No Bubbles, No Power Law?

The bubbles helped lock it in, but they are fading with time

As we await the next Bitcoin bubble, that many think is overdue after a couple of attempts to significantly break out last year…someone asked on TwiXter, “if there were no bubbles would there still be a power law?”

Yes, with high confidence there would be. What in fact we have seen is that the bubbles have stress-tested the power law and in fact have helped to lock it in more tightly over time. And the bubbles, and volatility in general, have been decreasing with time.

Figure 1 is a histogram showing the strength of the bubbles of 2011, 2013, 2017, and 2021; these are the largest bubbles to date.

The bubble power on the y-axis is the average percent deviation over the duration of the bubble, and each of the histogram bars is labeled by the inverse of age. The decline of the bubble power is almost proportional to the inverse age, which would indicate a reciprocal power law of index -1. The best fit regression is a power law of index -0.83.

Figure 1. Histogram the major Bitcoin bubble power, that is defined as the average percent rise above the median quantile regression within each bubble’s lifetime above that level. The bubble power has fallen monotonically as the 0.83 power of the inverse of the age of Bitcoin.

Bubbles push up the overall power law index and then troughs pull it down below long term behavior. The power law has become more stable with time and has locked into a power law index (Price ~ Age^k) between k = 5 and k = 6. This is in accordance with Metcalfe’s law for networks, and the value has been stable during the last decade, even as bubble power has decreased.

The figure below shows how the power law index (also called ‘slope’ when price and age are plotted in a log-log fashion) has evolved as a function of age. Block years of 52,500 blocks each are used in this plot. The power law index is shown as the blue line and it oscillates from k = 3 to above k = 6, back to k = 4 then returns to k = 6, around age 6, and after that it settles down toward a stable value. The orange line shows log10 of block monthly interval prices along with a best fit regression line of index k = 5.41 in red. And the green line is the uncertainty (standard error) in the power law slope that started out large, above 1.0, but has dropped to below 0.1. The current power law index (slope) is very well determined.

And throughout all this time, the R2 and F-test statistics of power law fits have continued to increase.

Figure 2 shows the evolution of the best fit power law regression slope parameter, that oscillated in early years but has settled down to over k = 5 for most of the last decade (blue curve, k slope). The green curve shows the uncertainty in the index (slope) which started out above 1.0 and has fallen to below 0.1. Also shown are the log10 price history (orange) and current best fit power law model (red) with index 5.41 when block years are used. Current age is 17 block years since we are one year past the April 2024 fourth halving.

Bubbles didn’t create the power law and their moderation does not end it. With larger and larger investment capital flows and a much deeper market cap reservoir one expects lower long term volatility and less powerful bubbles (in relative terms, not in absolute price variation terms).

The power law core and bubble zones can be fit by two well separated Gaussians of the log residuals from a power law (thus, log-normal distributions). These are separated by a linear price ratio of nearly a factor of 3 (there is a 0.44 in log10 terms difference in their centroids). And the bubble zone width is 3 times greater in log terms than that of the core power law.

Figure 3. Best fit Gaussians for the residuals (green, left chart, corresponds to power law core) and above (red, right chart, bubble zone) the median power law quantile regression. The two distributions overlap somewhat but their centroids are separated by 0.44 in log10 terms, a ratio of 2.75 in linear values. The standard deviation width of the bubble zone is nearly 3 times that of the power law core.

The bubbles in the power law index behavior (that result from the price bubbles) can also be fit with log periodic power law models as shown for the first three major bubbles in figure 4. Both the amplitude and frequency of these have been coming down considerably with time.

Figure 4. Log periodic power law (LPPL) fits (dashed lines) to the first three bubbles of 2011, 2013, 2017 (solid lines). The LPPL model of Sornette can be fit to bubbles in price as well. The amplitudes have come down considerably and the interval has grown as well.

The bubbles in the power law index behavior (that result from the price bubbles) can also be fit with log periodic power law models as shown for the first three major bubbles in figure 4. Both the amplitude and frequency of these have been coming down considerably with time.

Bitcoin is a meta network of interacting and mutually reinforcing power laws.

What the bubbles do is help to anneal in the power law. The analogy is similar to the process of tempering steel by raising and lowering the temperature repeatedly, or as well to the simulated annealing method used in some computationally intensive optimization problems to find the lowest energy (most stable) solution within a space of high dimensionality.

The bubbles seem to have helped lock in the power law, but they did not create the power law and as they fade in relative volatility, the power law has persisted strongly. Tea without bubbles is pure tea. Bitcoin without bubbles would be pure power law plus moderate volatility.

Bitcoin’s suitability as Third Millennial monetary technology grows with time.