Mathematical measures of societal polarisation

TL;DR

This paper introduces four mathematical measures to quantify societal polarisation using graph theory and information theory, tested through opinion dynamics simulations to monitor real-time social network polarisation.

Contribution

It presents novel, mathematically rigorous methods for measuring societal polarisation, combining graph-based and probabilistic approaches, with validation through simulation.

Findings

Spectral radius, Kullback-Leibler divergence, and Hellinger distance effectively distinguish polarisation levels.

The measures can monitor real-time social network polarisation indicators.

Min-max flow was less effective in capturing nuanced polarisation states.

Abstract

In opinion dynamics, as in general usage, polarisation is subjective. To understand polarisation, we need to develop more precise methods to measure the agreement in society. This paper presents four mathematical measures of polarisation derived from graph and network representations of societies and information theoretic divergences or distance metrics. Two of the methods, min-max flow and spectral radius, rely on graph theory and define polarisation in terms of the structural characteristics of networks. The other two methods represent opinions as probability density functions and use the Kullback Leibler divergence and the Hellinger distance as polarisation measures. We present a series of opinion dynamics simulations from two common models to test the effectiveness of the methods. Results show that the four measures provide insight into the different aspects of polarisation and allow real-time monitoring of social networks for indicators of polarisation. The three measures, the spectral radius, Kullback Leibler divergence and Hellinger distance, smoothly delineated between different amounts of polarisation, i.e. how many cluster there were in the simulation, while also measuring with more granularity how close simulations were to consensus. Min-max flow failed to accomplish such nuance.