The old telephone tower in Stockholm

One Law Many Networks

There is a long history behind Metcalfe’s Law which allows measuring network effect. It was discovered in the 1980s and initially applied to Ethernet. This law derives the “value of a network” from the number of nodes so the network value is proportional to the square of its size. The potential applications of this law are numerous.

For example, the adoption of the telephone demonstrates the proportionality of the square number of installed telephones to the revenue of Bell and ATT companies. If we assume that companies maximize their profits and peers are ready to pay costs that are equivalent to the value they receive in exchange for their money, Metcalfe’s Law confirms.

Chinese researchers Zhang, Liu, and Xu in “Tencent and Facebook Data Validate Metcalfe’s Law” compared also Sarnoff’s law (V/n), Odlyzko’s law (V/nlog(n)), and Reed’s law (V/2n) against Metcalfe’s law using corresponding revenue data. They concluded that Metcalfe’s law describes social network dynamics better than others.

From Zhang, Liu, and Xu “Tencent and Facebook Data Validate Metcalfe’s Law”

Another author Timory Peterson looked into Metcalfe’s law application for describing Bitcoin’s protocol dynamics. In his “Bitcoin Spreads Like a Virus,” he also considered Facebook data and concluded,

“that the growth and price of bitcoin and other cryptocurrencies are likely to proceed according to a relatively straightforward mathematical model similar to the growth curves of Facebook and other networks”.

Wheatley et al. in their work “Are Bitcoin Bubbles Predictable?” proposed a method for detecting price bubbles using derived from Metcalfe’s law indicator. I relied on their research while preparing a popular article in Russian* about MMV indicator and a demonstration example for how Bitcoin network value could be measured. I found out that Bitcoin value recently started deflecting from its market capitalization.

Value of Bitcoin

This MMV script I was developing is very simple. You can find it here. On the first step, I overlapped two plots on a single graph, one for Bitcoin market capitalization and another for Metcalfe value of Bitcoin derived from daily Active addresses (Coin Metrics dataset). The Figure demonstrates that the market capitalization of Bitcoin started diverging with network value in approximately 2020. For simplicity, I scaled network value with constant such that

MV = Constant*CoinMetrics Active Addresses²

Of course, the Constant may be arbitrary. However, we could either adjust it by pre-2020 peaks, or by the current post-2020 peak. The point is that it is either pre-2020 peaks will be “overvalued” or post-2020 will become “undervalued” in Metcalfe Value terms. My initial hypothesis for this divergence was about monetary expansion which lead to inflated market capitalization or unaccounted network effects because the date strangely coincided with launching Lightning Network in beta.

Coinmetrics Active Addresses and LN node count (different Y axis scales on left and right)

For Lightning Network to be included in the evaluation I needed a new source of data about its historical growth. For this, I used bitcoinvisuals.com which conveniently allows downloading datasets in CSV files. The Active Addresses metric is different from the Lightning Network node count. This is the major reason for any further comparison to be failed. But also I tried to adjust the Constant in Metcalfe’s Law for Lightning network value and it worked. While for Active Addresses the Constant value is equal to exp(-1), for Lightning Node count it should be equal exp(8) to get the same plot as below.

Capitalization, Metcalfe Value computed via Active Addresses and Metcalfe Value with added LN node count network effect


There is important research “Does Lightning Network drive meaningful base chain usage?”. The author concludes that “Public channels account for ~0.1% of Blockspace usage, plus some Blockchain spelunking”. The 0.1% is a rather small block space demand. However, very simple math and Metcalfe law indicate that we probably observe an influence of increasing adoption of Layer 2 on Bitcoin price.

Liquidity in the LN network is somehow “locked” if the node doesn’t belong to an exchange. It makes onchain bitcoins scarcer and more expensive. The LN Metcalfe Value can be measured and added to the Bitcoin network value. It is rapidly growing and the next peak in the LN node count and channel capacity may give a hint to the next bear cycle on the Bitcoin market.

Of course, my method is quite simplistic, maybe naive. Non-homogeneous data are being compared. Using something like the Glassnode Onchain Entities metric could be beneficial if one tries to add LN network value to onchain Bitcoin network value since a typical LN node should appear like a single onchain entity. However, one must think about avoiding double count in the Glassnode metric and some other metric providing LN node count. Active Addresses may be less sensitive to double-counting since ideally addresses associated with channels shouldn’t be active onchain.

There is a possibility that Metcalfe’s law coefficients may be re-adjusted in the future. If the LN node count and Coinmetrics Active Addresses are being used, they just illustrate the relative weight of these metrics in the total network effect. It seems that even in such primitive form the relative size of Bitcoin as a multipurpose general protocol may be compared to its truly massive application the Lightning Network.

If you like this post, please do not hesitate to donate some sats via LNURL or via Lightning Address notgeld@lntxbot.com.

[*] “Сетевой эффект Биткоина и его цена”

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