Metcalfe's Law

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THE DEBAUCHERY

OF CURRENCY

METCALFE’S LAW

TIMOTHY F . PETERSON, CFA, CAIA

Published by Freedom Fortune Press

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ABOUT THE AUTHOR With 30 years in institutional investment management, Timothy Peterson is a published author with expertise in alternative assets, including cryptocurrency investment and valuation.

Timothy authored the very popular “Metcalfe’s Law as a Model for Bitcoin’s Value,” which was the first peer-reviewed publication on how to value Bitcoin with Metcalfe’s Law. Based on downloads, his cryptocurrency research ranks in the top 0.05% of over 800,000 authors on SSRN. He has also written two books, “The Debauchery of Currency: A Bloody History of Money” and “Performance Measurement for Alternative Investments” (Risk Books: 2015). Timothy has spoken at conferences around the world on various investment topics.

He is a dual graduate of the University of Colorado with a Master of Science Degree in Finance and a Bachelor of Arts degree in Economics. He is a Chartered Financial Analyst and Chartered Alternative Investment Analyst and has also completed the Harvard JFK School of Government course on Investment Decisions and Behavioral Finance.

Timothy teaches “Digital Assets for Bankers” at the prestigious Graduate School of Banking at Louisiana State University.

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CHAPTER X

METCALFE’S LAW “Apple is a mobile network, Google is a search network, Facebook is a social network, Bitcoin is a monetary network.

All benefit from advances in software & hardware technology, as well as Metcalfe’s Law. Many understand the former, few understand the later.”

—Michael Saylor, CEO of Micro Strategy

S

aylor’s observation highlights a glaring gap in understanding: while many grasp the impact of technological advancements on society, few comprehend the profound implications of networks on the value imparted to an economy. This chapter aims to correct that deficiency, explaining how Metcalfe’s Law fundamentally shapes the value and growth dynamics of interconnected systems.

When it was first introduced, Metcalfe’s Law was considered a tautology. Tautologies are used in logic and mathematics to establish foundational truths and validate arguments. Empirical laws are used in the natural sciences to describe and predict phenomena based on experimental data. Observed laws are often used in technology, economics, and social sciences to describe trends and patterns that guide practical applications and future predictions. This chapter will demonstrate that Metcalfe’s Law has progressed beyond theoretical principle to the realm of empirical law found in everyday circumstances.

As societies transition from physical production economies to digital information economies, Metcalfe’s Law becomes more relevant.

This principle applies across various network types—from transportation networks to social and monetary networks. However, its application to Bitcoin as a monetary network is particularly intriguing and potentially revolutionary.

Metcalfe’s Law provides a compelling explanation and deeper insight into why Bitcoin, as a monetary network, has the potential to iii

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become increasingly powerful and valuable in our interconnected world.

Unlike traditional financial systems, Bitcoin’s network effects are not constrained by geographical boundaries or centralized control, allowing for global growth in utility and value as adoption increases.

Metcalfe’s Law is a critical concept for understanding how networks acquire value, especially in the context of digital and telecommunications systems. The law, named for Dr. Robert Metcalfe, states that a network’s value is proportional to the square of the number of connected users. This has the interesting characteristic that doubling the users in a network more than doubles its value.

It helps to think about how this works in a social network. If there’s only one user, the network has no value since there’s nobody else to interact with. But add just one more user and now you have one potential connection. With five users, the number of possible connections jumps to ten. And with a dozen users, would you believe there are sixtysix possible connections? This rapid growth in potential interactions shows why networks become so much more valuable as they grow.

In the classic diagram describing Metcalfe’s Law, two telephones can make only one connection, five can make 10 connections, and 12 can make 66 connections.

Now, it is no great revelation that things that grow larger typically increase in value. There are many formulae and theories as to how this happens. What sets Metcalfe’s Law apart from these is its specificity.

It doesn’t just state that networks accumulate value as they grow.

Metcalfe’s Law tells you how much value the network accumulates.

Metcalfe originally explained this law using things like fax machines and telephones. A single fax machine or phone is pretty much useless by itself. But the more devices that get added to the network, the more valuable each one becomes because there are more people to communicate with. Suppose you have a network of 1,000 users and it is valued at $1,000. What happens to the value when one more user is added?

With 1,001 users, the value of the network is not $1,001, but $1,002. For a more exaggerated example, suppose there is a network with 100,000 2

users valued at $1 million. 1,000 new users are added to the network.

Metcalfe’s Law states that the value of this larger network is $1,020,100.

Furthermore, in certain decentralized systems, this increase in value accrues to all users and (eventually) manifests in price appreciation over time.

This law isn’t just a theory. It plays out in the real world all the time. The growth and value of the internet and social media giants have all followed the patterns predicted by Metcalfe’s Law. The more users that join these networks, the more dramatically their value rises. That’s why early adopters benefit from growing the network quickly.

Bitcoin is an excellent example of Metcalfe’s Law in action.

Every new person that starts using Bitcoin makes the entire network more valuable, because each additional user means more utility and potential transactions happening on the network. In fact, Bitcoin’s price is primarily determined by the growth of its user base, just like Metcalfe’s Law predicts.

The same thing happened with Facebook. In its early days, Facebook’s value shot up rapidly as more and more users joined. More users meant way more connections between people, which made the platform increasingly valuable to both advertisers and users themselves. This network effect is the primary reason why social media companies have reached such massive valuations.

In the context of financial transactions, larger networks would seem to have more value than smaller networks. Suppose there is a network of four friends: John, George, Ringo, and Paul. John has tickets to a concert he believes is popular. He offers to sell the tickets for a large markup over face value to George, Ringo, and Paul. No one accepts his offer. What can John conclude about the asking price of the tickets? Perhaps none of his friends are free the night of the concert. Perhaps they don’t like that type of music. Perhaps they don’t like concerts. With such a small network, there is little information to go on.

John lists his tickets on a popular website where his offer is viewed by 40 would-be purchasers. Still, he receives no bids. Now John is more likely to conclude that his price is too high. The network has provided valuable information to John about his ask price. But everyone in the network receives valuable information: since all other participants see that the ask was not accepted, each participant receives 39 confirmations that his or her rejection of the ask price was justified. The important thing to note here is that all participants have gained value from the network, even though no transaction actually occurred!

Now suppose John is in the ticket sales business. He lists many thousands of tickets at various prices. Some ticket-price combinations 3

attract a large number of bids, and some ticket-price combinations attract a small number of bids. Thus, transaction volume at a specific price level also provides valuable information, and this value accrues to all participants, whether they actually engage in a transaction or not!

This illustrative example demonstrates that network size in terms of potential connections is what provides the driver behind value. The actual number of transactions which occur is important, but that is a secondary consideration.

Consider this simple example: Country A consists of large three cities only. Each city is connected by a road, so there are three connections in this network. These roads are heavily traveled. Country B consists of 20 medium sized cities. Each city is also connected by a road, resulting in 190 connections. Also assume that each country has the same population and resources.

One day, a major road closure occurs due to an accident. Suddenly, the seamless connectivity that the residents of Country A depend on is severed. Traffic grinds to a halt, and the remaining two roads become overwhelmed with the increased volume of vehicles. Commuters face hours-long delays, businesses struggle to receive supplies on time, and emergency services find it challenging to navigate through the congestion.

The economic impact is immediate and severe. Productivity plummets as workers are stuck in traffic instead of at their jobs. Perishable goods spoil in transit, leading to losses for businesses and shortages in stores. Companies reliant on just-in-time delivery systems face disruptions, causing a ripple effect throughout the economy. The closure of a single road in this tightly knit network reveals the fragility of Country A’s infrastructure, with economic losses mounting rapidly.

In contrast, consider Country B, a nation of 20 medium-sized cities, each linked by a complex web of 190 roads. The extensive network offers multiple routes for any given journey, providing a robust and resilient transportation system. On a typical day, the roads of Country B see a steady flow of traffic, but the system is designed to handle fluctuations and incidents.

One day, a significant road closure occurs here as well. But unlike Country A, the impact is far less dramatic. The vast network of alternative routes absorbs the displaced traffic, ensuring that congestion is minimal and delays are short-lived. Commuters easily find other ways to reach their destinations, and goods continue to flow with only minor interruptions.

