Let's build a simple model to understand (very unrealistically!) the impact of virality on user growth. That is, what is the impact of the spread rate on reaching a target number of users. The model will take several arguments as input: the number of starting users, the spread rate - how many friends a user will recruit, and the end target number of users. The output will be the number of cycles, let's say months, to reach the target. In our case we hope our little startup will IPO after reaching the target!

So if we want to reach, say, 5 million users, how many months will it take if we start with 100 users with a spread rate of 2/user?

[sourcecode language="python"]

Define a procedure that takes three inputs: the starting number of users, the spread

rate (how many new friends each user convinces to join per month),

and the target number, and outputs the number of months needed to reach

(or exceed) the target.

def users_to_ipo(n, spread, target): if n >= target: return 0 else: return 1 + users_to_ipo(n * (1 + spread), spread, target)

let's test some use-cases:

0 more needed, since n already exceeds target

print users_to_ipo(100000, 2, 40000) #>>> 0

after 1 cycle, there will be 100+ (100 * 2) users

print users_to_ipo(1000, 2, 3000) #>>> 1

months needed to match or exceed the target

print users_to_ipo(50000, 2, 150001) #>>> 2

only 13 months to exceed the current population of 7b!

print users_to_ipo(100, 2, 7 * 10 ** 7) #>>> 13

more friends means faster world domination!

print users_to_ipo(15000, 3, 7 * 10 ** 7) #>>> 10

#the original question: print users_to_ipo(100, 2, 5000000) #>>> 10

[/sourcecode]

So it takes 10 months to reach our target of 5MM. This is definitely an optimistic model and one that will need some refining before we secure funding for our venture!