Warning: the graph below looks like a pyramid but we just made a post explaining Bitcoin is not a pyramid scheme. It simply shows how the network grows, that is similar to a curbed exponential.

Power-law adoption is more sustainable than exponential (S-curve) adoption.
What is the difference?
Exponential growth works like this. Imagine a bacterium that doubles every five minutes. You start with one bacterium. After five minutes there are two. After ten minutes there are four, then eight, sixteen, and so on. The key feature of exponential growth is that the growth rate stays constant in time.
Now imagine these bacteria live in a closed container with limited resources. Suppose it takes three days for the bacteria to fill half of the container. How long will it take to fill the remaining half?
Only five minutes.
That is the paradox of exponential growth. Everything appears manageable for a long time, and then the system suddenly runs out of resources. Once the container is full, the bacteria have no food and the colony collapses. The system did not allocate resources in a sustainable way.
Many processes that follow exponential growth behave like this: they grow extremely fast and then crash.
Bitcoin, fortunately, does not follow that pattern. Its adoption follows something closer to a power law.
A power law still allows growth to depend on the number of people already in the system, but it contains a natural curbing factor proportional to 1/t, where t is the age of the system. As the system gets older, growth gradually slows down.
In simplified form, the change in the number of Bitcoin adopters can be written as
dN/dt = 3N / t
where
N is the number of people using Bitcoin
t is the age of Bitcoin (for example in days)
This equation means that adoption still benefits from network effects—more users attract more users—but the growth rate slowly decreases as the system matures.
Let’s illustrate this with a simple example.
Start with Satoshi as the first user.
On the first day:
3 × 1 / 1 = 3
So roughly three additional people join the network (Hal Finney can be one of them). Now we have 4 users.
On the second day:
3 × 4 / 2 = 6
Six more people join, bringing the total to 10 users.
On the third day:
3 × 10 / 3 = 10
Ten new users join, bringing the total to 20 users.
The network continues to grow, but the 1/t factor progressively slows the acceleration. Growth remains strong, but it becomes increasingly stable as the system matures.
Without that 1/t term, growth would explode exponentially and quickly become unsustainable.
The power law therefore describes a system where network effects drive expansion, but time naturally stabilizes the growth rate. This is why power-law adoption is far more sustainable than the explosive boom-and-bust dynamics typical of exponential systems.
Bitcoin’s long-term adoption appears to follow exactly this type of process.