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Are there any resources that look at the computational complexity of running Ethereum on a private network? I am not considering any smart contracts or even transactions. I simply am trying to find the computational cost and perhaps communication cost of implementing Ethereum's consensus protocol (i.e. ensuring each node has the same version.)

If each node is a miner, would this simply be the cost of Proof-of-Work and the logical operation of comparing chain length?

I have not been able to find anything of this sort. I am looking for something such as a complexity O(n) where n is the number of nodes. There has to be some overhead for just implementing the Ethereum blockchain in its simplest form.

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