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Are there simulations of scenarios taking into consideration that most payments in LN will go into specific nodes (say, Amazon or Ebay)? The majority of payments we make are not towards our friends, but towards big entities that sell stuff.

So the number of nodes making payments will always outweigh the nodes receiving them - this seems like the law of business if you ask me.

I'm not sure how the graph will behave when all the money is always pushed towards few nodes. Have simulations been done for these sort of scenarios?

EDIT

To clarify, my main concern is about how money would flow through hops if, like I said, we're mostly paying these big nodes in the graph. My concern is that a route may be harder to find if the typology of the network pushes all the money towards these specific areas of the graph -- we might end up with not many routing possibilities because most channels in a route might be exhausted in that direction.

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Payment routing could handle this. Any business has operational costs and will have to make outgoing payments.

If your concern is that channels will mostly consist of payments in a single direction (i.e. from customers to larger businesses), and that this will lead to a high frequency of channel closings or on-chain settlements, then we can consider the theory of Six Degrees of Separation.

The theory of Six Degrees of Separation postulates "that all living things and everything else in the world are six or fewer steps away from each other". In regards to the Lightning Network, we can postulate the theory that any user will be able to pay any other user through approximately six other users.

So if you have a source of incoming payments on the Lightning network (i.e. your wage), and the larger business has outgoing payments (i.e. their employee's wages), then through the six degrees of separation you may be able to pay the larger business through already existing payment channels you have open with your contacts.

Successful payment routing should ensure that payment channels operate bidirectionally and that payment channel closing or on-chain settlement should be of a low relative frequency.

Edit: Put Alternatively

Customers may never directly make payment channels with large businesses, but instead create routed payments through payment channels they have open with exchanges or other Lightning Network intermediaries.

Users may create payment channels with an exchange. A large business may create a payment channel with the same exchange. Users make payments to the large business routed through the payment channels they have open with the exchange. The large business decides to sell some bitcoin through the exchange. It makes a payment using the payment channel it already has open with the exchange. This process continues indefinitely such that there is a bidirectional flow of bitcoin along the open payment channels.

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    More than the actual connectivity, I was worried about how money would flow through hops if, like I said, we're mostly paying these big nodes in the graph. My concern is that a route may be harder to find if the typology of the network pushes all the money towards these big nodes. – Luca Matteis Dec 29 '17 at 21:35
  • In the case that these larger nodes are gathering such large amounts of bitcoin, then laws of supply and demand would suggest that prices in bitcoin would fall. Potentially leading to the scenario in which any outgoing payments these larger nodes make, could be large enough to cover the incoming payments. – Matthew Charles Stannard Dec 29 '17 at 21:52
  • I'm not sure you understand my question. I'm talking about channel exhaustion. Nothing to do with how much money people have. – Luca Matteis Dec 29 '17 at 22:08
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    Sure, but my answer was that any outgoing payments a larger node makes would create opportunities to prevent channel exhaustion. Or, through payment routing there will always another open payment channel that you can use. Everybody, including large businesses, have incoming and outgoing payments. – Matthew Charles Stannard Dec 29 '17 at 22:12
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    Amazon may want to sell their bitcoin for dollars, this could create a Lightning Network payment to an exchange. An Amazon customer may get payed in dollars but wants to purchase their product using bitcoin, this could create an Lightning Network payment to this customer from the exchange... etc. Just a hypothetical example. – Matthew Charles Stannard Dec 29 '17 at 22:27
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I think a solution to this problem comes in the form of developing routing algorithms that benefit the network as a whole.

Routing algorithms will look for paths where channels need help in rebalancing themselves -- they do this to help the network as a whole, so that the probability of finding a path is always high for everyone.

So even though money flows towards specific areas, the paths for sending the money will make sure everything is balanced out and that channels are always in a "healthy state" (ready to be used in both directions).

In other words, say I need to pay the Amazon node, instead of taking the shortest path there (maybe 1 hop), I will look at paths that are with unhealthy channel states -- this is incentivized because passing over these unhealthy channels will result in more fees grabbed along the way for myself.

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