I am expecting a linear relationship between difficulty and hash rate. If I generate a scatter plot between the two variables I can see a linear relationship, but also values that are not on the line (in red frame of figure). I tried to correlate this with protocol changes, but this was not successful. Why are this values of the line?

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The data is borrowed from https://etherscan.io/charts The information about the forks I have found here: http://list.wiki/Ethereum_Forks

You can find the code to generate the plot in https://gist.github.com/Slebi/1e1e614dcccc7e04424da5126c80c0a6


Without looking in too much detail, I'm going to speculate that it's due to the difficulty adjustments related to the originally planned Difficulty Bomb. (See: What is the "difficulty bomb" and what is the goal of it?)

The EtherScan chart to look at would be the one showing how the block time increased in a series of steps (due to the increased difficulty), up until the point of the Difficulty Bomb's cancellation (EIP-649), which is when it shifted back to your yellow path.

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