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What do mathematicians say about the ‘friendship paradox’?

by Sushree Mohanty

What do mathematicians say about the ‘friendship paradox’?

June 15, 2021

How many friends do you have? Two, three, or are you a social butterfly? How many friends do your friends have? More than you or less than you? Probably more, that's what the "Friendship Paradox" says. Mathematicians have developed a theory that takes this paradox further than 'averages'.

It's probably not appropriate to compare ourselves to our friends, but the friendship paradox is true!

-George Cantwell, a postdoctoral fellow at the Santa Fe Institute

Friendship Paradox

In the pages of a dictionary, friendship means a relationship between individuals who are friends or the state of being close enough to share feelings, thoughts, value each other, and in the process, strengthen the bond. However, there is a theory of the “Friendship paradox”. It says that your friends have more friends than you do.

In 1991, American sociologist Scott Feld, a scholar of mathematical sociology and social networks, was first to explain the idea of the “friendship paradox”. An article on the same was published under the title “Why Your Friends Have More Friends Than You Do” in a journal.

Researchers found a general idea that the number of friends our friends have is greater than the number of friends we have in-person through mathematical calculations.

However, taking out an average can also be highly misleading because of the unknown sample. George Cantwell said, “It can be biased and can also lead to the failure of the theory based on people’s experiences”.

freindship-paradox-scoolbuzz
Researchers found a general idea that the number of friends our friends have is greater than the number of friends we have in-person through mathematical calculations.

How is the 'friendship paradox' popularity equated?

Let’s imagine a person with two or three friends in contrast with a person who has a group of friends. Friends often shape our behavior and the way we deal with social issues. 

Society is a web of social relationships, and man is always found to be seeking a company. Therefore, we are more likely to be friends with someone with whom our values, thoughts, and actions match.

However, on the other side of the coin, we seek to be friends with a social butterfly. So here, we as individuals are one of the hundreds of friends’ the social butterfly possesses. 

Thinking of an introverted individual who is happy with just two or three friends can also become a friend of that social butterfly. However, the introverted individual has a social butterfly friend who has more friends in this social bubble.

But, not everyone is dependent upon the behavior of an individual for friendship. So, this is taken on average.

cartoon the friendship paradox evoliteracy 2016
An introverted individual who is happy with just two or three friends can also become a friend of that social butterfly.

The Assumptions

George Cantwell, with his colleagues, analyzed and developed new mathematical equations. These might help researchers to understand the paradox of friendship found in real-world social networks. 

Mathematical researchers and sociologists based their equations on assumptions relating to real-world studies. The first assumption says that there’s a significant degree of variation in the number of friends an individual has. This depends upon the social network in which he/she is living.  

The second assumption is that popular people are always found in clusters with other popular people or friends. In contrast, we can find unpopular people are more likely to be friends with unpopular people.

Generalized friendship paradox

The researchers, however, developed a new mathematical theory as a variation of the friendship paradox known as the “generalized friendship paradox”.

This new theory, on average, states that our friends are not only more popular but richer and better looking as well. But, again, this is because we look up to them. 

These new equations that emphasized the assumptions mentioned earlier explains approximately 95% of the variation faced in real-world situations.

Mathematical equations show that this friendship paradox tends to strengthen social networks made up of people with differences in popularities. 

 

A person with two friends in the same social network where the person’s friend has 100 more friends in general, here the friendship paradox will be stronger than a social butterfly or person having ten friends.

Again researchers have believed that their social circle samples are biased samples of the population. Again, however, it isn’t clear how bias samples may turn the equations of friendship and emotional bonds in specific cases. 

Certain aspects of society depend upon empirical data and even some mathematical equations that can help explain the other aspects of society. For example, the friendship notion of emotional ties often tends to get stronger with time, and hence our close friends become our family.

However, statistically, friends with whom we are in close contact and who have shared an emotional bond are more likely to be in close contact with other friends or many other people outside our social circles. 

Concluding the theory, Cantwell opined that in most cases, “it’s probably not appropriate to compare ourselves to our friends”, but the friendship paradox is true.

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