Tuesday, January 31, 2012

The Friendship Paradox

It is the phenomenon first  observed by the sociologist Scott L. Feld in 1991 and he explains the same in his paper "Why your friends have more friends than you do" that,

 Most people have fewer friends than their friends have, on average. 

It’s not specifically about friendship, but a mathematical fact about any relation which is symmetrical and which varies across a population.
It can be explained as a form of sampling bias in which people with greater numbers of friends have an increased likelihood of being observed among one's own friends.

Assuming that a social network is represented by an undirected graph , where the set V of vertices corresponds to the people in the social network, and the set E of edges corresponds to the friendship relation between pairs of people. That is, friendship is a symmetric relation: if X is a friend of Y, then Y is a friend of X. the average number of friends of a person in the social network can be modeled as the average of the degrees of the vertices in the graph.Then the average number μ of friends of a random person in the graph is 

   μ =  2|E| / V
The average number of friends that a typical friend has can be modeled by choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint. That is, mathematically, it is

sum of d(v) for every uv in E / 2|E|  =    μ + σ2/ μ

where σ2 is the variance of the degrees in the graph. For a graph that has vertices of varying degrees (as is typical for social networks), both μ and σ2 are positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.
After this analysis, Feld goes on to make some more qualitative assumptions about the statistical correlation between the number of friends that two friends have, based on theories of social networks such as assortative mixing, and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average of their friends' numbers of friends.

However, this conclusion is not a mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from a large complete graph) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees.

'Friendship Paradox' May Help Predict Spread of Infectious Disease

Nicholas Christakis, professor of medicine, at Harvard University, and James Fowler, professor of medical genetics and political science at the University of California, San Diego, used the paradox to study the 2009 flu epidemic among 744 students. The findings, the researchers say, point to a novel method for early detection of contagious outbreaks.

[1] Zuckerman, Ezra W.; John T. Jost (2001). “What Makes You Think You’re So Popular? Self Evaluation Maintenance and the Subjective Side of the “Friendship Paradox”“. Social Psychology Quarterly 64(3): 207–223. doi:10.2307/3090112.

[2] Nicholas A. Christakis, James H. Fowler. Social Network Sensors for Early Detection of Contagious Outbreaks.PLoS ONE, 5(9): e12948 DOI:

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