The process of polarisation as a loss of dimensionality: measuring changes in polarisation using Singular Value Decomposition of Random Dot Product Graphs

Social and political polarisation is an issue of increasing concern in New Zealand [1], as well as a wide range of countries [2]. [3] [4] [5] [6]. While it was initially thought that New Zealand had avoided the populist takeover seen in countries such as the USA [7], radical changes in the government makeup in both 2020 and 2023 have since called this into question. This paper will explore existing methods of measuring polarisation and propose a new measurement based on Singular Value Decomposition of social Random Dot Product Graphs.

New Zealand was considered to have escaped the development of polarisation which had developed in similar nations, particularly with the 2016 election of Donald Trump in the USA [7]. Analysis of voter polarisation in NZ from 2009-2018 showed little evidence of polarisation in that period [8]. Topic modelling of political party manifestos showed similarities between the manifestos of parties on the left and right [9], suggesting that despite their rivalries the parties are not deeply divided. Tan [10] argues that the lack of bipartisanship between NZ’s left and right indicates that the parties are polarised; however, they found that the perceived distances between left and right parties, and between those parties and voters, were stable over time, suggesting that polarisation is not increasing. Satherley et al. [11] find that there is a high level of stability in the partisanship of NZ voters (i.e. they do not frequently change which party they support) in years up until 2017 and note that this creates a risk of polarisation if voters stay committed to a party that becomes increasingly extreme. In another paper, Satherley et al. [12] find that in analysis of polarisation between the Labour, Green, and National parties, NZ European voters and voters of high socio-economic status exhibit more polarisation in their party preferences than other groups. Stanley et al. [13] find that climate change is not a polarising issue for New Zealand, with the voting public largely accepting its existence.

This history of political stability highlights how irregular the NZ election results in 2020 and 2023 have been, with significant implications for potential polarisation. The 2020 election saw the Labour party win more than 50% of the vote, the first outright majority since the change to Mixed Member Proportional government in 1996, and saw the right-wing ACT minor party increase their vote share from 0.5% to 7.58%. The covid-19 pandemic has subsequently become a focus for disinformation producers, prompting concerns about an increase in polarisation since 2020 [14] [15] [16]. The 2023 election gave further new results; despite the return to the right-wing, both the NZ First and ACT parties were required for the National party to form a coalition, creating the first instance of the National party having to negotiate with two coalition partners of this size. The 2023 Coalition Government has rapidly removed Māori language official Government documents [17], and removed references to the Treaty of Waitangi (a constitutional document that forms the basis of the relationship between the indigenous Māori people and the British settlers) from the law [18]. The ACT party have also introduced a parliamentary bill to redefine the “Treaty Principles” in ways that remove references to Māori. The Waitangi Tribunal (the permanent commission of inquiry tasked with investigating governmental breaches of the Treaty of Waitangi) has argued that “the Crown’s [Government’s] actions ‘threaten to disfigure or rupture’ the Crown–Māori relationship and ‘could set back the foundational relationships of Aotearoa New Zealand for decades’” [19]. A rise in general racist sentiment, as well as racist hate crimes and violence, has been documented [20]. As such, we think that questions about whether the polarisation may be happening along an axis other than party affiliation are particularly relevant in the current political atmosphere, since there is a risk that these policies will create a polarisation between Māori and pākehā (descendants of British settlers).

Polarisation scholarship in New Zealand is particularly influenced from research in the USA, so it is useful to cover some key analyses of their system as well as general data-based polarisation measurement methods. Heatherington [21] famously found that the Republican and Democrat parties have no overlap in their policies or voting patterns, finding a bimodal division by inspection (e.g. no overlap between the ideological groups). Most subsequent research uses significance using tests such as Hartigan’s Dip test [22] [23]. When investigating social networks, bimodality is expressed as strong in-group/out-group divisions [24], and is often characterised by hostility between the two groups; it is common to consider this hostility as a key sign of polarisation, in addition to the differences in policy positions between the two poles (usually the Republican and Democrat parties in the USA) [25] [26]. On social media, these hostile divisions usually take the form of ”echo chambers” that focus on one set of political views and exclude all others [27] [28].

We have some specific concerns about the use of bimodality in detecting polarisation. The most important is that use of hypothesis tests can be used to detect when a distribution is polarised, but after the first significant result it is difficult to robustly show that any further increase in polarisation is also significant. That is to say, while subsequent results may also be statistically significant, it is difficult to show that the increase from the first result to the second is itself significant. This means that we are not confident that studies which use bimodality are able to claim that We are therefore interested in developing measurements of polarisation that do not rely on significance testing in this way. A secondary concern is that this approach can struggle when assessing multi-party democracies that do not have a clear left- and right-wing split [29]. In the New Zealand context, the NZ First party makes it difficult to clearly reduce the field to left and right blocs, as they have formed coalition governments with both the Labour and National parties many times. As such, finding a method of measuring polarisation that can better handle this type of situation would be beneficial to researchers in New Zealand and similar countries.