The economic impact in Country B is contained. Businesses experience some delays, but the presence of alternative routes mitigates 4

significant disruptions. The economy remains stable, and the resilience of the transportation network ensures that life continues with minimal impact. The extensive connectivity in Country B not only prevents severe economic losses but also highlights the country's capacity to adapt and thrive in the face of challenges.

In all likelihood, Country B will have a higher economic output than Country A. In Country B, businesses operate with the confidence that their supply chains are secure, even in the face of unforeseen incidents. This stability encourages investment and expansion, as companies are less likely to face catastrophic losses from logistical failures. In Country A, the fear of such disruptions can stifle growth and deter investment, leading to a slower economic pace.

Secondly, the efficient flow of goods and services in Country B leads to lower transportation costs. With multiple routes available, congestion is rare, and delays are minimized. This efficiency translates into lower costs for businesses, which can then pass these savings onto consumers or reinvest in further growth. In Country A, higher transportation costs from frequent disruptions eat into profit margins and reduce the overall economic dynamism.

Thirdly, the resilience of Country B’s network supports a more productive workforce. Commuters spend less time stuck in traffic and more time at their jobs, contributing to higher productivity levels. The reduced stress and time savings also improve the quality of life, fostering a healthier and more motivated workforce. In Country A, the frequent traffic snarls reduce worker productivity and increase stress levels, ultimately dragging down economic performance.

If you have used the internet, you’ve likely benefited from the innovations of 3Com Corporation. Named after “Computer Communication Compatibility,” 3Com was born from Metcalfe’s vision to bring Ethernet technology, which he co-invented, into practical use. The company played a pivotal role in transforming Ethernet from a theoretical concept into a widely adopted networking solution. Under Metcalfe’s leadership, 3Com developed a range of products, including network interface cards, hubs, and switches, that facilitated the rapid deployment of local area networks in businesses and organizations worldwide. This democratization of networking technology was crucial for the growth of computer networks and the internet during the 1980s and 1990s.

Dr. Robert Metcalfe, known colloquially as Bob Metcalfe, is an American engineer and entrepreneur renowned for his contributions to computer networking. Born in 1946 in Brooklyn, New York, Metcalfe 5

pursued his education at MIT and Harvard University, earning degrees in Electrical Engineering, Industrial Management, Applied Mathematics, and Computer Science.

In the early 1970s, Metcalfe joined Xerox PARC, where he and David Boggs invented Ethernet. Ethernet revolutionized high-speed communication between computers and became the standard for local area networks. This innovation significantly contributed to the growth of the internet and transformed computer connectivity. In 1979, Metcalfe co-founded 3Com Corporation to commercialize Ethernet technology, leading the company to become a major player in the networking industry. After leaving 3Com, Metcalfe became a venture capitalist and an advocate for technological innovation and entrepreneurship. In 2010, he joined the University of Texas at Austin as a professor of innovation and entrepreneurship.

Bob Metcalfe has received numerous prestigious awards for his pioneering work in developing Ethernet technology. He was honored with the IEEE Medal of Honor in 1996 for his significant contributions to networking and the growth of the internet. In 2003, he received both the National Medal of Technology and Innovation from President George W. Bush and the Marconi Prize, recognizing his groundbreaking work in Ethernet and his role in founding 3Com Corporation. Metcalfe was inducted as a Fellow of the Computer History Museum in 2008 for his efforts in standardizing and commercializing Ethernet, and in 2013, he was inducted into the Internet Hall of Fame for his pioneering role in expanding internet accessibility.

Bob Metcalfe recently received the 2022 ACM A.M. Turing Award, often referred to as the “Nobel Prize of Computing,” for his invention, standardization, and commercialization of Ethernet. This prestigious award includes a $1 million prize and is a testament to Metcalfe’s profound impact on the field of computer networking. The Turing Award recognized Metcalfe’s pivotal role in developing Ethernet, a technology that remains fundamental to modern computer networks and the internet, connecting billions of devices worldwide.

Ethernet enabled high-speed data transfer between computers using a simple and cost-effective approach. The initial Ethernet system operated at 2.94 megabits per second, relying on coaxial cables to transmit data packets accurately across the network. The IEEE 802.3 standard formalized Ethernet as a networking protocol in 1980, allowing for widespread adoption and compatibility.

Over the years, Ethernet evolved significantly, with improvements in speed and reliability. Fast Ethernet increased speeds to 100 megabits per second, and subsequent developments led to Gigabit and 6

10 Gigabit Ethernet. These advancements supported the growing demands of data-intensive applications and high-bandwidth services.

Ethernet’s impact extends beyond technology, shaping the modern internet and business communication. Its simplicity, robustness, and scalability made it a preferred choice for networks of all sizes. Ethernet’s compatibility with wireless technologies, like Wi-Fi, further extended its reach and utility. Today, Ethernet continues to evolve, focusing on increasing speeds, reducing latency, and enhancing security features.

In 2010, 3Com was acquired by Hewlett-Packard for $2.7 billion.

The story of Metcalfe’s Law begins in the early days of Ethernet, when Metcalfe and his team at 3Com were working to commercialize this revolutionary networking technology. In 1980, as Ethernet was still in its infancy, Metcalfe faced the challenge of convincing potential customers of its value.

Metcalfe mused, “So in the course of selling Ethernet–now you have to remember the challenge here–you’ve all heard of Ethernet. A lot of people hadn’t in the 1980s...Most of our customers weren’t interested in a network for personal computers because they didn’t have any personal computers.”

Many businesses at the time were hesitant to invest in a technology they didn’t fully understand, and the benefits of connecting computers in a local area network were not yet apparent.

Metcalfe devised a compelling argument to illustrate how value via Ethernet is achieved. In a pivotal presentation to the 3Com sales force, he developed a slide (a 35mm slide,1 not a PowerPoint slide; PowerPoint did not exist yet) with a graph that would later become the foundation of Metcalfe’s Law. The graph depicted the relationship between the number of nodes in a network and its overall value, showing that as the number of connected devices increased, the value of the network grew quadratically.

When something grows quadratically, its growth rate is related to the square of a number. Imagine you have a network, and you count the number of devices (nodes) connected to it. If you have 2 devices, and you square that number (22), you get 4. If you have 3 devices, and you square that (32), you get 9. This is slightly different from exponential growth. When something grows exponentially, it increases at a much

1 A 35 mm slide is a small, transparent photo mounted in a 2-inch frame, commonly used

for screen projection. It uses 35 mm film and was popular for presentations and archival purposes before digital photography.

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faster rate than Metcalfe’s Law states. For exponential growth, you often see the term used as a power of a constant base, like 2n.

A log scale plot of exponential vs. quadratic functions shows differences in each.

On a log scale, exponential growth is linear and constant. Quadratic growth is curvilinear and non-constant, with exceptionally fast growth in early periods.

Metcalfe explained, “We sold a kit of three cards, because computers didn’t come with Ethernet built-in in those days...All of our customers told us the following thing: they said, ‘It works just like you said, but it’s not useful.’ And I was head of sales and marketing then, so that’s a challenge.

I gave this slide to my sales force. And we went back out to those people who had bought three node networks, [and] we made the following high-concept argument: ‘The reason that your network is not useful is that it’s not big enough. So therefore, you need to buy more of our products.’”

As Ethernet gained traction and more businesses embraced the technology, the validity of Metcalfe’s Law became increasingly evident.

The growth in the value of Ethernet networks provided the proof of concept that fostered the rapid adoption of the technology. By the late 1980s, Ethernet had become the dominant standard for local area networking. 3Com, riding the wave of Ethernet’s success, went public in 1984 and saw its revenue soar to billions of dollars by the end of the decade.

Metcalfe is quick to point out he did not name the eponymous law: “Fifteen years later, that slide was called by someone else–not me– Metcalfe’s Law.” The moniker “Metcalfe’s Law” entered public lexicon in the mid-1990s when economist and futurist George Gilder, who had earlier helped to popularize Moore’s Law in his book Microcosm, championed the law in his sequel Telecosm.

