How many gatherings of a billion people have you attended?
Nobody actually has direct relationships with a billion people. That's why there is no need to "solve" the "problem" of how to centrally coordinate a network with a billion nodes. The coordination happens spontaneously as people voluntarily interact with the people they directly know. In economics, this mechanism is called a "free market".
But for sake of argument, let's move N down to 10,000 (a small city or large crowd). How do you manage that without hierarchical organization and government?
So you have a billion Facebook friends?
> commerce, et cetera.
So you go to a billion different grocery stores? A billion different drugstores? A billion different restaurants?
The fact that, ultimately, there might be a billion or more people involved in the processes that bring you your food, household items, etc., or visiting the same social media sites that you do, does not mean you interact with all of those people directly. Nobody does. All of that activity is coordinated spontaneously, driven by individual voluntary interactions involving much smaller numbers of people.
In fact, talking about the social media case, centralization of power, as in the case of Facebook, has arguably done more harm than good.
> let's move N down to 10,000 (a small city or large crowd). How do you manage that without hierarchical organization and government?
You will most likely have hierarchical organization in these cases. But hierarchical organization is not the same thing as government. Hierarchical organizations evolve all the time through the voluntary interactions of people. Governments impose their rules on people by force. Those are very different things.
The functional health of an organization is directly determined by how closely its constituents are following a loose language of agreed-upon behaviors -- protocols. If you pretend such protocols don't exist, or explicitly refuse to enforce them, your organization will simply disintegrate.
This can work for large N because people fill roles. If you travel today to some other place and want to buy bread, you don't need to know the name of a bread seller in that place. You look for a store that looks like it sells food and ask if they have bread. If they don't, they will probably be able to direct you to a store that does.
If you do something wrong, this can either be tracked by something you're required to keep on your person such as an ID or a tattoo, or by some commonly accessible records.
Since this is a tech site, you can think of it like interactions between objects in object-oriented programming. A billion objects can interact with each other in all sorts of ways without requiring some central manager.
Somewhere, sometime, an true anarchist will complain about that.
I don't think most anarchists would complain about senior bakers, or senior/elder/educated people generally, sharing advice, if that's what you mean, as long as all relationships remain voluntary.
See the stateless democracy in Rojava, which is perfectly compatible with anarchism:
https://iasgazette.com/2021/01/14/democracy-of-the-stateless...
The day that CENTCOM leaves Syria is the day that the mythical anarchist paradise of Rojava ceases to exist.
Great, you've considered the case of N ~= 100. Don't need a state to manage that.
How about N ~= 1,000,000,000?
Anarchism (and any fully decentralized protocol) "works" for small N because the average node can keep all the other nodes of the society in memory. If you need something, you know exactly whom to ask. And if you do something wrong, everyone will remember and hold you accountable. This plainly doesn't work with millions of nodes. Same principle why Bitcoin will never take off without hacking on traditional centralized finance mechanisms like Lightning; at some N, the resources required for individual nodes to maintain an accurate representation of the entire network becomes impractical (either impossible or just way too slow).