Another common aspect of definitions of polarisation is that the polarisation is maximised when the groups are of equal size, as well as strongly divided. This was first proposed by Esteban & Ray [30]. Taking group size into account is intended to help tell polarisation conflicts apart from other major social conflicts, such as conflicts over wealth inequality. However, a consequence of this theoretical approach is that sets a rigid upper limit on how severe a polarisation can become if one of the groups is a small minority. This concerns us because of issues such as potential polarisation against groups such as refugees; they are by definition a small proportion of the population but are still considered to be a potential cause of polarisation, especially in countries that are seeing influxes of refugees due to war or climate change. We are interested in finding definitions of polarisation, and tools for measuring it, that are not as affected by the size of the groups.

Computational data science approaches to measuring polarisation are popular, especially for their speed when analysing millions of records from social media services. There are fast and simple algorithms for detecting communities that can then be investigated for polarisation, such as the Louvain method [31]. In some cases, clustering algorithms are used to measure polarisation; there are a number of ways of measuring distances between the clusters, which can indicate polarisation between the groups if the clusters also have low internal variation [32]. However, this has similar issues to bimodality testing insofar as it becomes difficult to interpret when there are more than two clusters. It is possible to argue that three or more groups can all be polarised against each other, but we are unconvinced by this; as the number of polarised groups increases, so does the number of dimensions required for their clusters to all be equidistant from each other, which increases the complexity of the space so much that it becomes more complex than what we would consider to be “polarised”.

In computational analyses focussed on bimodality instead of clustering, network data is usually processed using a dimension reduction algorithm, such as Principal Component Analysis or Canonical Correspondence Analysis, in order to create a single dimension that can be evaluated for bimodality. Measuring latent positions empirically means projecting them in lower dimensional spaces, and then assess the resulting first dimension for bimodality [33]. These analyses do not treat the correlation structure delivered by the PCA or CCA as informative in and of itself; they are simply used to discover the largest principal component or axis (i.e. the largest dimension). An advantage of this approach is that the issue driving polarisation is generated from the data, rather than researchers presuming what it is and risking choosing incorrectly. However, it does not entirely mitigate the risk of choosing the wrong dimension, since it presumes that polarisation is happening along the first dimension, and it therefore will still miss cases in which polarisation is happening along one of the other dimensions.

In order to comprehensively address the issue of correct dimension selection, Baldassarri and Gelman [34] propose a definition of polarisation as increasing correlation in the ideological space, i.e. that pairs of ideological issues asked about in the American National Election Study become more correlated as a result of polarisation, reducing political pluralism and restricting possible ideological opinions, until maximum correlation is reached and an oppositional binary is created. This polarisation stands in comparison to an integrated, non-polarised society which ”is not a society in which conflict is absent, but rather one in which conflict expresses itself through nonencompassing interests and identities”. In their tests of Pearson correlations of pairs of issues in the American National Election Study, they did not find that there was increasing correlation in the ideological field of US-American voters pre-2004. However, subsequent research by Kozlowski and Murphy [35] using the same methods found that correlation between pairs of ideological issues rapidly increased between 2004 and 2016. They noted that the increase in polarisation was strongest in the domains of economics and civil rights issues, rather than in the domain of moral issues that the ”culture war” framing of polarisation may suggest. Similarly, DellaPosta [36] conceptualises polarisation as similar to an oil spill, with the increasing correlation in ideological positions spreading polarisation to previously apolitical members of society. The article analyses how the ”belief network” of US-American politics has changed over time, concluding that the network has developed clusters which have reduced the prevalence of cross-cutting ideological positions; this means that pluralism has decreased and polarisation has increased.

Incorporating post-structuralist political theory allows us to expand on this understanding of polarisation as correlation. Ernesto Laclau & Chantal Mouffe [37] begin from the same understanding of pluralism and polarisation as scholars such as Baldassarri and Gelman, but they develop this concept beyond just they ideological field. They argue that ”In a colonized country, the presence of the dominant power is every day made evident through a variety of contents: differences of dress, of language, of skin colour, of customs […] the colonizer is discursively constructed as the anti-colonized.” (p. 128). This is to say that it is possible for elements of a society’s culture that are not explicitly political to become associated with ideologies and drawn into the correlation, further reducing opportunities for political pluralism. Two poles are constructed that are mutually exclusive and have nothing in common, sustained by segregation in all layers of society (e.g. the South African regime of racial apartheid). Notably, the polarisations that they examine do not occur primarily in the division between political parties, but along fault lines such as race and ethnicity, economic class, the urban-rural divide, and the division between coloniser and colonised. The starkest demonstration of this principle is in the actions of the 1970s Argentinian military dictatorship, which banned Venn diagrams from being taught in primary schools because they ”were feared to encourage subversive models of collectivity” [38].