Metcalfe argued that if a network was too small, its cost would exceed its value, but once it reached a critical mass, the value would skyrocket. This powerful insight resonated with the sales team and armed them with a compelling argument to convince customers to invest in Ethernet and expand their networks beyond the initial “starter kits” of three nodes.

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The actual visualization from a 35mm slide that Bob Metcalfe showed the 3Com sales team.

In a fully connected network, each user (denoted as n for node) can connect with every other user. The number of possible connections (denoted E, because in network topology these connections are called edges), where you’re choosing 2 users out of n to form a connection, can be calculated using the combination formula: 𝐸=

𝑛 × (𝑛 − 1) 2

Here’s how it works: Total Users (n): The total number of users in the network.

Possible Connections: Each user can connect to (n − 1) other users. Therefore, for n users, there are n × (n − 1) total connection possibilities if we count each connection twice (once for each direction).

Divide by 2: Since each connection is counted twice (once from user A to B and once from B to A), we divide by 2 to get the actual number of unique connections.

Each node or connection carries a cost per user, denoted C. The total network cost is C × n. At some number of users n, there is a critical mass where 𝐶×𝑛=

𝑛 × (𝑛 − 1) 2

And as more users are added after that, E > C × n. The net value of the network is V = E – (C × n).

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If n is very large, then the value V starts to grow at the same rate as n × n = n2. Sometimes Metcalfe’s Law is written V = n2, but this is incorrect. It is more correct to say V ∝ n2 which means value is proportional to n-squared. This shortcut acknowledges that there are other factors beyond just users-squared necessary to estimate network value.

How large is ‘very large’? Surprisingly, it does not have to be very large. It depends on other factors, but ‘large n’ could be as few as 100 or even 10 users.2

A fundamental consideration for Metcalfe’s Law, and the derived relationship between network value and user numbers, is that precise definitions and exact counts are less crucial than capturing the overall trend. Any reasonable proxy that correlates with user growth will provide sufficient insight for estimating network value changes, making this approach both practical and effective.

Given that Metcalfe’s Law demonstrates a proportional relationship between the value of a network and the square of the number of its users, the exact definition of ‘user’ or the precise count of users becomes less critical. Any reasonable proxy for the number of users (or network size) will suffice.

For example, when considering the internet, proxies such as bandwidth usage or the number of active IP addresses can provide valuable insights into network size and growth. These proxies are easier to measure and often correlate well with actual user numbers. Similarly, website traffic statistics can serve as another proxy, reflecting the overall usage of the internet without needing to count every single user.

What is a ‘user’ of Bitcoin? Recall that money can serve three functions: store of value, medium of exchange, or unit of account. A unit of account simply means pricing goods and services in Bitcoin, so that doesn’t directly involve network participation. However, using Bitcoin as a store of value or medium of exchange would qualify. If an entity holds Bitcoin as a savings instrument, that is a user. If they buy a good or service with it, that is a user. If they trade it for profit, that is a

2 Defining V1 as Metcalfe’s Law and V2 as the approximation n2, we need to find an approximate value where the slopes resemble each other. As n increases, the slopes and begin to exhibit similar behavior, with both increasing linearly as n

grows large. The ratio ln approaches 2.0 at n = 100. For practical purposes, you can start to see a resemblance for n in the range of 10 and above (94% of the full network effect is captured at this n).

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user. And if they have some for any purpose whatsoever, even if they don’t know what it is, that is also a user.

In practical terms, obtaining an exact count of users can be difficult and costly. However, it is often feasible to measure proxies with greater ease and accuracy. By leveraging proxies, one can still make informed decisions about network growth and value without needing perfect data. The key is to choose a proxy that reliably reflects the underlying trend in user numbers.

In the context of road networks, proxies like number of charging stations or cities (or even the number of McDonald’s restaurants) can be informative. These metrics can be more practical to obtain than attempting to track every individual vehicle on the road or the exact number of road connections.

In banking networks, proxies such as the number of branch locations or ATMs can offer insights into network value and user growth.

These proxies allow for a practical assessment of the network’s health and expansion without needing precise user counts.

For Bitcoin, the number of addresses or network difficulty can serve as effective proxies for network size and demand. These indicators are more readily accessible and can provide a clear picture of the network’s growth and value.

Metcalfe admitted that his Law was not empirically validated.

That is because, at the time, there simply was not enough data available on which to perform an analysis.

“I knew these networks had to be useful. I just had to explain why these little ones weren’t…This is very controversial...I provided no data to support this law. This was just a vision thing," said Metcalfe.

It wasn’t until decades later that Metcalfe obtained some network data with which to test his law.

Metcalfe explained, “I took Facebook’s data. From inception [through the] first ten years of Facebook, and I looked at two things. I looked at their monthly average users as a measure of n, the number of people using it. And then as a surrogate for value, I looked at their revenue...I managed to figure out how to put four sliders, and three of which are the parameters of the netoid function,3 which I then use as the red line fitting the growth of users of Facebook...And then the more aggressive

3 Metcalfe uses the invented term netoid to mean a sigmoid function applied to growth of

users in a network.

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thing was that the take the MAU (monthly average users) and relate it to revenue.”

Peterson’s Facebook slide shown to a joint session of CFA and CAIA Charterholders in Texas in 2018 predicted Bitcoin would grow to $50,000 in about five years.

About five years later, Peterson recreated Metcalfe’s test and compared the results to Bitcoin’s growth. This slide was presented to a gathering of investment managers in 2018. At the time of the presentation, the price of bitcoin was about $8,000. On the chart, year 9 for Bitcoin is about 2019, which means year 14 corresponds to 2024. The forecast was for Bitcoin to be at about $50,000. This translated into a forecast of +35% annual return, something people considered laughable at the time.

Recall that in 2018, Bitcoin was widely derided by established banks and governments as “magic internet money” with a value based purely on illicit conduct and the speculative trading activity of computer geeks. Yet for the first half of 2024, Bitcoin traded between $40,000 and $60,000 most of that time. This is one of the longest, most accurate, and most public forecasts Peterson made using Metcalfe’s Law. Despite this and other evidence, Metcalfe’s Law is frequently declared to be wrong.

In 2006, a cover story titled “Metcalfe’s Law is Wrong” appeared in IEEE Spectrum. IEEE Spectrum is a monthly magazine published by the Institute of Electrical and Electronics Engineers, one of the world’s 12

largest technical professional organizations. This article sparked a fierce debate in the tech world.

Robert Bricoe and his colleagues Andrew Odlyzko and Benjamin Tilly (“BOT”) argued that Metcalfe’s Law was not only incorrect but also dangerously misleading. The catchy title garnered significant attention, its shock value lying in directly challenging a long-held belief that– they allege–shaped the strategies of numerous companies during the dotcom boom. The provocative nature of the title ensured that the critique reached a broad audience, compelling stakeholders to reconsider the assumptions underlying network valuation.

Odlyzko, a professor at the University of Minnesota, alongside researcher Benjamin Tilly, had published a detailed paper expanding on their critique. They alleged several specific flaws in Metcalfe’s Law: All nodes in a network were not equal in value: One of their primary criticisms was that Metcalfe’s Law assumed all nodes in a network were equal in value. This assumption is fundamentally flawed because, in reality, the connections within a network vary greatly in their intensity and utility. Not all connections are used equally; some connections are much more valuable than others.

n (users) grows to infinity: The formula contains no constraint on growth. This unchecked exponential growth assumption is unrealistic and “dangerous,” according to BOT. Therefore quadratic growth rate of n-squared proposed by Metcalfe’s Law was “too fast.”

Their criticism extended to the notion that the unrealistic growth assumptions of Metcalfe’s Law were not only flawed but also unsubstantiated. During the dot-com boom, they claimed, many startups relied heavily on these assumptions for their business models. These companies anticipated that once they reached a critical mass of users, exceptional growth in network value would follow. However, when this exceptional growth did not materialize, it led to widespread financial losses and the eventual collapse of many ventures.

