**A mathematical argument for forging friendships has been found by researchers, reports Roger Highfield, Director of External Affairs and coauthor, with Martin Nowak, of Supercooperators.**

A revealing insight into the lightweight nature of many Facebook friends has emerged from the effort by mathematicians to understand how social networks affect the way that we cooperate with each other.

The Harvard University mathematical biologist Martin Nowak has spent decades studying the evolution of cooperation, not the struggle for existence but what he jokingly refers to as the ‘snuggle for existence.’

This week, with colleagues at Emmanuel College and Harvard University he has come with mathematical insights into basic questions about the influence of social networks on cooperation: does it help if I join more social networks, since that means I will spend less time in each one? Or does it help if I join smaller ones, since their size should make it easier to meet other people?

The team reports in the journal Nature a new algorithm that predicts whether a particular social structure is likely to favour cooperation and reveals that strong one-to-one relationships – not loose networks – are the most conducive to cooperation.

“What we are able to do is calculate the critical benefit-to-cost ratio for cooperation to thrive on any fixed population structure,” explained Martin Nowak, Professor of Mathematics and of Biology and Director of the Program for Evolutionary Dynamics, and the senior author.

“And what we find is truly interesting. We can take any social network (mathematicians use the term graph), and if it has strong pairwise ties, that is what is most conducive for cooperation. This is mathematical argument for stable families, for stable partnerships or for stable friendships.”

The team used the new tool to analyse real-world social networks, including those between animals like dolphins and rhesus macaques. Of the networks examined, though, they found that by far the lowest cooperation score came from Facebook. This shows that having hundreds of Facebook friends does not mean you have more people to rely on.

“While global interconnectedness has increased rapidly over the past few decades, there are downsides to that,” added study first author Benjamin Allen, an Assistant Professor of Mathematics at Emmanuel College and a researcher at the Program for Evolutionary Dynamics.

“More connectivity won’t necessarily promote people being good to each other,” he said. “It’s not that global connections are bad, but they are no substitute for a small number of strong local connections.”

“That stems from the fact that people have a lot of connections on Facebook, and in that data set we only have the raw connections,” Allen said. However, he added that if the Facebook social networks were weighted so “that close friends count for more, it could have changed the outcome.”

Or to put it another way, large social networks foster connections by overcoming national, geographic, and even linguistic barriers. But when it comes to fostering cooperation, global connectivity leaves something to be desired.

Scientists for decades have sought to understand the interplay between population structure and evolution, beginning with idealised (mathematically tractable) examples that they call “well-mixed” populations.

In such populations – where every individual person equally with every other person – evolution selects against cooperation, causing it to eventually die out.

In 1992, however, Martin Nowak and Robert May (now Lord May), then collaborating at Oxford University, published a landmark paper in Nature where they showed that cooperators can survive depending on their neighbours, in what they called spatial cooperation.

In 2006, Nowak and colleagues at Harvard showed that, in certain “special” populations – notably those in which all members had the same number of social connections – cooperation is able to thrive if the benefit-to-cost ratio of cooperation exceeds the number of friends.

“If you are a cooperator, then your friends can also act as cooperators, so you form a cluster and receive a positive payoff from each other,” Nowak said. But they had difficulty in drawing conclusions about more realistic social networks, say one where one person had three friends, and one of their friends had five, and someone else had eight. This was an unsolved mathematical problem.

This problem was so complex that solving it would result in accidentally solving one of the most important open challenges in computer science. Known as P versus NP, the challenge asks whether every solved problem in which the answer can be quickly checked by a computer can also be quickly solved by a computer.

The scientists however realised that a special answer could be possible for the case where fitness differences between individuals are small. Experts talk about the ‘limit of weak selection’ where they know that evolution is mostly driven by probabilistic events, which are mildly influenced by natural selection.

To crack the problem, Allen and Nowak enlisted the help of Harvard mathematicians, including Fields Medal winner Shing-Tung Yau, the William Caspar Graustein Professor of Mathematics and Professor of Physics, Gabor Lippner and Yu-Ting Chen. Also an Iranian husband-wife team, Babak Fotouhi and Naghmeh Momeni joined in the pursuit.

The team realised that the answer lay in finding a way to calculate the possibility of people on the network encountering each other. With the resulting algorithm, researchers can now not only find out whether a given social network would foster cooperation, but could even predict whether alterations to the social network might encourage cooperation to evolve. Or to put it in precise mathematical terms, says Nowak, the new discovery allows them to calculate any evolutionary game on any fixed population structure.

Nowak said the study also helps to paint a fuller picture of the interplay between evolution and society. “The traditional, Darwinian theory of evolution is basically that everybody fights against everybody else – it’s just an all-out competition,” Nowak said. “But in the last 20 years, people have understood more and more that the other half of the story is about cooperation, the snuggle for survival. Natural selection can also favour cooperation, and the winners in that situation are those who cooperate with each other. This is a major extension of the story of evolution.”