In this paper we present a novel method of measuring polarisation that follows from both Laclau & Mouffe and Baldassarri & Gelman, namely using Singular Value Decomposition to determine the correlation structure of social networks. Using social networks allows us to capture interactions between people that are not explicitly political — being neighbours, sharing a workplace, etc — but which become politicised and segregated during extreme polarisation. As such, we are able to determine whether correlation is increasing not just among possible political positions, but whether it is increasing among social determinants of interaction as well. Our method gives a value that corresponds to the network’s capacity for complexity, and is inversely related to its level of polarisation. Incorporating these additional layers of a society into the analysis should make it easier to detect whether polarisation is occurring.

We represent the conversation happening on a social platform (Twitter, Facebook, Instagram, etc.) as a network. Each user that took part in the conversation is mapped to a node. We add an edge between two nodes if the two respective users have communicated in the time window of the observation. Depending on the chosen social platform considered and the specific research question, a communication can be given by a reply, a mention (”tagging” them in a post), a quote, a repost/retweet/share, or a set of these. These networks can be directed (as is more common) or undirected. Here, we consider communication networks only as unweighted graphs, although the generalization to weighted graphs doesn’t present any fundamental challenges.

We apply our computational framework for polarisation to three different data sets: two from Twitter, and one consisting of simulated interaction networks. These have been chosen in order to show that our method works well, and to explore some common beliefs about polarisation in a new way.

We have demonstrated a novel method for measuring polarisation through the embedded dimensionality $\hat{d}$ of random dot-product graphs. This is a reliable and straightforward implementation of the correlation-based approach to polarisation suggested by Baldassarri and Gelman. Our method captured the presence of polarisation in all the scenarios where it was expected and had been found by other researchers, in both simulated data and real social media networks. In particular, the ability to observe the changes in $\hat{d}$ over time in data from the COP conferences provided a useful demonstration of how this method can be applied to longitudinal data. The RDPG approach also allows us to easily see that the process of polarisation is occurring in a network over time, through its embedded dimensionality reducing, rather than relying on a binary test of whether the network was polarised or non-polarised.

Another advantage of the RDPG approach is that it is computationally light; the main bottleneck is the computation of the first singular values of a large matrix, but this is well known in computer science literature and has already been strongly optimised. We found that the SVD was feasible even when used on networks with millions of nodes. Bimodality-based methods typically use SVD or correspondance analysis to determine the dimension they will test for bimodality, so our approach is at least as efficient.

Our approach is also highly interpretable, without forcing the latent ideological distributions into an artificially unidimensional space. Rather than creating a unidimensional space and then interpreting its political meaning (such as pro- and anti-climate, or left- and right-wing), the dimensionality method instead focuses on the number of dimensions rather than what those dimensions are. In high-dimensional spaces, we do not need to know exactly what ideologies the dimensions correspond to; the important part is that they signal that there are ideological connections being made between nodes that would not be possible if the network was polarised.

The SVD entropy of the network did not relate to $\hat{d}$ as closely as we expected, though it did reflect major changes in the networks when they occurred. As such, we think it is best to use $\hat{d}$ of the network to measure its polarisation.

A major limitation of this method which could be improved is that it does not capture affective polarisation very well. Our method functionally considers any interaction between two nodes to be “good”; this means that it is not capable of capturing antagonistic interactions between nodes, and as a result it may overestimate $\hat{d}$ of the network by mistaking brief antagonistic reactions for positive social bonds. There is a great deal of scope for integrating the concept of affective polarisation into our model, through methods such as using signed matrices and classification systems such as sentiment analysis to determine whether interactions in a social network are positive, negative, or neutral before determining $\hat{d}$.

Another possible extension of this method would be to implement a nonparametric two-sample hypothesis test[47], since this would allow a hypothesis test of whether the two networks are significantly different as additional evidence of polarisation having occurred. We believe that being able to observe the embedded dimensionality of the graph alone is useful; however, we understand that sometimes a hypothesis test is demanded, and we believe this would help demonstrate that changes in the embedded dimensionality of the network are significant. In our tests using stochastic block models, we found an expected result that $\hat{d}$ increases as the probability of edges forming between the groups increases. This is evidence for the common belief about polarisation that connections between the groups decrease their polarisation. We also found an unexpected result that $\hat{d}$ does not straightforwardly increase when one group is much larger than the other, and in fact decreases greatly when one group is 100 times the size of the other. While this is somewhat a property of the size of the network and its sparsity, since the small group was very small and thus unable to make a large number of edges with the big group, we believe that it is an important case to have tested and that closer investigation of this scenario is warranted. We think that properties sparsity are important for modelling the real conditions of social networks at risk of polarisation, such as communities who have recently received refugees, and should be carefully considered in the model design to ensure that real effects are not being mathematically counteracted.

It would also be productive to experiment more with stochastic block models, since we only tested two possible scenarios. For example, this paper has only explored the two-block case; many instances of online ”echo chambers” have a large number of groups who all hate people different to them, and it would be useful to see what happens to the embedded dimensionality in such cases. Similarly, our testing on group prevalence showed a decrease in embedded dimensionality when one group was 100 times the size of the other, but we did not follow the smaller group’s size all the way to zero so we do not know whether the embedded dimensionality remains low or rebounds to a higher value.