BOT argued that their n log(n) model provided a more realistic framework for evaluating networks, one that accounted for the diminishing returns of adding new users. They asserted that their model better explained historical patterns and the economic forces at play, offering a tempered and accurate perspective on network valuation. It recognized the complexities of real-world networks and the varying value of connections, providing a critical corrective to the overly optimistic projections of Metcalfe’s Law. So effective were these criticisms that BOT’s refutation is still frequently cited today. There was just one problem.

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As it happens, none of the so-called refutations posed by BOT are demonstrably true. In the two decades since “Metcalfe’s Law is Wrong” was published, much more data has come to show that, in fact, Odlyzko’s thesis is wrong, and Metcalfe’s Law is correct.

Let’s start with the inadequacies asserted by BOT. All nodes in a network were not equal in value: While the basic Metcalfe formula assumes equality in user size, so does BOT’s. Users are simply defined as “n”, with each n being equal.

It is reasonable to assume that this assumption does not hold.

The question is, “Does it matter?” Models are necessarily incomplete and often oversimplify the real world through a set of assumptions. For example, the Capital Asset Pricing Model (CAPM) assumes stock returns are the result of a single factor: risk. Should we really think that is true in the real world? No, but it even so, CAPM is surprisingly robust in its explanatory power, nonetheless.

An adjustment to Metcalfe’s formula (or BOT’s for that matter) assigning a weight to each n solves the problem quite easily. The details how of this is done and the result are found in the chapter on Zipf’s Law. Spoiler alert: as it turn out, this adjustment doesn’t even matter; it has almost no impact on the outcome predicted by the simple version of Metcalfe’s Law.

n grows to infinity: While this was an assertion by BOT, Metcalfe’s Law says nothing about how big n gets. It only says that, at a certain point, value exceeds cost, and grows proportional to n-squared. It doesn’t assume n grows to infinity, as BOT states. Oddly this criticism is ignored for BOT’s proposed model. Like Metcalfe’s Law, BOT also says nothing about how big n gets. On that basis, BOT’s appraoch n log (n) is also equally flawed. So, BOT’s proposed solution doesn’t really solve this “problem.”

By 2013, Metcalfe, however, did propose two solutions to the problem: First, there is a distinction between total network value and the value perceived by an individual user. “Who hasn’t received way too much email or way too many hits from a Google search?

There may be diseconomies of network scale that eventually 14

drive values down with increasing size,” said Metcalfe in 2013.

Metcalfe correctly modified his formula to account for the diminishing marginal returns to a user, termed Affinity 𝐴 = 𝑘 𝑛. 𝑉=

𝑘 𝑛 × (𝑛 − 1) × 𝑛 2

A is value per user and k is a proportionality factor that declines over time. This factor represents the Law of Diminishing Marginal Returns and decreases over time. At some point, the impact of A is probably greater than the impact of n2.

Second, he again correctly proposed that n grows like an S-curve.

“Don’t change my formula, change n,” said Metcalfe. “I went shopping for a function. And there it was on the web, it has its own Wikipedia page. It’s called the sigmoid. And the sigmoid function is cool. It’s an S-curve. Meaning, imagine you have some population, you begin by winning adoptions and eventually it flattens out. And the really nice thing about this law is that …this particular S-curve has the feature that the derivative is proportional to the number of members of the network times the number of people or devices that have yet to join the network.” The takeaway is that growth is at its peak when about half the people have joined the network.

An S-curve represents a pattern of growth characterized by three phases: initial slow growth, rapid growth, and then a leveling off or plateau as the growth reaches maturity. This type of curve is common in many natural and human-made systems, where growth is initially slow due to initial resistance or limitations, accelerates as the system gains momentum, and eventually slows again as it approaches a saturation point or encounters limiting factors. While Metcalfe used a formula called a logistic function, research shows that technology adoption more commonly follows a Gompertz function or a Bass Diffusion function discussed in the earlier chapter.

The Dot-Com Bubble: The dot-com bubble was driven by the widespread adoption of the Internet and World Wide Web, which attracted significant venture capital investment and resulted in soaring valuations of new dot-com startups. Low interest rates in 1998-99 facilitated an increase in capital availability, while the Taxpayer Relief Act of 1997, which lowered the top marginal capital gains tax, encouraged speculative investments.

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Additionally, the Telecommunications Act of 1996 spurred technological innovations that investors hoped to capitalize on.

Many companies prioritized rapid growth over profitability, engaging in excessive spending on marketing and infrastructure.

This speculative environment, bolstered by the enthusiasm of investors overlooking traditional valuation metrics, led to an 800% rise in the NASDAQ index. However, when the bubble burst in 2000, most startups failed after burning through their venture capital without achieving profitability, leading to a significant market crash.

There is scant evidence that Metcalfe’s Law was a contributing factor to the high-tech stock valuations of the late 1990s. This is in part because, as mentioned earlier, there was almost no data available on user growth at the time. More importantly, valuation is a tricky topic, encompassing not just mathematics and finance, but psychology, economics, and regulation. A review of earnings transcripts and analyst reports on such companies reveals that Metcalfe’s Law is almost never mentioned, let alone understood in a manner that would drive speculative investment.

Even today, Metcalfe’s Law, network effects, and network economics are not covered in professional investment education programs like the Chartered Financial Analyst or Certified Financial Planner programs. As recently as 2020, courses in these topics were taught in only a handful of universities around the world (this author was one of the first). In light of the dearth of education in the financial and investment industry, ironically, BOT’s claim itself appears to be unsubstantiated speculation.

Lack of Empirical Validation: BOT also failed to validate their purported law with actual data. Moreover, they offered no economic explanation or mathematical derivation for their “law,” which was not even in fact a law (meaning a mathematical tautology or empirical observation). In fact, BOT explicitly refer to their “law” as “a rule-of-thumb valuation.” This is quite an unscientific rationale, and hardly different than Metcalfe’s characterization of his law as “a vision thing.” They justify it on the basis that it “grows faster” than Sarnoff’s Law (where value is a linear proportion of users) but slower than Metcalfe’s Law (value is a quadratic proportion of users). In other words, this proposed replacement is better because it just seems better.

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Econometric Deficiencies: Neither Metcalfe nor his detractor competitors were versed in the field of Econometrics. Consequently, each in trying to justify their own laws did so in a flawed manner. In a way, the dispute is like two blind people arguing over who has better night vision.

Econometrics uses statistical methods to analyze economic and financial data, helping to understand and explain relationships.

Most financial data is time series data. These are observations recorded over time, used to identify trends and patterns in various fields like economics and weather forecasting. This data is challenging for econometric analysis because it often has correlated observations and requires specific techniques and considerations when conducting an analysis. Neither Metcalfe nor BOT did this.

Addressing these deficiencies is well beyond the scope of this book. However, suffice it to say that in subsequent research conducted on Metcalfe’s Law, these special considerations were addressed.

In 2013, researchers António Madureira, Frank den Hartog, Harry Bouwman, and Nico Baken conducted a study validating Metcalfe’s Law. Analyzing Eurostat data from 2002 to 2009, they identified 13 ways users derive economic value from digital networks, such as coordination and information search. Their findings confirmed that the value created by these networks increases quadratically with the number of users, providing strong evidence for investing in digital infrastructure to boost economic growth.

In March 2015, Xing-Zhou Zhang, Jing-Jie Liu, and Zhi-Wei Xu published a study in the Journal of Computer Science and Technology demonstrating that data from Tencent and Facebook validate Metcalfe’s Law. Initially proposed by Robert Metcalfe in the 1980s, the law posits that the value of a network is proportional to the square of its size. Their research analyzed over a decade of real data, concluding that Metcalfe’s Law provides a better fit than Sarnoff’s, BOT’s, or Reed’s laws. Additionally, they discovered that the costs for both companies scaled quadratically with network size, contradicting the linear cost assumption.

In August 2016, Van Hove published a paper examining the validity of Metcalfe’s Law. Van Hove highlighted methodological flaws in previous studies, such as those by Madureira et al. (2013) and Zhang et al. (2015), who tested this law using data from social networks like Facebook. He pointed out that accurate assessments require correctly identifying network effects, defining market boundaries, and accounting for network quality changes over time. Van Hove’s re-examination 17

suggested that while Metcalfe’s Law often fits real-world data better than alternatives like BOT’s model, careful empirical testing is necessary to avoid overestimating network value.

