Title: The process of polarisation as a loss of dimensionality: measuring changes in polarisation using Singular Value Decomposition of network graphs

Abstract: The increasing polarisation in our societies is a major international concern. Current approaches to defining and detecting polarisation largely rely on finding evidence of bimodality in social networks or voter opinion surveys. It is difficult to detect temporal trends in polarisation, as the results usually fall into a binary of polarised or non-polarised, which cannot robustly show that subsequent increases in bimodality are statistically significant.
Our work is aligned with Baldassari and Gelman's theory that polarisation should be defined as increasing correlation between positions in the ideological field. We also draw from post-structuralist work which argues that polarisation is the process of both the ideological and material layers of society being segregated into two poles, as in cases of apartheid. Thus, in order to measure the polarisation in a society, it would be beneficial to be able to assess social networks directly.
In this paper we use Random Dot Product Graphs to embed social networks in metric spaces. In the case of a social network, the embedded dimensionality corresponds to the number of reasons any two people may form a social connection. A decrease in the optimal dimensionality for the embedding of the network graph, as measured using truncated Singular Value Decomposition of the graph adjacency matrix, indicates increasing polarisation in the network.
We apply this method to two different Twitter networks based on discussions of climate change, and show that our methods agree with other researchers' detection of polarisation in this space. We also use networks generated by stochastic block models to explore how an increase of the isolation between distinct communities in a network, or the increase in the predominance of one community over the other, are identifiable as polarisation processes.