Can everyone believe that the majority of people will agree on one preference when, in fact, the opposite is true? Surprisingly, the answer is yes – and mathematics can reveal when and why.

This is one of many examples where mathematics is used to model real-life scenarios involving networks: from traffic flow modelling to information spread in social networks.

In this particular scenario, we are interested in networks where individuals only have access to limited information in order to reach a conclusion on what will be the most agreed on preference.

Imagine a community where individuals are asked to agree on two preferences, A and B, with their friends. You agree on A with some friends and on B with others (for example if I have seven friends, I may agree with four friends on A and with three on B).

Suppose that in the whole community, the most agreed upon preference is A. You, however, do not have access to this information – instead, by asking all your friends (including what they agreed between them), you find out that the most agreed-upon preference was B, leading you to expect that B will win the majority. Amazingly, this mismatch can occur for everyone: the entire community may believe B will win when polling locally, even though A dominates globally.

This mismatch can occur for everyone

This phenomenon, called a ‘local versus global majority flip’, arises in certain mathematical models of social networks. The accompanying diagram illustrates this.

The left side shows the whole community, where clearly A (red edges) is agreed on more than B (blue edges). The right side focuses on a single person (vertex), where we see that locally each individual sees more agreement on preference B (7) than A (6).

This phenomenon can even occur if we allow each individual to access more information, such as asking friends of friends for what preferences they agreed on, or if we allow individuals to express more than two preferences!

The example considered exhibits a high degree of symmetry, which is generally not the case in real life. However, computer simulations have shown that certain mathematical models of real-life social networks, which use randomness to break symmetry, can exhibit such majority flips. This fascinating phenomenon implies that we should be more careful when extrapolating from our friends’ opinions – sometimes it is not of help just to know what they agreed on between themselves!

Xandru Mifsud is a doctoral student at the University of Oxford, having previously done a BSc and MSc at the University of Malta.

Sound Bites

•         During 2024, a research group from the University of Malta (Josef Lauri, Xandru Mifsud, Christina Zarb), along with Yair Caro (University of Haifa ‒ Oranim) and Raphael Yuster (University of Haifa), published extensively on ‘local versus global majority’ problems.

•         The Mathematics Genealogy Project, run by the Department of Mathematics at North Dakota State University, is a collaborative project that builds a ‘family tree’ of mathematicians, where two persons are siblings if they shared the same supervisor. After exhaustive research and interviews, a project by Xandru Mifsud extended this work to document the development of mathematics in Malta. A poster may be found here.

For more science news, listen to Radio Mocha on www.fb.com/RadioMochaMalta/.

DID YOU KNOW?

•         On average, an individual’s friends have more friends than the individual – this is known as the friendship paradox and has a mathematical underpinning.

•         The formula for the summation of the first n integers, namely that 1 + 2 + … + n = n (n + 1) / 2, is attributed to Carl Friedrich Gauss.

•         In Arthur Conan Doyle’s Sherlock Holmes, the fictional character of Professor Moriarty was a mathematician. Some believe that certain aspects about Moriarty were inspired by Gauss.

•         Records show that Mark Appleby, an academic member at the Royal University of Malta in the 1930s, was a member of the Association for Symbolic Logic during its formative years when members included the likes of Alonzo Church, Alan Turing, Alfred Tarski, and Haskell B. Curry.

For more trivia, see: www.um.edu.mt/think.