In his 2017 paper “Metcalfe’s Law as a Model for Bitcoin’s Value,” Peterson found that Bitcoin’s medium- to long-term price movements align with Metcalfe’s Law “exceptionally well.” This research accounted for the special time series considerations necessary for appropriate analysis.

Evidence overwhelmingly supports the real-world applicability of Metcalfe’s Law. Across diverse contexts and time periods, from early telecommunications to modern digital networks, we observe the principle in action, shaping the growth and value of interconnected systems.

Historical data from Bell Telephone Company in the early 20th century reveals a quadratic relationship between user growth and revenue, aligning perfectly with Metcalfe’s Law—decades before its formal articulation. In the present day, Tesla’s Supercharger network demonstrates a similar pattern, with company revenue closely correlating to the network’s expansion in a way that Metcalfe’s Law precisely predicts.

Even in the realm of cryptocurrency, Bitcoin’s long-term price behavior is best explained by a model based on Metcalfe’s Law, adjusted for network growth patterns.

These case studies, spanning over a century of technological evolution, provide robust empirical evidence that Metcalfe’s Law is not just theoretical but a fundamental principle actively governing network value in the real world. Three such studies are described in detail below.

Appendix C provides a graphical summary of several other validations.

Consider the fascinating case of Bell Telephone at the turn of This finding is presented to the public here for the first time ever, and categorically confirms Metcalfe’s Law in a surprising way.

Peterson stumbled upon an intriguing dataset by sheer chance.

While sifting through historical financial records, he found data on the number of telephone users and the corresponding revenue generated by Bell Telephone Company during the early 20th century. This serendipitous discovery led him to apply Metcalfe’s Law to this dataset, yielding fascinating insights.

The number of Bell Telephone users followed a linear trend, much like the initial slide developed by Dr. Metcalfe while at 3Com. Revenue, however, appeared to grow quadratically. the 20th century.

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A chart of pre-WWII Bell Telephone data mirrors Bob Metcalfe’s 1980 slide perfectly.

Peterson proceeded to calculate a Metcalfe network value using the number of telephones against the company’s revenue over several decades, creating a visual representation that revealed a clear correlation.

This relationship seemed to align perfectly with Metcalfe’s Law, suggesting that the value (or revenue) of Bell’s network increased quadratically with the number of users.

However, simply looking like a quadratic function is not the same as exhibiting quadratic growth. The human eye is ill equipped to discern the mathematical nuances of different growth formulae. Subsequent statistical tests found that Metcalfe’s Law was the best explanation for Bell Telephone revenue over the period and that BOT would have grossly underestimated Revenue (See Appendix A).

Peterson’s work prompts a deeper question: “Was Bell Telephone pricing its services based on an understanding of network effects?” The answer is a resounding “No!” Bell Telephone couldn’t have been using Metcalfe’s Law to set its prices because Robert Metcalfe was born in 1946, long after these early telephone networks were established. The company’s pricing strategies were likely influenced by other economic factors of the time.

The manifestation of Metcalfe’s Law in early telephone networks, even before its formal conceptualization, offers a compelling illustration of market efficiency and Adam Smith’s “invisible hand” principle. Despite the absence of explicit knowledge about network effects, the market intuitively recognized and valued the expanding telephone network in a manner that precisely aligned with Metcalfe’s Law.

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Metcalfe’s Law existed before Bob Metcalfe did.

This phenomenon resonates with the efficient market hypothesis, which suggests that asset prices reflect all available information and that market price reflects economic value in the long run. As individuals made self-interested decisions to join the telephone network based on its perceived utility, they unknowingly contributed to a collective outcome that followed Metcalfe’s Law. The resulting growth in network value and company revenue demonstrates how decentralized market forces can lead to efficient outcomes and accurate valuations, even in the absence of a formal understanding of the underlying principles. This real-world example not only validates Metcalfe’s Law but also underscores the power of markets to aggregate information and behaviors in ways that reflect fundamental economic truths.

As mentioned earlier, we expect that adding users increases the value of a network (or any enterprise for that matter.) What is revealing is that this growth follows a specific pattern we now recognize as Metcalfe’s Law. This phenomenon is reflected in how markets value networks today. But that value is not some haphazard percentage increase: it follows Metcalfe’s Law. Peterson’s findings about Bell Telephone underscore a truth about networks: their value grows quadratically with the number of users, a principle that holds whether we’re discussing early telephone systems or modern digital platforms. Yet, it’s important to note that while Metcalfe’s Law provides a useful framework, the real world often introduces complexities that the law doesn’t fully capture. These are addressed in later chapters.

The historical data from Bell Telephone serves as an early, compelling case study of Metcalfe’s Law in action, even before the law was 20

formally articulated. It shows how markets and networks operate on principles that often precede formal recognition or theoretical understanding. Additional quantitative study has concluded that Metcalfe’s law is a real phenomenon, and as such it has transitioned from theoretical tautology to empirical law.

However, the notion that Metcalfe’s Law only applies to telecommunication networks is a fallacy. The law has applications far beyond telecommunications. While the law originated in the context of telecommunications, its fundamental principle–that the value of a network grows non-linearly with the number of users–has proven applicable to many types of networks and systems.

Even if the assertion that Metcalfe’s Law only applies to telecommunications networks, it would still apply to Bitcoin. Bitcoin is a telecommunications network. Money is simply data that is communicated across a network and stored on a shared ledger. In this system, users exchange transaction information, which is then validated and recorded by miners on the blockchain. Much like traditional telecom networks facilitate the exchange of voice and data between users, the Bitcoin network transmits financial data, allowing seamless peer-to-peer transactions. This robust communication infrastructure enables the efficient transfer of monetary information, where the network’s value and utility grow with the number of active participants.

Tesla, Inc., founded in 2003 by Elon Musk, is a company known primarily as an automaker, manufacturing electric cars, as well as solar panel manufacturing, energy storage, and energy generation. The company name Tesla originates from Nikola Tesla, a Serbian-American inventor, electrical engineer, and futurist best known for his contributions to the design of the modern alternating current (AC) electricity supply system.

While Tesla has become famous for its electric vehicles, the company has more recently begun to move in the direction of an energy/electricity distribution company, or a power company. This shift in direction is similar to Amazon’s move from being an online bookseller to becoming a global provider of products and services.

Tesla’s shift from just an automaker to an energy/electricity distribution company has been a gradual one, beginning in earnest with the introduction of its Powerwall battery in 2015. This battery allowed customers to store energy generated by solar panels and use it later when the sun wasn’t shining. This was a major step in the direction of an energy/electricity distribution company, as it provided customers with a way to store energy for later use.

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Since the introduction of the Powerwall, Tesla has continued to expand its offerings in the energy/electricity distribution space. In 2016, Tesla acquired SolarCity, a company that designs, manufactures, and installs solar panels. This acquisition gave Tesla a foothold in the solar energy market and allowed it to offer customers a complete energy solution from generation to storage to use. In addition, Tesla has developed a network of “superchargers” where customers can charge their electric vehicles quickly, as well as a network of “destination chargers” where customers can charge their electric vehicles more slowly. This is similar to the way early power companies, such as JP Morgan’s Edison Electric Light Company, provided electricity to customers in the late 19th and early 20th centuries.

In addition to providing a way for customers to generate, store, and use electricity, Tesla is also beginning to offer energy services such as demand response, energy efficiency, and energy trading. These services allow Tesla to become an energy/electricity distributor, similar to a traditional power company, as it can now help customers manage and reduce their energy usage.

Finally, Tesla is also investing in energy infrastructure, such as battery storage and power lines, which will help it to better distribute energy/electricity to customers. This is similar to the way power companies invested in power lines and generators in the 19th century to provide electricity to customers.

Tesla’s Supercharger grid is an ambitious and rapidly growing network of electric vehicle charging stations. It is one of the first of its kind and has grown to be the largest in the world. The Supercharger grid is comprised of several thousand charging stations located in various places around the world, from cities to highways.

