Mining centralisation correlations

4th April 2015

0. Introduction
Yesterday I had a twitter conversation with @Bitcoin_Watcher  about whether one could expect centralisation to increase or decrease with changes in price. This is something I'd never thought of before, but eventually I hypothesised that increases in network hashrate (and difficulty) or decreases in price might lead to increased centralisation as miners move to larger pools to reduce income variance.

This sort of armchair guesswork is not very satisfying, so I decided to see if historical data supported my guess.

1. Data
I decided to use the Theil index as a measure of centralisation as it's easier to calculate, but both the Theil and Gini indices have highly correlated in the past so it doesn't matter which one is chosen. I did not use either of the mining centralisation indices since they only measure inequality between the larger and smaller block makers,rather than inequality generally. I of course used the log of the network hashrate and the USD price.

2. Correlations, cross correlation and differencing
From here I performed the usual calculations - cross correlations after various amounts of differencing or detrending in order to make the series' stationary. However, I found no correlations between the stationary data. 

As a last try, I detrended the log of the USD price using the by subtracting the estimate of the linear model:

detrended log price = log price - (weeks * a + b)

This resulted in something interesting:

The Theil index and the detrended log USD price correlate quite closely. The highest cross correlation coefficients were when the Theil index lagged by zero, one or two weeks (0.84, 0.84, 0.83) or lead by one week (0.83). 

Assuming a lag of zero, the result is a linear model with correlation coefficient of 0.84 and an r-squared of 0.7:

Theil index = 0.045 * Detrended log USD price + 0.243

            = 0.04484 * log(USD price) - 2.2e-9 * unixtime + 3.0594

So, I did some minimal data manipulation, and ended up showing the opposite of what I thought was the case - that price correlates with mining inequality. I'm not completely convinced that this is a significant relationship - the detrending I used is not a typical method, and I haven't checked to see if overfitting is a possibility. However, assuming the results are (and will continue to be) valid, one of the following may be true:

• Exponentially increasing USD price causes increasing mining pool centralisation and exponentially decreasing USD price causes decreasing mining pool centralisation. USD price changes lead mining pool centralisation changes by zero, one or two weeks.
• Increasing mining pool centralisation exponentially increases USD price and decreasing mining pool centralisation exponentially decreases USD price.  Mining pool centralisation leads price changes by zero or one week.
• Some other unknown factor is causing both effects.
• An interaction between price and another factor, or centralisation and another factor, is causing the correlation.

3. Another approach

The last point in part 2 lead me to consider my initial premise again - that losses of income might cause miners to move to larger, low variance pools. An increase in price might not lead to a decrease in centralisation if the network hashrate increases at a pace that makes the price increase irrelevant. 

So rather than just a relationship between price and centralisation, a centralisation might be caused by an interaction between price and the network hashrate. The advantage of this is that we don't need any unusual detrending methods, just a simple linear model of:

Theil index = log(hashrate)*a + log(price)*b + log(hashrate)*log(price)*c + d

where a =  -0.0438, b = -0.104, c = 0.004731, d = 1.463

The log of the Theil index could be used to ensure the model only produces positive values. I only thought of this later.

This model has well behaved residuals, highly significant variables and no unusual methods. The correlation coefficient is 0.85, and r-squared = 0.73, actually a little better than the detrended log price model.

The confidence intervals for the mean and the data are similar to those from the previous detrended log price model, but result from a much more intuitively understandable interaction between price and hashrate. This model is simply more informative, and tells us that a high price alone correlates with high centralisation, and a high hashrate alone correlates with lower centralisation, and a higher hashrate at the same time as a higher price has tend to average centralisation.

3. Summary

Based on the hashrate and price interaction model, 

• A high price alone correlates with high centralisation
• A high hashrate alone correlates with lower centralisation, and
• A high hashrate at the same time as a high price has tended to average centralisation

The fact of an interaction between price and hashrate correlating with centralisation is a bit more acceptable to me, and the points above make it more understandable. However it is once again the opposite of what I expected.

I had thought high hashrate and low price would correlate with higher centralisation; I realise now that I was really thinking of rapid changes in price or hashrate causing changes in centralisation, rather than just high prices or hashrate. So, does anyone have any explanations why these correlations exist?

Further, I still don't follow what @Bitcoin_Watcher was talking about ("Economies of scale" relating to centralisation). Anyone have any clear explanations? I did google "Economies of scale" and inequality (as suggested by  @Bitcoin_Watcher but didn't find any clear explanation of the claim that an increasing number of miners would decrease centralisation.

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