The power-law model:
what it measures – and what it does not

Anyone who follows Bitcoin charts knows the picture: the price swings wildly, yet on one particular kind of chart it hugs a straight line remarkably closely – over more than fifteen years. That line is known as the “power law”.

The charts are often superbly made. What they rarely explain is what you are actually looking at. That is what we set out to do here – in three steps: What is a logarithmic scale? What is a power law? And what does it tell us about Bitcoin – and what does it not?

Step 1: The same data, three views

The most important ingredient of the model is not Bitcoin but the way the data is drawn. So before we look at the price, here is a neutral worked example with two curves, both of which grow from 1 to roughly 1,000:

• Power law: the quantity grows like time cubed. After 2 time steps it is 8; after 10 it is 1,000.
• Exponential: the quantity doubles with every time step. After 10 steps it is 1,024.

Switch between the three views and watch which curve becomes a straight line, and when:

Normal scaleLogarithmic heightLog–log

On a normal scale both curves look similar. With logarithmic height, the exponential curve becomes a straight line. When both height and time are drawn logarithmically, the power law becomes a straight line.

What this example shows is the core of the entire model:

• On a logarithmic scale, each equal step means not “plus ten” but “times ten”. Enormous differences thus become visible in a single picture.
• When both axes are drawn logarithmically, a power law becomes a straight line. Its slope is the exponent – in the example, 3.
• A power law grows ever more slowly than exponential growth: the percentage increase per year declines over time.

A law found throughout nature

Power laws are not an invention of the Bitcoin world. They appear wherever systems grow and organise themselves:

• Metabolism: the metabolic rate of animals rises roughly with body mass to the power of three quarters – from the mouse to the elephant (Kleiber’s law).
• Earthquakes: strong earthquakes are rarer than weak ones, following a fixed power rule (the Gutenberg–Richter law).
• Cities and language: the size of cities or the frequency of words often follows a power law (Zipf’s law).
• Networks: how strongly networks grow and interconnect can often be described with power laws.

The power-law model applies this idea to Bitcoin: it uses a simple mathematical function to describe how the price has developed over time.

Step 2: The law applied to Bitcoin

The power-law model for Bitcoin plots the price against the time since the first block on 3 January 2009 – both logarithmically. On this chart, past prices lie close to a straight line. A best-fit line is calculated through all the points to date; in the common versions its slope is just under 6. Put simply: in the past, the price has grown roughly in line with time to the power of just under 6.

The approach became known around 2018 and 2019, through the physicist Giovanni Santostasi among others, and a widely read article by Harold Christopher Burger. Network effects are often cited as a possible rationale: with every new user, a network becomes more valuable to all the others – a kind of growth that power laws describe well.

Step 3: Looking back at the past

Two observations make the model interesting – and at the same time call for caution:

• The line describes the past remarkably well. Across many orders of magnitude, the price stays close to it.
• The deviations are nonetheless huge. On a logarithmic chart, a gap that in reality amounts to a multiple looks small. Time and again the price has sat far above and far below the line – for months and years.

Where things stand

The model has been around for some years and has been refitted with every new data point. An honest assessment therefore covers two questions: how well does the line describe the data so far? And how much does its slope change as new data is added? We recalculated the best-fit line ourselves three times – using data only up to 2016, 2020 and 2024:

Charts 3 and 4 use completed years up to 2024; Chart 2 shows the price up to today.

Where today’s price stands relative to the line is shown in Chart 2. We deliberately draw no interpretation from it – such as “cheap” or “expensive”: that would already be a statement about the future price.

What the model cannot do

• A good fit is not proof. Almost any series that grows strongly over a long period can be approximated well by a straight line on log–log axes. That it fits says little about why it fits.
• The chart distorts distances. Log–log axes visually compress large swings. What looks close to the line can be a factor of three or more.
• The result depends on the data selected. Which years are included changes the slope – and with it every conclusion drawn from it.
• There is only one history. A model calibrated on exactly this one history will inevitably describe it well. Whether it describes the future can only be tested in hindsight.
• A cautionary tale: the stock-to-flow model became similarly popular in 2019 and struck many as similarly convincing. From 2022 onwards, the price spent a long time far below what that model suggested. Elegant models can break.

Looking ahead: indicators instead of a forecast

Chart 2 draws the power law up to 2030 – as a mathematical function. The bitcoin price, by contrast, ends today. No one knows whether it will follow the law in future. These signs would indicate that the model no longer holds:

• The price moves persistently further away from the line than it has ever done in the past.
• With each new year of data, the slope changes markedly – the line “wanders”.
• The network-effects rationale cannot be confirmed by other measures, such as actual usage.

What this means for you

The power-law model is an instructive example of how much the choice of presentation matters. Once you understand how a logarithmic scale works, you can read such charts for yourself – and test claims rather than take them on trust.

For your security, what matters is not any price model but this: that your bitcoin truly belong to you, that your recovery has been tested, and that your family knows what to do if the worst happens.

Further reading

• Complex topics, simply explained · Part 1: The four-year cycle
• Bitcoin: A Peer-to-Peer Electronic Cash System