The Supercharger network has enabled Tesla owners to travel long distances with the assurance that they will have access to charging their vehicles. This has been a major selling point for Tesla, as it allows owners to take trips that would otherwise be impossible with a traditional gas-powered vehicle. The Supercharger network also provides Tesla owners with a convenient way to charge their vehicles, as they no longer have to worry about finding an available charging station while on the go.

The network economics of Tesla’s Supercharger grid is also quite impressive. The number of superchargers has grown at an average rate of about 34% per year.

Tesla’s network of Supercharger stations is a great example of network economics. In the case of Tesla, the company has created a network of supercharger stations that enable owners of its electric vehicles 22

to charge their cars quickly and easily. The greater the number of stations, the easier it is for Tesla owners to find a place to charge their vehicles, thus making them more likely to use the cars regularly. This, in turn, increases the value of owning a Tesla, making it more attractive to potential buyers.

Tesla’s Supercharger network has grown at an exponential rate.

By increasing the number of supercharging stations, Tesla would be able to connect with more customers and thus increase its network and its value. Additionally, more supercharging stations would enable Tesla to provide better service to its customers by providing them with a more convenient way to charge their vehicles. This would lead to increased customer satisfaction and loyalty, which would in turn lead to increased revenue for Tesla.

Tesla’s shift from just an automaker to an energy/electricity distribution company has been a gradual one, beginning in earnest with the introduction of its Powerwall battery in 2015. For that reason, our analysis begins in 2015, which also permits examination of the network size to be sufficiently large such that effects Metcalfe’s Law are noticeable.

With Supercharging stations as a proxy for network size (users), we can calculate the Metcalfe Value of Supercharging stations over time and compare that to Tesla revenue.

While this fit appears obvious, some statistical tests are necessary to achieve confidence that the relationship is what we expect it is.

Those tests are provided as part of the Appendix. We also did a comparison with BOT’s n log n model, and the conclusion was that Metcalfe’s Law offered the better explanation for revenue.

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Tesla’s revenue is a function of the network value proxied by number of Superchargers.

Tesla’s revenue and Metcalfe Value exhibit a consistent relationship.

When viewed as an energy network, rather than a car manufacturer, the value of Tesla becomes apparent. This distinction is crucial because the value of Tesla’s energy network can be explained by Metcalfe’s Law. For Tesla, this means that every new Supercharger station doesn’t just add a little value—it significantly boosts the entire network’s utility. And importantly, it is Metcalfe’s Law, and not some other metric like auto sales or n log n, that best describes this value. So, as Tesla continues to expand its Supercharger network, the value of the company

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grows quadratically, driven by the increased convenience and attractiveness of owning a Tesla vehicle.

There’s another important point to make. Revenue growth for Tesla for this period was about 65% per year, roughly twice (1.91×) the 34% growth rate in Supercharger stations. This relationship is incredibly important, because the mathematical first derivative of Metcalfe’s Law is 2 × % change in n. This relationship is discussed later in “The Rule of 2” and is one of the ways that offers mathematical proof that it is Metcalfe’s Law at work, and not some other relationship like n log n or a generic power law.4

Bitcoin is best analyzed as a digital token. Some history regarding a popular Italian telephone token−the gettone−is necessary because Metcalfe’s Law, upon which our work is based, originated from a description of telephone networks.

The word gettone (pronounced “jet-TONE-ay”, plural: gettoni) literally means “token.” The first Italian telephone token was created in 1927. It was a little disc made of an alloy of copper, nickel and zinc, or bronze. Production stopped in 1983 when it was replaced with magnetic phone cards. It is estimated that 600 million such tokens were produced.

Gettoni

Gettoni were commonly used as and interchangeable with a 50 Lira coin until 1980, when its value (and the cost of a phone call) suddenly doubled to 100 Lira. The doubling occurred again in 1984, to 200 Lira, again a result of a price increase associated with pay-phone calls. It remained at that value until 2001, when the Euro was introduced and the gettone suddenly lost its money-like nature in the Italian economy.

The parallels between the gettone and bitcoin are many. Interestingly, during the periods in which the token’s price was increasing or

4 A power law describes a relationship between two quantities where one quantity varies

as a power of another. For example, if you double the size of a system, some properties might increase fourfold, or more generally, as some power. The mere presence of a relationship, however, does not imply predictive ability. Metcalfe’s Law is a specific example of a power law. If power law is a category of fruit, then Metcalfe’s Law is an apple.

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expected to increase, Italians hoarded gettoni. Gettoni were readily exchanged into Lira, but not other currencies. Both serve only limited roles as a literal form of currency, and as fiat money both are intrinsically worthless. It was not necessary to have a gettone to make a phone call; one could use a phone at the home or office to do that. Likewise, one is not required to use bitcoin to make purchases, but can choose to do so for convenience or other reasons. People carried both gettoni and Lira, in the same way people hold bitcoins and their currency of domicile. Like bitcoin, the cost to counterfeit a gettone, relative to its value as a medium of exchange, was so high it was ridiculous to even consider it. And, like bitcoin, a user could do one of three things: spend it, exchange it for government currency, or hold it.

The holders of gettoni and the payphones themselves are a network. The value of a gettone to someone in that network, when spending the coin, is one of convenience and the value of the information relayed over the network. If we assume a growing number of pay telephones and callers, and then apply the constraint of a fixed number of gettoni, we have mirrored the key elements of Bitcoin’s supply and demand characteristics.

Bitcoin’s price is best modeled as a network. Metcalfe’s Law, adjusted for the creation of new Bitcoins over time, is best suited to this task. This approach provides insight into the long-term value of Bitcoin, but it does not attempt to explain short term price movements, which we accept can be driven by a multitude of factors.

Critics of Bitcoin, knowing that supply is essentially fixed in the short term, generally point to changes in demand as responsible for all price changes. That may be true in the short term, but it is also an oversimplification. Demand-side approaches are often misspecified because they ignore the non-proportional value added through the addition of a new user.

A viable model requires only three datasets: a proxy for users, an estimate of the number of bitcoins available, and bitcoin price. To value bitcoin, we need to undertake 3 steps: 1. Calculate the Metcalfe value; 2. Divide by the number of bitcoins; and 3. Scale to price.

To calculate Metcalfe value, we used Bitcoin addresses with a non-zero balance as a proxy for number of users. Finding a suitable proxy is part of the art of asset valuation. Other data could have been used, for example, number of wallets, number of Bitcoin ATMs, number

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of addresses with a balance greater than 0.01 BTC (or .10 BTC), or number of nodes.

Using that data, we used software to determine the shape of the adoption curve. Statistical analysis of the data confirmed that the growth in bitcoin users followed a Gompertz function. The Gompertz function is a mathematical model used to describe growth patterns that start slowly, speed up, and then slow down again as they approach a maximum limit. The Gompertz function captures this S-shaped curve, making it useful for predicting things like population growth, the spread of diseases, or how quickly people adopt new technologies. It’s a handy tool for understanding and forecasting scenarios where growth isn’t constant but follows a more natural, accelerating then decelerating, pattern.

Using this data, we applied Metcalfe’s Law. To calculate a value on a per-coin basis, we divided by the number of coins available, which is smaller than the total number of coins mined due to irretrievably lost coins. Lastly, an adjustment was made to scale this value to Bitcoin price.

Bitcoin’s price and Metcalfe Value. The deviations are mostly a function of shortterm changes in macroeconomic environment and not halvings as widely believed.

Bitcoin’s price behavior has been a source of much speculation and debate since its inception. Many observers have attempted to explain its volatile price movements through simple events like halvings.5 However, these approaches often fall short in capturing the full complexity of Bitcoin’s value proposition.

5 Bitcoin halvings refer to the event where the reward for mining new blocks is halved,

which occurs approximately every four years. This process reduces the rate at which new bitcoins are generated, leading to a decrease in the rate of production. While halvings are significant, they do not fully explain Bitcoin's price swings. Research has shown that the effect on price is not as strong as suspected, and that macroeconomic condition is a better explanation of Bitcoin’s price ‘cycles.’

