### Network Effect

Posted:

**Wed Aug 06, 2014 12:26 am**Some time last year I thought of this problem and posted it on a wall at my workplace, but nobody wanted to solve it... and then I remembered this forum. So here I am with this problem. Enjoy! (Later I find out there are papers written about this problem, but I couldn't find one which gave an exact formula.)

Some people attend a networking event, which is a fancy way of saying "a place where people shake hands." Oddly enough, each person initiates a handshake with exactly one other person over the course of the event; this other person is randomly chosen with uniform probability -- and of course can not be himself.

We consider two people to have shaken hands if either or both of them initiated a handshake with the other. Thus many people could have shaken hands with the same person, and two people may have shaken hands with each other twice. We consider two people A and B to belong to the same network if they have shaken hands, or if A has shaken hands with someone belonging to the same network as B.

For example, suppose 7 people attend. We denote each person using a number. The handshakes may be as follows:

1 shakes hands with 2

2 shakes hands with 1

3 shakes hands with 7

4 shakes hands with 5

5 shakes hands with 1

6 shakes hands with 5

7 shakes hands with 3

This results in the networks {1, 2, 4, 5, 6} and {3, 7}.

Suppose 100 people attend the event. To 6 decimal places, what is the expected number of people in the largest network?