Valuing Bitcoin and Ethereum with Metcalfe’s Law

Bob Metcalfe’s 1972 sketch of his vision for the original “ethernet.” Credit: PARC.

Keywords: valuation, metcalfe’s law, NVT, PMR, bitcoin, ethereum

Data sources:,, current to Feb 9/10 (BTC/ETH) of 2018


  • Here we report that Metcalfe’s Law, and similar laws, nearly perfectly correlate with ETH and BTC’s USD price
  • This suggests that cryptoassets that have reached critical mass behave like online social networks such as Facebook with their value defined by their usage (as measured by daily transactions or active addresses)
  • We examine the Network Value-to-Transactions (NVT) ratio — a popular metric for potentially identifying when the USD price of a cryptoasset has exceeded or fallen below the value suggested by its underlying daily USD transaction volume
  • We propose a similar formula — the Price-to-Metcalfe Ratio (PMR) — and propose that it may have some advantages over NVT; however, we recognize that more research is needed before an optimal PMR is determined
  • Note: an earlier version of this article incorrectly stated the formula for Metcalfe’s Law as M^(1.5). This was a typo in the formula only. M² was used for all calculations denoted as M.

In the 1980s, Robert Metcalfe, the inventor of Ethernet, proposed a formula:

the value of a telecommunications network is is proportional to the square of the number of connected users of the system (n²)

Since its inception, “Metcalfe’s Law” (M) as it is now known, has become an influential formula for studying network effects and valuing online networks.

In 2015, Zhang et al. found that Facebook’s revenue — a proxy for its value — was proportional to the square of its monthly active users (MAU). In fact, M best fit Facebook’s value when compared with other similar laws (Sarnoff’s, Zipf’s, and Reed’s). Zhang also found that M best fit Tencent’s revenue — Tencent is China’s largest social network. What’s more, Zhang found that the growth trend of Tencent’s and Facebook’s MAU fit the netoid function first defined by Metcalfe in 2013 (more on this later).

Building off this work, Ken Alabi applied this thinking to the blockchain and found that Bitcoin, Ethereum and Dash also fit M.

Here, we’ll look closely at Bitcoin and Ethereum and show just how closely M fits their respective prices in USD. We’ll also examine modifications of M (M1-M3) and other competing laws, Sardoff’s and Zipf’s, to explore how well these fit prices compared with M. We’ll also examine how daily transaction volume in USD — which we’ll term V — correlates with prices since this measure is the denominator of NVT. Finally, we’ll explore whether we can use M or its variants to generate a ratio similar to NVT.

First, our formulas:

Sardoff’s Law (S)
Metcalfe’s Law (M)
Zipf’s Law

N = # daily transactions (or # unique active addresses)
S = current supply

Let’s start by looking at the Pearson correlation of these 6 formulas with the USD price of BTC.

Because we have data on both transactions and unique active addresses for Bitcoin, we’ll examine both. TX will be used to refer to formulas that use transactions and ADR addresses. Also note that we only have data for every other day as opposed to daily which may affect precision. Finally, we’ve removed any early data points at which the USD price was $0.

Table 1 displays the Pearson correlation coefficients (r) of BTC’s USD price with our formulas unadjusted (row 1), smoothed using 30-day backward-facing moving averages (row 2), or smoothed using 90-day backward-facing averages (row 3). Only our formulas are smoothed, price is not. Rows 4 to 6 measure the correlation of the natural log of USD price with the natural log of our formulas.

Abbreviations: ln, natural log; MA, moving average

All formulas show near perfect correlation with BTC’s USD price, particularly on a natural log scale. In any other field, such a correlation would be considered witchcraft. V, S, and M perform best across all rows. However, the difference in correlation between the formulas is so small, particularly on the natural log scale, that they can all effectively be considered equal.

Now let’s see what the data look like when using active addresses.

