The network of links between individuals—their social network—has long fascinated social scientists. These networks are neither random nor entirely ordered. Instead, they occupy a middle ground in which people are strongly linked to a few individuals they know well, with weaker links to a larger group of friends and coworkers plus extremely weak links to a wide range of casual acquaintances.

Social scientists measure the strength of these links using a variety of indicators, such as how often a person calls another, whether that call is reciprocated, the time the two people spend speaking, and so on. But these indicators are often difficult and time-consuming to measure.

So network theorists would dearly love to have some way of measuring the strength of ties from the structure of the network itself.

The bow tie structure in a social network reveals the strength of friendships.

Today, they get their wish thanks to the work of Heather Mattie at Harvard University in Massachusetts and a few pals, who say they’ve found a special pattern in the links between individuals that reveals the strength of the ties between them. The pattern forms a topological structure that resembles a bow tie.

The team’s method is straightforward. Mattie and co study the strength of links in two social networks from very different settings.

The first is the network of links between almost 70,000 people in 75 rural villages in India. Social scientists reconstructed the network using surveys designed to gather relevant social information. This survey asked which friends and relatives visit the respondent’s house, which people the respondent would borrow from or receive medical advice from or go to the temple with, and so on.

This allowed the researchers to reconstruct a social network consisting of 37,000 edges between 17,000 people for whom they had complete information. They then used the number of social links from the surveys as a measure of strength for the social ties between people. So if two people were relatives who visited each other at home, went to the temple together, and lent each other money, each of these factors would contribute to the link between them.

Mattie and co also constructed the social network between mobile-phone users from an unnamed European country. Because of the size of the network, the team chose 500,000 people at random and then constructed the social network using the number and length of calls between them. They assumed the link was stronger if the calls were reciprocated and if the total amount of time spent talking was high.

Next, the team examined the structure of both networks. For each pair of individuals, they constructed the network of friends they had in common and the network of friends that they did not have in common. It is this structure that looks like a bow tie (see diagram). They then analyzed these bow tie networks.

The results make for interesting reading. The team found that the number of friends that pairs of individual have in common is strongly correlated with the strength of the tie between them, as measured in other ways. That’s regardless of whether people are linked by mobile-phone records or by social ties in rural Indian villages.

This result captures the findings of two of the first researchers to work on social networks. The first comes from Elizabeth Bott, an influential anthropologist who published a book in 1957 called Family and Social Networks.

In this book, she hypothesized that the degree of clustering in an individual’s network could draw the person away from a tie with somebody else. In other words, if you are part of a group of close friends or relations, you are less able to make strong links outside this group.

The second comes from Mark Granovetter, an American sociologist who in 1969 wrote a hugely influential paper called “The Strength of Weak Ties.” In this paper, he suggested that the stronger the tie between any two people, the higher the fraction of friends they have in common.

The bow tie framework captures both ideas. “Both data sets provide evidence to support the weak ties hypothesis and the Bott hypothesis,” say Mattie and co.

The new finding has interesting implications. It suggests that it should be straightforward to measure the strength of links between individuals, simply by looking at the structure of their social network. “This local structure allows for analyses that are computationally feasible for networks of any size,” say Mattie and co.

And it should allow a proper test of Bott’s original hypothesis, which she applied to married couples and families. “This would enable testing of the original version of Bott’s hypothesis, rather than a generalized form as we present here,” say Mattie and co.

The work also shows how social networks can reveal more about people than they might otherwise wish to make public. Mattie’s work implies that the shape of your social network reveals not only who you are friends with, but how strongly you are linked to them. Not everyone is going to be happy with that.

Ref: arxiv.org/abs/1710.04177 : The Social Bow Tie