reveals the natural space that best describes the social landscape, which does not necessarily correspond to traditional categories. We illustrate the application of this framework in quadripolar and pentapolar real-world democratic processes, finding non-trivial ideological structures with clear connections to the underlying social context. Our methodology offers a comprehensive perspective of multilateral social tensions, as it incorporates complementary aspects of polarization: network segregation, opinion extremeness, and issue alignment.

Figures (4)

• Figure 1: Illustrating the ideological space and the opinion inference process. A: Diagram of the opinion spaces of dimension $1$ (i), $2$ (ii) and $3$ (iii). While in 1D the poles can be intuitively labeled as against ($-1$) or in favor ($+1$) of something, in higher dimensions they are labeled with numbers from $0$ to $n-1$ where $n$ is the number of poles. The neutral point (the barycenter of the simplex's vertices) is marked with a black cross. B: Examples of the opinion position of a listener node for three different weighted networks of a tripolar system. Top: network of the tripolar system with one listener node in gray (L) connected to three elite nodes corresponding to the three poles. The numbers on the links are their weights. Bottom: the corresponding opinion simplex showing the opinion position of the listener node. C: Evolution of the opinions of the nodes driven by Eq. \ref{['eq:degroot_model']}. The nodes of the networks in the top row are colored according to their closeness to each pole in the ideological space of the bottom row. Interactive versions of panels B and C can be found in https://vis.csh.ac.at/multipolar-viz.
• Figure 2: Visualization of multi-polarization measures. Synthetic opinion distributions of tripolar systems and the direction of their corresponding first (blue) and second (orange) PCs shown as arrows with a length proportional to the fraction of the variance they explain. In the three central panels the second PC explains so little variance that is not visible in the plot. The $TV$ of the system (measure of opinion extremeness) and the fraction of explained variance by each PC (measure of pole constraint) is also shown.
• Figure 3: Quadripolar opinion distributions of the 2015 Spanish elections. A: Cloud of users' opinions with its center of mass marked as a red square and the principal components as arrows with length proportional to their respective explained variance. An interactive version of this panel can be found in https://vis.csh.ac.at/multipolar-viz. B: Projections of the distribution onto the simplex faces shown as heat maps and contour plots. The centers of mass of the projected opinion distributions are represented as white squares and the projection of the direction of maximum polarization (PC 1), as a double headed arrow. The 1D projections onto each edge of the simplex are shown on the sides of the triangles. C: Diagram showing the projection of the poles onto the first two principal components. D: Opinion distribution projected onto PC1 and PC2. Each point has been projected as in the diagram of panel C, which shows the poles' projections.
• Figure 4: Pentapolar opinion distribution of the 2019 Spanish elections. A: Contour plots corresponding to the 2D projections of the opinion distribution of the 2019 Spanish elections onto the faces of the simplex. The centers of mass of the projected opinion distributions are represented as white squares and the direction of maximum polarization as a double-headed arrow. The 1D projections onto each edge of the simplex are shown on the sides of the triangles. As in Figure \ref{['fig:2d_od_eg_1516']}, we have filtered out users with low activity (tweets $<30$) to avoid noise, and in each projection we have only considered the opinions of the users that are closer to the poles of interest. B: Projection of the opinion distribution and the poles onto the first two principal components. C: Citizen placement of parties on the left/right scale according to a nationwide survey carried out in May 2019 cis_data. The distribution of the answers is shown for every party as a bar plot and the average as a vertical dashed line.