Model created by CSH scientist Samuel Martín-Gutiérrez and colleagues captures social division in multiparty democracies. The US could also benefit from it.  

Austria, India, Mexico, and Spain. As multiparty democracies, these countries present a more complex picture of what brings people together – or divides them. Using social media data, a group of researchers proposes a novel approach to measuring polarization in nations with multiple political parties.


Complexity Science Hub scientist Samuel Martín-Gutiérrez and his colleagues developed a model that infers opinions from social networks, and measures the left-right political division, as well as other more nuanced sources of tension. They analyzed Twitter data from the Spanish elections of 2015 and 2019.


“It’s important to understand polarization in our society, and our findings shed new light on how political beliefs are shaped,” says Martín-Gutiérrez. “Our approach can lead to useful insights when applied to real-world debates happening on social media.”


The findings of the study were recently published online in the journal Chaos, Solitons & Fractals.


The approach could be used in a variety of situations to identify the main points of disagreement. It could also be used in countries with two-party systems, such as the US. The primaries, a key part of the American voting system, are an excellent example, points out Martín-Gutiérrez. “There are several candidates in the presidential state primaries. Our model could be helpful in tracking and understanding how party members’ opinions are distributed” says the CSH researcher.

Neutral and unbiased

A key characteristic of the model, according to Martín-Gutiérrez, is its neutrality and unbiasedness. “We assumed we didn’t know anything about the [main Spanish] political parties in the study. And our findings demonstrate that a rigid classification of parties into pre-defined dimensions may not paint the most complete and accurate picture.”


Using Twitter data from two elections, Martín-Gutiérrez and his colleagues from the Universidad Politécnica de Madrid tested the proposed approach in practice. First, they explored the Spanish general elections of December 2015. In a second moment, they dissected the Spanish general elections of April 2019.


“In 2015, we had four opinion poles corresponding to the four main political parties: People’s Party (PP), Spanish Socialist Workers’ Party (PSOE), Podemos, and Citizens (Cs). In 2019, we have five opinion poles, with the addition of Vox,” explains Martín-Gutiérrez. 


As a first step, the team identified the opinion leaders, and their ideological positions, on Twitter. “The second step was to analyze the listeners, the interesting part of the network, and assign an opinion to a user according to the average opinion of the people they retweet”, says Martín-Gutiérrez.


Since the social spectrum is more complex and colorful than black and white, the researchers determined the same distance between every pair of poles to avoid bias. “In some contexts, it may seem more appropriate to place certain poles close together. For example, in an election certain political parties may be more ideologically aligned with each other than with the rest. However, parties that share ideology also compete for the same electoral base, often leading to even greater antagonism between them,” explain the authors of the study.

New knowledge

According to them, the opinion patterns of both analyses were consistent with the underlying Spanish social reality and could be interpreted taking into account their context. “There is a clear picture of the Spanish political spectrum, and we can see, for instance, where the opinion distribution is most stretched or where the average opinion is located,” says Martín-Gutiérrez.


“Our findings seem to be in line with what we intuitively know about Spanish politics. From this we can start building new knowledge, for instance, about citizens’ perceptions of politics and their behavior.”


In the visualization created by Liuhuaying Yang, CSH visualization expert, you can explore the opinion distribution of the 2015 Spanish elections: 

(c) CSH
Opinion distribution of the 2015 Spanish elections: a cloud of users’ opinions with its center of mass marked with a red square and the principal components as arrows with lengths proportional to their explained variances

The proposed method could also assist policymakers in dealing with polarization and healing social rifts. “It could help policymakers devise de-escalation interventions by addressing the issues that cause the strongest tensions,” argue the authors of the study. 


Furthermore, it could be used to combat online misinformation spread by bots and trolls. The model could be used to evaluate the influence automated bots have on social media as compared to real human accounts.


The study “Multipolar social systems: Measuring polarization beyond dichotomous contexts” is available online at Chaos, Solitons & Fractals

What do buttons, dartboards, a Black Forest cake and the Colosseum have in common?


Exactly, they are round. More precisely, circular. Or even better: If you measure their circumference and divide it by their diameter, you always get one number: π, pi.


Pi (π) is one of those famous constant numbers that keep popping up in all sorts of different fields. Many calculations, many formulas, and most things that feature a circle ultimately depend on its 3.141592…. Even those who are searching for Nessie bump into .

Finding Nessie

Imagine a person trying to take the best photo of Nessie, the Loch Ness monster. Hundreds of people have tried it. The lake itself is quite large (a surface of 56 km2). So, not an easy task. Where would you start looking for Nessie? The best technique: where it was last seen. At least you know that Nessie once stayed  in that part of the lake. 


World Pi Day at Complexity Science Hub


The issue is that the monster moves, and even if it moves slowly, the area that we have to search grows pretty quickly. For example, if Nessie moves at a speed of 1 km per hour, what is the area we need to search? Just one hour after the last known report, the monster could be anywhere within a circle with an area of 3.14 km2. And in two hours, the area will be 12.6 km2. So the area where the monster is sure to be after h hours since its last sighting is πh2.

Pi (π) in cities and people

Let’s leave Nessie alone again, nevertheless appears almost everywhere. Besides circles and countless mathematical formulas, you can find it playing pool. Even in the Bible. As well as in cities. And it works like this:


If you randomly distribute points in a circle with radius r, you you may be (more or less) stunned (depending on whether you are stunned by at all) to find the following: The average distance between points is 128r/45. The same formula is used over and over to model various aspects of urban transportation and traffic flow. For more than 40 years, we have been modeling cities using the same formula.

Commuting by pi

If you know the area of a city, say A, then traveling from one location to another will grow according to 128√A/45π. This formula holds for cities that are perfectly circular (perhaps a very simple city!) and where you can travel in a straight line between any two points. Of course, no city is like this, but it gives us an idea of how long trips will be and how long the commute will take.


Vienna, for example, has an area of roughly 415 km2. So if we take any two locations in the city, they are on average 18 km apart. However, Innsbruck has a surface of 105km2, so on average the places are 9 km apart. This is shown in a recent study by Rafael Prieto-Curiel of the Complexity Science Hub, who, together with colleagues, measured about 183 million buildings in African cities. He used a similar formula to the one for the distance between points inside a circle can also be used to measure distances between buildings in a city.

World Pi Day at Complexity Science Hub

There are other aspects of cities, including their irregular shapes, the space between its buildings, the many roads or its fractal patterns. However, keeps appearing! 


Last but not least, does not even stop at the city boundaries. Instead, the fascination with lives in the heads of people. This culminates in worldwide competitions  in which people try to outdo each other in enumerating more digits of . The official world record is currently 70,030 digits. It took Suresh Kumar Sharma 17 hours and 14 minutes.


This was in 2015, and no one has officially beaten it since. But developer Emma Haruka Iwao announced another record on June 8, 2022: a program from Google calculated the circular number to 100 trillion digits. That took 157 days. By way of comparison, in 1940 scientists were only able to calculate around 1,000 decimal places.


Happy day! A day for mathematicians, people who like math, or even people who suffered from in school but can still remember the first digits of 3.14!