Question

Friendship Paradox

In the year 1991, the sociologist Scott L. Feld made an interesting discovery. He realized that on average, most people have fewer friends than their friends have. This phenomenon is called the friendship paradox. Do some online search. Describe the friendship paradox using graph theory, try to understand and explain the mathematical proof for it. Find two examples for possible applications. Can you think of how this paradox can be used to monitor COVID-19 outbreak?

Answer 1

Friendship Paradox really an interesting point of discussion from the sociological point of view eventhough it has less to do with engineering. Hovever let us proceed. As mentioned in the question, on average, most individual’s friends have more friends than the individual. This phenomenon is called the friendship paradox.

First let us see the concept of friendship. Consider two person A and B. If the person A has friendship with B then its vice versa (person B has friendship with A) is also true. But it may not be always true but usually they are. Some people think they are friends with others but this thought may not always mutual.

Gaph theory of Friendship Paradox

Considering the same indivisuals A and B. They can be represented as the circles (This proccess is called vertical graph theory.) assuming that they are unknowns or little known to each other (They are not friends) as shown in following figure.

Now assume that A and B are got to know each other and became friends. This can be represented graphically by connecting A and B by a straight line.

Let us expand the network by adding more indivisuals.

Mathematical Proof for Friendship paradox

For the given diagram, let V be the number of vertices and E be the number of edges. The social network we want to model is a function of V and E. i.e G = f(V,E). The number of friendship each indivisual vertice v has is d(v).

The number of vertices V = 5

Number of edges E = 7.

Node A has d(A) = 4 friends.

Node B has d(B) = 3 friends.

Node C has d(C) = 2 friends.

Node D has d(D) = 3 friends.

Node E has d(E) = 2 friends.

Application of friendship paradox

• The observation of friendship paradox has been used effectively as a way to predict and slow the course of epidemics and disasters, by using this random selection process to choose individuals to immunize or monitor for infection while avoiding the need for a complex computation of the centrality of all nodes in the network.
• This paradox applies to other charecteristics as well. For an example for an author, his co author may be more likely to have published more books or publication and collaborations
• Similarly this paradox fits well to social medias and social network analysis as well.

Applying paradox to monitor the COVID 19 outbreak

As discussed earlier in the application of friendship paradox in monitoring epidemics, COVID 19 is a kind of epidemics. It can be utilized effectively due to fact that obtained from the study of earlier pandemics. Succh analysis concluded that the friends of a random group of people got infected earlier than the random group. Thus, by applying the friendship paradox and observing the friends of random people it might be possible to recognize contagion outbreaks earlier.

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