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They spin up each node on your device, not on digital ocean. It doesn't cost anything to run software on your phone.
Many lightning wallets run a node on your phone, including breeze, phoenix, and simple bitcoin wallet. Mutiny just runs several. It helps that a full lightning node does not have huge storage and bandwidth requirements like a full bitcoin node does.
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How can they make this work financially?
They claim to "spin up a node for every channel".
Say every node is on DigitalOcean, where one instance costs ~$11 PER MONTH.
How on earth are they paying for that?