Abbreviations: ln, natural log; MA, moving average

Using unique active addresses increases the correlation of our formulas with BTC’s USD price. V still performs best on a non-log scale whereas S, M and M2 perform best on the natural log scale. As above however, the differences in correlations are so small they can effectively be considered equal.

To illustrate the point of just how correlative these formulas are with BTC’s USD price, we’ll overlay the 30-day moving average of BTC USD with that of M2, both on the natural log scale. As seen below, they are nearly identical (note: M2 was chosen because it fits the price curve best and does so without having to adjust for scale).

Abbreviations: LN, natural log; MA, moving average

Now let’s look at Ethereum.

Once again we’ll look at the Pearson correlation of these 6 formulas with the USD price of ETH. However, because etherscan does not have data on unique active addresses, we’re forced to rely only on the number of daily transactions as a proxy for network usage. As above, we’ve removed any early data points at which the USD price was $0.

Compared with BTC, our formulas show very strong correlation with ETH USD when using transactions even on a non-log scale. This may indicate that these formulas lose some correlative strength over time (we have nearly 8 years of data on BTC vs. <3 years on ETH) or it may suggest that there are features unique to BTC transactions such that active addresses is a better measure of network usage (more on this here).

For ETH, V, S and Z show the best overall performance. However, as with BTC, the difference in correlation between the formulas is so small that they can effectively be considered equal.

Once again we’ll overlay the charts of M2 with USD price using log-transformed 30-day moving averages.

Abbreviations: LN, natural log; MA, moving average

Given that price correlates so closely with transactions (or active addresses), we can question what current network usage suggests about current price. To do so we will compute a Price-to-Metcalfe Ratio (PMR). Stephen Powaga of ETF Momentum Investing suggests that such a ratio may be somewhat analogous to a price-to-book ratio in equity analysis.

The idea here is to study the relationship between the price (or value) of a cryptoasset and its fundamentals (as suggested by network usage). This type of ratio analysis is gaining popularity with Network Value-to-Transactions (NVT) being the most widely studied. For a background on NVT please see recent articles on rethinking NVT by Dmitry Kalichkin and a further expansion of its use as a trading signal by Willy Woo.

Let’s start by looking at NVT and defining its formula:

Kalichkin’s recent revision uses a 90-day backward-facing moving average (previous versions used a 14-day backward-facing and 14-day forward-facing moving average).

While data on daily market capitalization of cryptoassets is easily obtained, both and have this for example, daily transaction volume in USD is not readily available. has this data freely available for both ETH and BTC but they acknowledge some difficulty in estimating it here. What’s more, other researchers calculate it differently as Willy Woo explains here. This underscores one of the potential limitations of NVT: obtaining reliable data.

Below is a chart of the natural log of BTC’s USD price against its revised NVT ratio.

Abbreviations: LN, natural log; MA, moving average

At the “yellow zone” line that Kalichkin suggests (at an NVT of 20) we can see a few areas where NVT may have been useful in predicting major corrections— moments in time where the price of USD far exceeded its underlying fundamental value as determined by USD transaction volume and resulted in a rapid downward correction.

Applying NVT to ETH, the metric is less convincing but still appears useful. The NVT of ETH throughout its early history pre-2017 was in the hundreds and therefore way above our threshold of 20. Zeroing in on 2017 and beyond however, we see a couple spots where NVT may have helped predict corrections when using a threshold of 20.

This is probably a good time to mention that barring formal research the choice of cutoff is somewhat arbitrary, so it’s possible that different assets may benefit from slightly different cutoffs or that more optimal thresholds may be later identified.

Abbreviations: LN, natural log; MA, moving average

While NVT appears to be a useful metric for predicting major corrections, it does feel like there’s room for improvement.

NVT is limited by the fact that on-chain USD transaction volumes are difficult to calculate and different approaches at calculating NVT’s denominator can lead to slightly different conclusions. The other issues is that the numerator and denominator are both a function of the USD price of a cryptoasset. It’s typically not good practice to rely on a predictive ratio that includes what you’re ultimately trying to predict in both its numerator and denominator.

