The hash rate, as a power law

What the hash rate is

Miners compete to add the next block by brute force. There is no calculation to solve and no shortcut: each miner takes the block it wants to add, adds a number, and runs it through a hash function, which returns what is in effect a random number. If that number is below a target, the block is valid; if not, change the number and try again. It is closer to rolling dice trillions of times a second than to working anything out. The hash rate is how many of those guesses the whole network makes each second, measured in hashes per second. Today that runs to zettahashes: a zettahash is 10²¹, a thousand billion billion. Nobody reports the figure: it is estimated from how hard the target is (the difficulty) and how fast blocks are actually found, so a single day’s figure is noisy. This page uses weekly averages.

It matters because it is the cost of attacking the network. To rewrite recent blocks, an attacker would need more computing power than everyone else combined, and every rise in the hash rate raises that bar. That work is paid for in two ways. The first is the block reward: a fixed number of new bitcoin per block (3.125 since April 2024), halved about every four years, whose value to the miner is that number times bitcoin’s price in dollars or other currency, the way most participants judge it. The second is the fees on the transactions the block picks up from the mempool, the queue of transactions waiting to be confirmed. A higher price can mean higher fees, but not necessarily: fees depend mostly on how crowded the mempool is and how urgently people want their transactions confirmed. Both streams, and so the hash rate miners can afford, rise and fall with the price.

A law that keeps slowing

A power law grows quickly at first and more slowly later, but at a steady rate in the right units: the exponent should be the same whatever stretch you fit. Fit the hash rate from three starting years and the exponent falls each time.

Most of the early speed came from the machines. Mining moved from ordinary processors to graphics cards to chips built for nothing else, and each step multiplied the hashes a dollar of electricity could buy. Those leaps have run their course: chips still improve, but by smaller steps, and the hash rate now depends more on how much miners spend than on leaps in efficiency. That is why the exponent fell, and why the hash rate today sits well below a line drawn through its fast early years.

So is it a power law?

Not one power law. Bitcoin’s price, fitted the same way, keeps roughly the same exponent as the starting year moves; the hash rate’s falls steadily. Here are both, from the same starting years:

This page’s reading: the hash rate is not a law of its own. It is downstream of the price. A miner’s revenue is the reward and fees, valued at the price, and over time miners spend roughly what mining pays on machines and electricity. When chips were getting dramatically better each generation, the same spending bought ever more hashes, and the hash rate raced ahead of the price. As those gains shrink, the hash rate grows more nearly in step with what is spent on it, which follows the price. If that is right, the hash rate’s exponent should settle near the price’s, plus whatever chips still improve, rather than keep falling, and the next few years of data will test it. It is a reading of the record, not a law; the halvings, which cut the reward in half every four years, pull the other way.

Price and hash rate

Each dot below is a month: bitcoin’s price against the hash rate that month, both on log scales. A straight band of dots means one grows as a power of the other.

Which moves first?

Take each month’s change over the year before, in price and in hash rate, and line the two up with one shifted against the other. The taller a bar, the more closely the two changes track at that shift.

That order is what the economics would predict. A higher price raises what each block pays, miners buy and plug in more machines, and the new machines take months to order, ship and power up. The hash rate is the security the price pays for, and it follows the price; it does not forecast it. A rising hash rate says the network is getting harder to attack, and that miners expect mining to pay. It says nothing about where the price goes next.