Unexpectedly, Nakamoto's coefficient was not mentioned by Bitcoin's creator. It was published in 2017 by Balaji S.Srinivasan, former CTO (Chief Technical Officer) at Coinbase and General Partner at the renowned Andreessen Horowitz fund (known as a16z). This coefficient is important because blockchain users and developers need to be sure that any valid transaction they submit to the network will be included in a block and then confirmed by consensus.

If a group of consensus nodes (validators on the network) is compromised or acts in a malicious, coordinated way, it may try to modify or prevent the network from reaching consensus on new blocks. The Nakamoto coefficient is a way of measuring a blockchain's resilience to such behavior. In other words, it enables us to estimate its vulnerability to the risk of a malicious consensus.

In simple terms, this coefficient represents the number of nodes (validators on a network) that must intentionally want to prevent a blockchain from functioning correctly. The higher this coefficient is in relation to the total number of validators, the lower the risk of corruption, and therefore the more decentralized the network becomes. It is calculated on the basis of 6 indicators:

  • Mining (by reward)
  • Customer (by code base)
  • Developers (by software commits)
  • Nodes (by number)
  • Exchanges (by transaction volume)
  • Ownership (by address)

The level of risk differs according to the underlying consensus model (proof-of-stake or proof-of-work) of the blockchain network studied.

Nakamoto coefficient - Proof-of-Work (PoW)

In Bitcoin, and in all blockchains operating under the Proof-of-Work consensus, any person or group of people with 51% of the network's computing power (hash rate) can control it at will. In other words, this person or group can double-spend and alter the course of the chain. For example, on Bitcoin, two mining pools (grouping together several miners to accumulate computing power) alone concentrate 56% of computing power.

Hash rate distribution on Bitcoin (May 28, 2024)
Source: Blockchair.com

There are several possible interpretations of Nakamoto's coefficient on Bitcoin. We could say that if Foundry USA Pool and AntPool (graph above) got together to coordinate an attack on the network, they would possess the computing power needed to corrupt Bitcoin (i.e. over 51% of the computing power). Nevertheless, behind each of these pools are miners, individuals like you and me, who have decided to contribute to the system by pooling their computing power in order to have a better chance of validating a block and thus being rewarded with Bitcoins. If they validate a block together, they share the reward in proportion to each other's contribution.

This mutualization is made possible by a function built into the Bitcoin client software that enables a mining group to exploit blocks cooperatively by pooling the computing power of a group of miners.

To put it in layman's terms, we could imagine it as if you logged on to a server with a username, password and payment address, and then started working your machine to solve calculations with other machines from other users on the same server. Once the algorithmic equation of the block has been found together, the reward is divided between the different machines according to the amount of work done (per proof of work).

Let's return to the Nakamoto coefficient. As mentioned above, we could estimate that this coefficient is 2 by assembling Foundry USA Pool and AntPool, which together hold 56% of total computing power. In other words, all it would take is for two players to get together to corrupt the network. But as miners are free to leave the pools, they have the ability, at any moment, not to follow the will of certain other miners within these pools who would like to corrupt the system. As a result, the coefficient for Bitcoin is generally calculated by the number of individual nodes. At the time of writing, there are just over 19,000 such nodes.

Number of nodes on the Bitcoin network
Source : Bitnodes

The coefficient is therefore generally estimated at 51% of the total number of nodes on the network, or 9601. However, this number should be taken with a grain of salt, as not every node has exactly the same computing power, depending on the number and quality of the machines accumulated by the miners behind them. In other words, a miner A with 2 mining machines owning its own node will have far less computing power than a set of miners in a pool B with 100 machines. By taking the nodes with the greatest computing capacity, the number of 9061 would therefore be reduced, since this number is initially calculated as if each node had the same computing power. However, even with this not inconsiderable subtlety in mind, Bitcoin remains the most decentralized network by far.

Here, we're interested in Nakamoto's coefficient for a proof-of-work blockchain, but it works quite differently for those operating under proof-of-stake.

Nakamoto Coefficient - Proof-of-Stake (PoS )

In proof-of-stake blockchain networks, the Nakamoto coefficient that must not be exceeded is different from that seen previously under proof-of-work.

For a proof-of-stake network to function correctly, a node must not hold more than 33.4%, or one-third, (compared with 51% for proof-of-work) of the locked supply on the network, otherwise it could single-handedly corrupt the chain. Thus, preventing a node from holding more than 33.4% of the voting rights is essential to maintain the smooth running of the network and ensure its resistance to censorship. On the other hand, using the same procedure as for proof-of-work, nodes can pool their "stake" to jointly hold more than 33.4% of voting rights.

Note: It's important to emphasize that an attacker or group of attackers doesn't need to hold a third of the total outstanding supply, but rather a third of the active stake. Indeed, most investors buy and hold their crypto-currencies (which operate on a PoS blockchain) in wallet without staking them on the network (i.e. locking them) in order to become validators and earn returns.

Example: let's imagine that only 25% of the total supply is locked (staked / staked), so the amount required to corrupt the network is only 1/12 of the total supply, i.e. 1/3 x 25%.

We understand here that the "weight" or voting "power" of a validator is proportional to the amount of participation associated with it. As a result, validators with more at stake (and therefore more staked cryptocurrencies) can have a greater influence on the outcome of the consensus process, and block the production, securing and validation of validators with less at stake.

Let's take a few concrete examples:

Nakamoto coefficient of various PoS blockchains

Typically, looking at the graph above, we can see that the Nakamoto coefficient is 31 for the Solana network. This means that the minimum number of validators that would have to agree to censor the network is 31. In other words, the 31 addresses holding the most staked SOLs on the network would have to agree and unite to reach 33.4% of the total staked supply and thus be able to censor the network. We could have the same intellectual path for the Avalanche, Binance and Polygon networks, but with their associated coefficients (see graph above).

An increase in this coefficient over time is a sign of good health for network decentralization. Conversely, a decrease.

Note: Validators on a network have no interest in getting together to hold 33.4% of the network. This could lead to a total loss of investor confidence in the network, and therefore in the associated underlying cryptocurrency. This could lead to massive sell-offs and a drastic fall in the asset's price. As a result, validators would still hold 33.4% of the network (through holding the staked cryptocurrency supply), but its economic value would automatically become zero.

Ultimately, Nakamoto's coefficient enables us to appreciate the decentralized nature of a blockchain network. A feature so important to the creator of Bitcoin. By contrast, the crypto-blockchain initiatives that quickly followed in the wake of Satoshi's libertarian philosophy are entrepreneurial in nature. Over 20,000 crypto-currencies have been added to bitcoin in the space of 10 years, promising decentralization, social revolution and a new trustless economic paradigm. A 3.0 marketing veneer that, in many cases, has been cleverly and dishonestly introduced into the white-papers of crypto-blockchain projects leaving fast-money-hungry novices to be purged with unfungible sauce. Thanks to Nakamoto's coefficient, we can now at least quickly appreciate the decentralized nature of a blockchain, although we need to appreciate other elements to ensure the reliability of a distributed network.