Extending my previous article, here, I discuss whether PoW, PoS, and hybrid protocol-based cryptocurrencies could be considered complex or chaotic systems, risking entering a chaotic regime, with catastrophic consequences.

Before proceeding, it must be understood that there is no concise definition of a complex system, which has been oft identified with complicated or random (stochastic) systems. Besides, the term “complexity” has been so much used by so many authors, both scientific and non-scientific, that the word has, unfortunately, almost lost its meaning.

Another remark that should be made is on attempts of understanding chaoticity in terms of high volatility. Volatility is usually defined as a statistical measure of dispersion around the average of any random variable such as market parameters etc. assuming that its price follows a Gaussian random walk or a similar distribution and that some sort of regression toward the mean will always happen. It should be noticed, however, that Mandelbrot showed long ago that financial markets are characterised by “wild randomness,” in the sense that the price changes do not follow a Gaussian distribution, but rather Lévy stable distributions having infinite variance. Therefore, if some market enters a chaotic regime, there will be nothing predictable about it, and the prices can go anywhere. In other words, ordinary volatility is expected and even desirable to some degree as it provides profitable opportunities; chaoticity is deadly.

Consequently, as done in the previous article, the systems’ complexities was analysed by means of the Crutchfield’s Statistical measure of complexity, as it was shown that it was the measure that best captures the qualitative notion of complexity of systems.

As shown in the previous article, Bitcoin blockchain can be seen as an infinite-string-production є-machine (see figure above) that oscillates between two states about every 10 minutes: the mining state (m) and the broadcasting state (b), in which the desired nonce was found and the validated block is broadcast to the P2P network for inclusion into the blockchain.




These extremely low statistical complexity results lead us to the conclusion that PoW-based blockchains, in general, can hardly be considered complex, confirming and extending to all these cryptocurrencies Nakamoto’s statement about Bitcoin that “the network is robust in its unstructured simplicity”. The functioning of these blockchains may be regarded as algorithmically complicated, but not complex.

As the PoW-based cryptocurrencies are computing-power-intensive nature and suffer from significant energy consumption cost overhead in the operation of such networks, there was recently a burst of popularity of the cryptocurrencies that used an alternative algorithm known as “Proof-of-Stake” (PoS) for choosing the block creators. In purely PoS-based cryptocurrencies, the creator of the next block is selected in a deterministic (pseudo-random) way, and the probability that an account is chosen depends on its current wealth (the stake) and not its computational resources. For this reason, in PoS cryptocurrencies, the blocks are usually said to be forged (in the blacksmith sense of this word) or minted.

Therefore, PoS-based blockchains can also be seen as infinite-string-production є-machines that oscillate between two states about every minute: the targeting state (t) and the broadcasting state (b).

Consequently, applying to Nxt the same procedure, the Crutchfield’s Statistical Complexity measure is obtained as 0.122, a value of that is 10 to 20 orders of magnitude bigger than those of Table 1 and that, consequently, raise serious concerns about the possibility of Nxt entering a chaotic regime at any time without notice. For details of the calculation and a discussion on this result, see an extended version of this article.

There are also other protocols PoS with conceptually different implementations. For example, the forging probability may also depend on the so-called coin age, which is simply defined as currency amount times holding period the time the coins were in the account without being transferred.

Another example of PoS variation is Reddcoin’s Proof of Stake Velocity (PoSV), which intends “to encourage both ownership (Stake) and activity (Velocity) which directly correspond to the two main functions of Reddcoin as a real currency: a store of value and a medium of exchange.”

There is also hybrid PoW+PoS implementations, in which PoW mining works as both a steady distribution channel for the cryptocurrency and a fall-back network security mechanism. As PoW block rewards go down over time, the PoS protocol has enough time to move to the spotlight.

For example, in King and Nadal’s Peercoin design, a new minting process is introduced for PoS blocks in addition to Bitcoin’s PoW minting, and blocks are separated into two distinct types: PoW blocks and PoS blocks. However, a significant difference is that the hashing operation is done over a limited search space (more specifically one hash per unspent wallet-output per second) instead of an unlimited search space as in PoW. Thus, no significant consumption of energy is involved.



The figure above exhibits a comparison of the calculated complexities according to the consensus protocols. One observes a dramatically higher value of the complexity measure of the Nxt PoS protocol in comparison to the other protocols.

The higher difficulty of the other protocols in being selected to mine/forge a block, and consequently needed higher hashrate, lead to their lower complexity value. In contrast, as the time interval between blocks forging in Nxt is kept around 60 seconds, the probability of any node to be selected is hugely higher than other currencies.

This may be not a feature of PoS protocol itself, however, but a characteristic of Nxt. It is conceivable, therefore, that some different implementation of the basic PoS protocol could have a higher competition among ‘forgers’ and, consequently, a lower complexity.
Conclusion

This work suggests that Crutchfield’s Statistical Complexity may be used as an effective analysis tool to evaluate the viability of proposed high-performance network cryptographic methods from the available quantitative data.