Was the decision of the UK electorate to leave the EU truly surprising?
Our research suggests that wide variations in the Remain vote were always going to make it difficult for the Remain campaign to mobilise voters around any consistent message.
By contrast strong similarities across the swathes of Leave voting constituencies made the Vote Leave campaign’s task significantly easier.
Complexities in voting data often make understanding results like that of the 2016 referendum hard to visualize; and, indeed, make it hard to predict how future elections might go.
A vast diversity of factors drive the results.
Simplifying the picture can help understand the strategies of the major parties.
Our work shows how Labour face a twin challenge of preserving their strong presence in Remain constituencies whilst making inroads into the Leave areas.
We show too how Conservative strongholds overlap heavily with Leave.
For the Conservatives, preventing Labour making progress in these areas will be the key to a majority government.
On Brexit, the standard interpretation has become that lower qualifications, lower income and lower mobility were all associated with voting Leave.
All this contributes to a prevalent narrative of the Leave voting areas being ‘left behind’, having missed out on the growth of the UK economy.
To see such relationships we might draw scatter-plots, discuss correlations and potentially add trends.
For example, graphs of education and voting suggest we may expect a constituency with a lower level of education to have a higher Leave vote.
Such simplifications are headline grabbing, but they do a clear disservice to the detailed information that lies behind voting data.
In turn they do a disservice to our understanding of the referendum result.
Consider instead a giant scatter-plot with many axes, one for every variable that might explain the Leave vote.
We cannot draw such a plot – we can’t draw beyond three dimensions, clearly – but like the simple two-axis version if we could see it then it would contain useful information.
Being unable to draw in multiple dimensions has been a barrier to our understanding
The Topological Data Analysis Ball Mapper methodology produces a visualization of just such a multi-dimensional plot, and we have used it to help to understand the 2016 referendum result.
Our research analyses the results from the 631 parliamentary constituencies (excluding Northern Ireland and the Speaker’s constituency) in a multi-dimensional point cloud.
We use information on the residents that was recorded in the 2011 census across a series of headers: the proportion who own their own home, the proportion with a mortgage, the proportion living alone, the proportion who are married, and so on.
In all 16 variables are used to create a virtual 16-axis scatter-plot.
To make that multi-dimensional cloud readable we create a set of balls, centered on randomly selected constituencies, such that all constituencies are within at least one ball.
In order to cover the whole dataset with balls there will be overlap between some of the balls.
Where constituencies are similar in these 16 characteristics they will fit within the same ball, and where constituencies differ greatly they will be distant in the space.
To capture this we draw connections between any pair of balls that have constituencies in common.
Our voting data then becomes an abstract diagram of collections of balls, some connected and some left out in space.
Because we are compressing dimensions into a two-dimensional plot distances based on just the horizontal and vertical distances have no meaning.
Visualising the constituency voting behavior dataset in this way immediately conveys the core message of our research: there were 26 constituencies which were similar across the 16 characteristics, and these all had Leave majorities.
Conversely, the constituencies with Remain majorities (in yellow and red) also tended to differ from one another.
And there’s another pattern: there are essentially two groups of Remain constituencies.
One is at the top right of the plot, the other towards the bottom right.
By thinking about the balls according to the 16 characteristics it is possible to see just what separates these groups.
The bottom-right area is primarily Scotland.
This group of balls has characteristics similar to those Leave constituencies to the lower left.
Here is where Remain voters with families, who own their own homes and who have low- and mid-level social status are found.
Importantly, the data also show that these are typically Leave voting characteristics.
To the top-right are the heavily student areas, where respondents are usually single, renting and have high qualifications.
These are also areas with the highest social classifications.
For Remain the challenge was to mobilise both groups, whilst also trying to appeal to those who share most, but not all of these characteristics in the Leave areas.
For Leave the large balls sitting close together in the space made a consistent message much easier to relay – and thus the referendum result was less surprising.
Next, consider what happens when we colour according to the majority Labour enjoyed over their nearest rivals in the 2017 general election.
Securing votes in those constituencies which were also Leave voting will be critical to their chances of winning any election – as will be maintaining strength across the Remain areas.
Contrast this with colouring by Conservative majority and the position of both main parties in the Brexit policy space is clear.
Further attempts to attract voters in those Labour voting Leave areas to vote Conservative is obviously driving the strategy seen at the recent Conservative party conference.
Much of this intuition had been hidden in the temptation to consider variables independently.
Understanding more about the splits in Remain may yet decide the final outcome of the Brexit saga.
By Dr Pawel Dlotko, senior lecturer in Mathematics at Swansea University, Wanling Qiu, doctoral candidate in Finance at the University of Liverpool, and Dr Simon Rudkin, senior lecturer in Economics at Swansea University.