Friends at infinity

Probability Level 4

Take a simple model of a social network where friendships form at random between individuals. Each person forms some number of friendships with other people, kik_i. The average number of friendships that any given person makes is then 1Niki=k\frac{1}{N}\sum\limits_ik_i = \langle k \rangle.

We call a friendship island (FI) a group of people such that everyone in the FI can reach anyone else in the FI by passing a note through mutual friends. If two people cannot send notes through a series of mutual friends, they must be in different FI.

At some value of k\langle k \rangle, kc\langle k\rangle_c, the expected size of the largest FI becomes \infty. What is the value of kc\langle k \rangle_c?

Notes and assumptions

  • There are infinitely many people in the population.

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