new to this forum and studying network/graph theory on my own. Math has never been my strong point, and I need help with how the author came to this equation:

Imagine a network of people. Suppose that for some small probability p, each common friend that two people have gives them an independent probability p of forming a link each day. So if two people have k friends in common, the probability they fail to form a link on any given day is (1 - p)^k: this is because each common friend fails to cause the link to form with probability 1 - p, and these k trials are independent. Since (1 - p)^k is the probability the link fails to form on a given day, the probability that it does form, according to our simple baseline model, is : Tbaseline(k) = 1 - (1 - p)^k.

Why isn't it p^k if p is the probability of a link forming and k is # common friends?

p^k is the probability that a link is created with all of the k common friends. The probability of k equally probable independent events to occur is , and the probability of the occurance of none of these events are therefore . And now to the probability of forming at least one link, well thats just the complement to the probability of not forming a link at all: .

I don't know if this made it any clearer, but I hope so