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|
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
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.
References
[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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