That brings us back to Metcalfe and its variations. We’ve already seen that i) these formulas are slightly more correlative than transaction volume, at least on a natural log scale for BTC, ii) they are simpler metrics that rely on data easily obtained from and (in the case of Bitcoin, we can use active addresses and move away from transactions and its limitations), and iii) they are not a function of the USD price.

So what would a PMR look like?

To be honest, this is where things got a bit experimental as we relied largely on trial and error rather than formal testing. While we ended up identifying an interesting metric, more formal research may arrive at an even better metric.

Given how well our formulas correlated with price on a natural log scale, we decided to focus on log-transformed data and ultimately arrived at the following:

To clarify the above, when we say 30-day moving average of M2, we first take the 30-day backward-facing moving average of unique active addresses (or transactions for Ethereum) and then use those values for our daily M2 calculation. 60- or 90-day moving averages could also be used and perform similarly (they are a bit less sensitive but potentially slightly more predictive). And as described above, M2 correlates highly with USD price on a natural log scale and amazingly fits on the same scale as the log-transformed USD price without any additional adjustment.

Let’s see how PMR fits with Bitcoin.

Abbreviations: LN, natural log; MA, moving average

At first glance, we have what appears to be a rather strong leading indicator that accurately predicted 3 of bitcoin’s largest corrections when using a PMR of 1.0 as the cutoff. Using a PMR of -1.25 we can also identify potential accumulation zones.

One specific point to highlight is the recent correction we just experienced in early 2018. Bitcoin shot well above a PMR of 1.0, reaching as high as 1.5, before correcting. What’s interesting here is that even after its rebound, it’s still dangerously close to a PMR of 1.0, which is in stark contrast to its NVT. Something worth monitoring.

Now let’s turn to Ethereum.

Abbreviations: LN, natural log; MA, moving average

As with Bitcoin, a PMR of 1.0 accurately predicted 4 of ETH’s major corrections while a PMR of -0.25 appears to identify strong buying opportunities.

Also of note is that PMR did not predict the most recent correction in early 2018 — in fact, according to PMR, ETH is currently in a strong buy zone. Does this mean that PMR failed? No. We believe the reason for ETH’s crash was Bitcoin’s over-inflated NVT and PMR — both of which indicated that Bitcoin was due for a major correction. And because Bitcoin is the dominant cryptoasset, when it corrects, the entire market tends to move with it.

Limitations of PMR

Despite the fact that PMR appears to offer some advantages vs. NVT, it’s not without its limitations.

  1. We can’t explain why M2 works as well as it does or where it came from.
  2. Because PMR relies on transactions or unique active addresses, it cannot capture off-chain transactions. Off-chain solutions like Lightning Network, Raiden, other side chain or state channel scaling solutions that move transactions off the main chain may reduce the predictive power of M2 and its variants over time. Similarly, some proportion of transactions reflects exchanges moving money around (in terms of USD volume this may represent a decent chunk of total volume but in terms of raw numbers of transactions, it may be less an issue).
  3. In the case of Ethereum, ERC-20 transactions now represent upwards of one third of all on-chain transactions. Because ERC-20 transactions do not directly relate to trading of ETH, it’s possible that we’ll need to discount transactions in the future by some function to account for the rise in non-ETH transactions. Similarly, as dapps begin to launch and Ethereum sees increased non-speculative usage, further discounting may be needed. On the other hand, perhaps this increased usage will not affect M2’s correlative power. Time will tell.

Because of these limitations, we believe that both NVT and some variant of PMR will be useful metrics moving forward.

There is no single indicator that can accurately predict the price of a speculative asset like ETH or BTC as there are too many variables to consider; however, if we accept the premise that blockchain networks that are predominantly in the speculative stages of adoption behave like online telecommunications networks, then Metcalfe may help us to better understand where usage and price intersect and when one has significantly outpaced the other.

