“If a person is looking for friends, they should basically be active in as many communities as possible,” says Anshumali Shrivastava, assistant professor of computer science at Rice University and coauthor of the study, which the researchers presented at the 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining. “And if they want to become friends with a specific person, they should try to be a part of all the groups that person is a part of.”
The finding is based on an analysis of six online social networks with millions of members. Shrivastava says its simplicity may come as a surprise to those who study friendship formation and the role communities play in bringing about friendships.
‘Birds of a feather’
“There’s an old saying that ‘birds of a feather flock together,’” Shrivastava says. “And that idea—that people who are more similar are more likely to become friends—is embodied in a principal called homophily, which is a widely studied concept in friendship formation.”
One school of thought holds that because of homophily, the odds that people will become friends increase in some groups. To account for this in computational models of friendship networks, researchers often assign each group an “affinity” score; the more alike group members are, the higher their affinity and the greater their chances of forming friendships.
Prior to social media, there were few detailed records about friendships between individuals in large organizations. That changed with the advent of social networks that have millions of individual members who are often affiliated with many communities and subcommunities within the network.
If two people are active in the same community at the same time, they have a constant, usually small, probability of forming a friendship. That’s it.
“A community, for our purposes, is any affiliated group of people within the network,” Shrivastava says. “Communities can be very large, like everyone who identifies with a particular country or state, and they can be very small, like a handful of old friends who meet once a year.”
Finding meaningful affinity scores for hundreds of thousands of communities in online social networks has been a challenge for analysts and modelers. Calculating the odds of friendship formation is further complicated by the overlap between communities and subcommunities. For instance, if the old friends in the above example live in three different states, their small subcommunity overlaps with the large communities of people from those states. Because many individuals in social networks belong to dozens of communities and subcommunities, overlapping connections can become dense.
In 2016, Shrivastava and study coauthor Chen Luo, a graduate student in his research group, realized that some well-known analyses of online friendship formation failed to account for any factors arising out of overlap.
“Let’s say Adam, Bob, and Charlie are members of the same four communities, but in addition, Adam is a member of 16 other communities,” Shrivastava says. “The existing affiliation model says the likelihood of Adam and Charlie being friends only depends on the affinity measures of the four communities they have in common. It doesn’t matter that each of them are friends with Bob or that Adam’s being pulled in 16 other directions.”
That seemed like a glaring oversight to the researchers, but they had an idea of how to account for it based on an analogy they saw between the overlapping subcommunities and the overlapping similarities between web pages that internet search engines must take into account. One of the most popular measures for internet search is the Jaccard overlap, which Google scientists and others pioneered in the late 1990s.
The model offers a simple explanation of how friendships form: overlap between communities.
“We used this to measure overlap between communities and then checked to see if there was a relationship between overlap and friendship probability, or friendship affiliation, on six well-studied social networks,” Shrivastava says. “We found that on all six, the relationship more or less looked like a straight line.”
“That implies that friendship formation can be explained merely by looking at overlap between communities,” Luo says. “In other words, you don’t need to account for affinity measures for specific communities. All that extra work is unnecessary.”
The math behind making friends
Once the researchers saw the linear relationship between Jaccard overlap of communities and friendship formation, they also saw an opportunity to use a data-indexing method called “hashing,” which organizes web documents for efficient search. Shrivastava says he and Luo developed a model for friendship formation that “mimicked the way the mathematics behind the hashing work.” The model offers a simple explanation of how friendships form.
“Communities are having events and activities all the time, but some of these are a bigger draw, and the preference for attending these is higher,” Shrivastava says. “Based on this preference, individuals become active in the most preferred communities to which they belong. If two people are active in the same community at the same time, they have a constant, usually small, probability of forming a friendship. That’s it. This mathematically recovers our observed empirical model.”
He says the findings could be useful to anyone who wants to bring communities together and enhance the process of friendship formation.
“It seems that the most effective way is to encourage people to form more subcommunities,” Shrivastava says. “The more subcommunities you have, the more they overlap, and the more likely it is that individual members will have more close friendships throughout the organization. People have long thought that this would be one factor, but what we’ve shown is this is probably the only one you have to pay attention to.”
The National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research supported this work.
Source: Rice University