# Friends at infinity

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, $$k_i$$. The average number of friendships that any given person makes is then $$\frac{1}{N}\sum\limits_ik_i = \langle k \rangle$$.

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

Notes and assumptions

• There are infinitely many people in the population.
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