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Bitcoin’s long-term price behavior is best explained by Metcalfe’s Law, adjusted for the network’s growth pattern using a Gompertz function. This model recognizes Bitcoin not just as a digital asset, but as a network whose value grows non-linearly with its user base. While short-term price fluctuations may indeed be influenced by various factors including changes in demand and macroeconomic conditions, the underlying trend follows the growth and interconnectedness of the Bitcoin network.

By understanding Bitcoin through the lens of network effects, we can better appreciate why its price behavior is often misunderstood or misattributed. The value of Bitcoin isn’t solely determined by its scarcity or by speculative demand, but by the utility and reach of its network.

As the network grows, each new user adds value not just for themselves, but for all existing users, creating a compounding effect that is captured

by Metcalfe’s Law. This perspective offers a more nuanced and potentially more accurate way to assess Bitcoin’s long-term value. It suggests that rather than focusing on short-term price movements or singular events, investors and analysts should pay attention to metrics that reflect the health and growth of the Bitcoin network. As we continue to observe Bitcoin’s evolution, this network-centric approach may prove invaluable in understanding its true potential and future trajectory in the global financial landscape.

The Rule of 2 is a powerful corollary to Metcalfe’s Law, providing a practical application of the network effect principle in a more accessible form. A corollary is a proposition that follows directly from a proven theorem or statement. It’s essentially a natural consequence or result of an already established principle. In mathematics and logic, a corollary is often considered simpler or more specific than the original theorem it stems from.

The Rule of 2 is the first derivative of Metcalfe’s Law. The first derivative of a function in mathematics tells us how that function changes at any point. Think of it like measuring how your speed changes as you drive. If we know how your position changes over time (your speed), we can predict how your journey progresses. In the same way, if we know how the size a network changes as more users join, we can predict its change in value.

This corollary stands as a cornerstone principle in understanding how network value evolves, particularly in the realm of cryptocurrencies and digital platforms. While Metcalfe’s Law states that the value of a network is proportional to the square of its users, the Rule of 2 distills this concept into a more intuitive relationship between user growth and value 28

increase. The Rule of 2 elegantly captures the relationship between user growth and value increase, stating the percent change in a network’s value is approximately twice the percent change in its number of users.

In mathematical terms, this is expressed as: % change in value ≈ 2 × % change in users.

The 2 in this equation comes from then n2 in Metcalfe’s Law. Put another way, the ratio of changes in users to price should be about 2: % 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 ≈2 % 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑢𝑠𝑒𝑟𝑠

To validate Metcalfe’s Law, one only need to compare changes in value (or price) to changes in users (or network size). The result should be close to 2, and not some other number like 1 or 4.6 Peterson did this for several cryptocurrencies, including Bitcoin, and for periods that covered several years. While there is certainly variation in the short term, the results centered around (or averaged) 2 in every case. For Bitcoin since 2015, the median value of 60-day periods of the lognormal7 ratio of price-change-to-user-change is 1.70; for 90-day periods it is 1.82; for 6-month periods it is 2.08; and for 12-month periods it is 2.02. This is exceptionally compelling evidence of Metcalfe’s Law at work.

The rule’s applicability extends beyond cryptocurrencies. Social media platforms provide another compelling example of how this corollary to Metcalfe’s Law manifests. If a platform like Facebook experiences a 10% increase in its user base, we might anticipate its value to grow by around 20%. This dynamic illustrates why social media companies place such a strong emphasis on user acquisition and engagement strategies, leveraging the network effects captured by Metcalfe’s Law and quantified by the Rule of 2.

Messaging applications and e-commerce platforms also demonstrate the rule in action. A messaging app seeing a 5% boost in users might expect a corresponding 10% increase in network value. Similarly, an online marketplace with a 7% growth in active users could potentially

6 For BOT’s model, this value converges to 1.0 at about 1,000 users.

So, if we want to know whether a network is following Metcalfe’s Law or BOT’s model, we calculate ln over many periods. If the result is 2.0, it is Metcalfe’s Law at work. If it is 1.0, it is BOT or some other model at work.

7 A log scale displays numbers on charts using equal multiples (e.g., 1, 10, 100, 1000)

instead of equal steps. This allows representation of very large and small numbers together, shows a huge range on one graph, makes exponential growth appear linear, and facilitates comparison of rates of change.

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see its overall value climb by about 14%. These examples show how the Rule of 2 provides a tangible way to estimate the impact of network growth across various digital ecosystems.

The Rule of 2, as a corollary to Metcalfe’s Law, helps explain the rapid value growth often observed in successful networks, which can sometimes outpace traditional economic models. It quantifies the power of network effects, demonstrating how each new user contributes disproportionately to the network’s overall value, a key insight from Metcalfe’s original observations. This principle provides network operators with a framework for strategic decision-making, emphasizing the crucial nature of user acquisition and retention efforts in harnessing network effects.

Money is a network and follows Metcalfe’s Law and therefore the Rule of 2. If 5% of Japanese Yen users leave the network and opt for Bitcoin (or something else), the value of the Japanese Yen would fall 10%. If 10% of the world’s fiat currency users stopped using it and instead saved or spent in Bitcoin, the value (measured as purchasing power) of that fiat would fall 20%. However, the purchasing power did not leave the economy. It was simply transferred to another currency, one that is not under the control of banks or governments. This principle helps explain why governments and banks are threatened by Bitcoin adoption. Relatively small changes in network size can translate to large changes in network value.

By providing a straightforward yet powerful tool for understanding the relationship between user growth and value in networked systems, the Rule of 2 offers valuable insights into the dynamics driving the growth and valuation of modern digital networks. It illuminates why seemingly small changes in user numbers can trigger significant shifts in network value, a phenomenon particularly evident in the digital and cryptocurrency landscapes, and directly tied to the network effects described

by Metcalfe’s Law.

Value in finance is not simply a subjective concept. The belief that value is inherently subjective often stems from conflating personal value with economic value. Personal value is indeed subjective, influenced by individual preferences and emotional attachments. Though value may be unobservable and imprecise, it is not arbitrary. Economic value is that which improves society in some way.

The production economy of the 20th century recognizes types of value—book, intrinsic, market, appraisal, and liquidation. They are determined through objective, systematic, and quantifiable methods. Each type of value—book, intrinsic, market, appraisal, liquidation, and 30

network—relies on specific metrics an These provide a comprehensive framework for evaluating assets based on different criteria and contexts.

The emergence of network value in the Information Age highlights the evolving nature of value assessment, recognizing the profound impact of digital networks and connectivity in the 21st century. Network value has become an essential addition to the traditional types of value. Network value captures the additional value created through network effects, where the worth of a product or service increases as more people use it.

Time value of money–a concept associated with intrinsic value– is a fundamental tenet of finance that recognizes that money available today is worth more than the same amount in the future. This is because money you have now can be invested to earn returns, while future money cannot.

To understand this better, imagine you’re offered $1,000 today or $1,000 a year from now. The $1,000 today is more valuable because you could invest it and potentially have more than $1,000 in a year’s time.

Typically referred to as ‘compounding’ or ‘compound interest’, this is expressed as 𝑉 = 𝑉 × (1 + 𝑟)

Where V1 – V0 is the change in value, r is the growth rate, and t is time. The percentage change in value (return on investment) is then %∆𝑉 =

𝑉 − 𝑉 𝑉

This concept is crucial in many financial decisions, from evaluating investments to planning for retirement. It’s why we have tools like compound interest, which allows money to grow exponentially over time, and why early saving and investing are often emphasized in personal finance advice.

Metcalfe’s Law can be directly tied to TVM by the relationship between growth rate in n and the required return of an investment. Assuming growth is constant and adoption is less than 50%, The Rule of 2 says 𝑟 = %∆𝑉 = 2 × %∆𝑛

or 𝑉 = 𝑉 × (1 + 2 × %∆𝑛)

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Asset valuations during the dot-com era, particularly for tech companies and telecommunications networks, faced significant criticism.

According to BOT, these valuations were primarily based on observing and extrapolating user growth numbers. By applying Metcalfe’s Law, analysts forecasted steeply rising values. This approach, as critics like BOT pointed out, often resulted in inflated asset values.