Whether this continues to hold true is difficult to predict, particularly as networks such as Ethereum begin to see usage beyond speculation. During these early speculative stages, however, the number of transactions or number of unique active addresses likely serves as a close proxy for demand.

Ken Alabi addresses this nicely in his paper:

Due to the fact that the assets we consider have very little actual use currently — they are not really used much as a transactional medium for payment systems, nor even significantly as a payment rail — it may be reasonable to interpret their adoption to be based on their perceived future potential. If that potential or value proposition is presented to a selected number of people, the expectation is that the same ratio in that set would convert into acquiring the assets. That group would then communicate that same value proposition to their own social network of friends and acquaintances, and on. Therefore, the basic adoption process might not be very dissimilar from that of a social network.

Indeed, when prices rise, demand rises and when prices drop, so too does demand. Price and usage therefore share a symbiotic relationship such that neither can be used to predict one without the other, but what PMR and NVT are likely telling us is that there is a threshold to the extent that one may outpace the other and that reversion to the mean will eventually take place.

Kalichkin refers to this concept as reflexivity.

Taken in context with other indicators and external factors, both NVT and some variation of PMR may serve as useful additions to the conscious investor’s toolkit. We caution however that thresholds should not serve as markers for immediate action. Instead, we would suggest a conservative approach: when an asset crosses above an upper threshold one could cease buying (as opposed to selling in hopes of timing the top and re-buying at a lower cost) and when an asset crosses a lower threshold one could start buying. The former suggestion is important because our charts show that a cryptoasset may continue rising in price after an upper threshold is breached and may even remain valued at a higher price even after a correction. In other words, it’s difficult to perfectly time the top.

Ken Alabi’s paper suggests a new formula for valuing blockchain networks derived from Metcalfe. It is a more complex function that requires curve-fitting. Future work might involve fitting this function and comparing its performance to our variants tested here.

Ken Alabi’s revised model for network value

Mr. Alabi’s paper also details a model for predicting future growth of a network — the netoid function. As with Alabi’s value function, application of this growth function requires curve-fitting. Using the netoid function, Alabi predicted that Bitcoin would reach its maximum growth rate in October of 2017 and that Ethereum would reach its sometime in 2018. The netoid function for these networks should be re-fitted based on recent data.

If such a function is a useful predictor of future growth (in terms of transactions or active addresses) then it follows that it may be an accurate predictor of price given how well price correlates with network usage. This could make it a very powerful tool for speculation.

Of course, for this to be true, we need to make some assumptions and assert some precautions. For starters, this would require that Metcalfe and its variants continue to maintain their correlative power which might be problematic (see “Limitations of PMR” above). Second, no model of growth can account for external events. A black swan event or the rapid rise of a superior cryptoasset competitor could see network usage dramatically shift in ways that cannot be modeled.

As Alabi writes:

In as much as the value of the network correlates with the number of users actively participating in it, that value can also prove to be as fleeting as the ease with which those users can move to a different network or cease to participate. The idea behind some of the assets designated for use on the networks studied here is one of fungibility. That fungibility also means that users can create new addresses on different networks and move their assets there easily, or simply just pull their assets out. In short, the ease with which users can move from the blockchain networks of the types studied here exceeds that of networks such as social media, where the user may have cherished items including pictures, conversations, social contacts, and other historical items that may not be as easy to move.

We however disagree with Alabi’s assertion that users will readily move between different blockchain networks — at least more readily than between social media networks. As blockchain protocols grow and capture significant network effects, particularly through mainstream dapps built on top of their protocols, users will naturally become more vested within the network and its ecosystem making it harder to exit. Social circles are also generated both online and in meatspace which further constrains individuals to their respective blockchains.

Finally, future work is needed to formally test PMR, its thresholds, and its variants.

Jacob Franek

Thanks to Kevin for insights and inspiration for this article.

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Co-Founder @coinmetrics. Crypto data. Former clinical epidemiologist and evidence-based medicine specialist.