However, professional asset valuation does not use this approach. Asset valuation is the art and science of discounting future values to their present value using TVM, not extrapolating present value into the future. Here, the discount rate r plays a crucial role, reflecting both the time value of money and the risk associated with the asset’s future value. It is the inverse of the compounding formula: 𝑉 =

𝑉 (1 + 𝑟)

The discount rate in financial terms is a rate used to determine the present value of future cash flows. It reflects how much future cash flows are worth today. Now, if a network’s value grows rapidly (as suggested by the Rule of 2), the future cash flows from this network (like revenue from ads or subscriptions) are expected to be significantly higher. 𝑉 =

𝑉 (1 + 2 × %∆𝑛)

Why does this lead us to a discount rate of 2 × % change in users? High rates of growth are associated with high rates of variability and uncertainty. To account for this uncertainty, one must use a higher discount rate, which results in a lower (hence more conservative) present value. This approach incorporates a “margin of safety” into the valuation.

Using the OBT model requires that investors discount future values by %∆n—a number which is too small and therefore results in higher asset values. In all likelihood, the dot-com era investment industry was using %∆n (a modest growth expectation) as the discount rate rather than 2 × %∆n (which accounts for quadratic growth in value). This led to inflated present values, contributing to sky-high valuations that were incorrectly calculated.

The irony is that it is OBT’s method, not Metcalfe’s, that results in higher present values. By using only r = %∆n, analysts and investors were underestimating the true rate at which the network’s value was growing according to Metcalfe’s Law. Had analysts applied Metcalfe’s 32

Law, they would have discounted future values more heavily, resulting in lower valuations.

Metcalfe’s Law is now widely regarded in the professional investment community as the most compelling explanation for Bitcoin’s long-term price trend, offering both theoretical foundation and empirical support. The core principle–that network value grows proportionally to the square of its users–aligns well with Bitcoin’s distributed nature, capturing how its utility and value increase with adoption.

Empirical evidence supports this application, showing strong correlations between Bitcoin’s price and network size proxies like active addresses or transaction volumes. This relationship holds true over both short-term fluctuations and long-term trends.

Compared to alternatives, Metcalfe’s Law stands out. Simple power laws lack specific explanatory gravitas, while speculative theories fail to account for persistent growth trends and underlying utility.

Metcalfe’s Law uniquely explains Bitcoin’s dramatic price increases during rapid adoption periods and increased resilience as the network grows.

While the conventional Metcalfe model has limitations, such as not addressing node addition costs or user size differences, it provides a powerful lens for understanding Bitcoin’s fundamental value drivers. Despite imperfections and external influences on short-term movements, this framework offers a valuable tool for assessing Bitcoin’s long-term potential in the global financial ecosystem.

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Using data from 1916 to 1937, we applied the “Rule of 2” to test whether the lognormal ratio of revenue R to users n was 2×. This is the first derivative of Metcalfe’s Law, where ln ln

𝑅 𝑛

𝑅

≈2

𝑛

This test was conducted for rolling 1, 2, 3, 4, and 5-year periods.

The Wilcoxon Signed Rank Test was used to determine if the median value for each dataset was significantly different from 2.0.

For each dataset, the median value was 1.94, 2.07, 2.14, 2.04, 2.05 with p-values of 0.5459, 0.5459, 0.9591, 0.9591, and 1.0000 respectively. These p-values correspond to the results of the Wilcoxon signedrank tests performed on each of the five datasets (1-year, 2-year, 3-year, 4-year, and 5-year rolling data). The p-values are all greater than the conventional significance level of 0.05, indicating that for each dataset, we fail to reject the null hypothesis that the true population median is equal to the given median value. Interpreting these results: For the 1-year and 2-year datasets, the p-value of 0.5459 suggests that there’s no strong evidence to conclude that the true median differs from the given values (1.94 and 2.07 respectively).

For the 3-year and 4-year datasets, the higher p-value of 0.9591 indicates even less evidence against the null hypothesis, suggesting that the data are very consistent with the given median values (2.14 and 2.04 respectively).

For the 5-year dataset, the p-value of 1.0000 suggests that the data are perfectly consistent with the given median of 2.05.

These results suggest that the given median values for each dataset are plausible estimates of the true population medians, based on the available data. However, it’s important to note that failing to reject the null hypothesis doesn’t prove that the true median is exactly equal to the given value; it only indicates that we don’t have sufficient evidence to conclude otherwise.

For periods of 6 years and greater (not shown), the median value was 2.69. With Signed Rank Test concluded that the true value was likely not 2.0 but higher. This stands in stark contrast to the proposition of BOT, which states that growth should be lower (close to 1.0), especially over longer periods.

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ChatGPT was used to perform an econometric analysis comparing network value under Metcalfe’s Law vs. BOT’s competing model.

Data consisted of the number of Tesla Supercharging stations from 2015-2022, and quarterly revenue for that period.

To understand the relationship between network value and revenue, we need to decide between using a VAR (Vector Autoregression) model and a VECM (Vector Error Correction Model). This decision hinges on whether the variables we’re studying move together in the long run, a concept known as cointegration.

VAR is suitable if our variables are stationary, meaning their statistical properties don’t change over time. However, if our variables are non-stationary but show a long-term relationship, VECM is more appropriate because it accounts for both short-term fluctuations and long-term equilibrium.

In our case, both Metcalfe’s and BOT’s values and revenue are non-stationary but cointegrated.

Cointegration in econometrics is like the relationship between chickens and eggs; it’s difficult to separate them to determine which causes the other. Over time, chickens and eggs move together in a stable, long-term relationship—more chickens lead to more eggs, and more eggs eventually lead to more chickens. Similarly, in financial data, if two variables like network value and revenue are cointegrated, they are intertwined in such a way that they consistently influence each other, making it hard to disaggregate them to determine cause and effect.

When variables are cointegrated, we use a VECM (Vector Error Correction Model) instead of a VAR (Vector Autoregression) model.

VECM accounts for both short-term adjustments and long-term equilibrium, providing a clearer picture of how these interdependent variables move together over time. making VECM the right choice. This model helps us see how quickly the variables adjust back to their long-term relationship after a change.

When it comes to deciding which model better explains the relationship between network value and revenue, the comparison between Metcalfe’s and BOT’s values offers intriguing insights. Think of the ec1 coefficient as a measure of how quickly things return to normal after a disruption. It’s like observing how swiftly two friends, who get separated in a crowded place, find their way back to each other.

The Metcalfe model shows that both Metcalfe’s values and revenue adjust reliably and quickly. Specifically, Metcalfe’s values have an ec1 coefficient of 0.2352, indicating a prompt adjustment. Meanwhile, revenue also responds with a significant ec1 coefficient of 0.3158. These low p-values highlight the strong and dependable nature of these adjustments. This balance suggests that both the network value and revenue 35

are resilient and interconnected, returning to equilibrium effectively after any disturbance.

On the other hand, the BOT model demonstrates an exceptionally rapid adjustment of revenue, with an ec1 coefficient of 0.7492, signifying a very quick return to normalcy. However, BOT’s values adjust more slowly, with an ec1 coefficient of 0.1595. While this model excels in showing how quickly revenue reacts, it lacks the balanced adjustment seen in the Metcalfe model.

BOT’s n log n also lacks a theoretical basis, unlike Metcalfe’s Law which describes value in the parsimonious and intuitive context of network connectivity. In light of this, and as a comprehensive and balanced understanding of how both network value and revenue stabilize after changes, the Metcalfe model emerges as the better choice. It offers a more holistic view, highlighting the strong, mutual adjustment mechanisms at play between network value and revenue.

Statistic

36

Metcalfe

BOT

ec1 Coefficient

0.2352

0.1595

ec1 p-value

0.000

0.001

L1.Revenue Coefficient

0.0674

0.078

L1.Revenue p-value

0.048

0.098

L1.Metcalfe Coefficient

-1.3047

-0.9436

L1.Metcalfe p-value

0.234

0.171

ec2 Coefficient

-

-0.0747

ec2 p-value

-

0.154

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