What Is Metcalfe’s Law? An Easy Primer
Lesson F: Covering What the Other Articles Don’t Explain Well
Ever heard of the phrases “network value” and “network effect”?
They convey the idea that as a network grows so does its utility and value. Think about social media sites, for example. The more users who are active members, the more utility there is for connecting and sharing; and hence the more value there is for users (in terms of friendship, sentiment, and connectivity) and the people who own the network (e.g. advertisement revenue and data monetization).
Metcalfe’s Law is the formal way of expressing network value exponentially based on the number of a network’s nodes or users. In determining this value, Metcalfe’s Law takes into account:
- the number of nodes or users
- the number of connections
Nodes are basically hardware operated by users of the network — such as, computers, smart phones, even fax machines.
Before we delve into more detail, I just want to announce a little caveat about some existing internet resources on Metcalfe’s Law:
In my research, I came across quite a bit of plagiarism. Much of the content is simply copied and pasted from the Wikipedia page on Metcalfe’s Law, while others simply just mirror comments made by other authors. One of the key giveaways was how the Wikipedia page does not really explain the law for non-specialists in mathematics. The same lack of explanation was prevalent in the other sources.
Ok, enough of the rant. This article is not about accusations of plagiarism in any detail. I want to repave the landscape, as it were, by providing a clearer account of Metcalfe’s Law for the beginner. I will elide any discussion of the history of Robert Metcalfe, which you can find on the aforementioned Wikipedia entry.
To start, let’s think about Metcalfe’s Law in two parts. Each part is a formal expression of the law under different, respective conditions.
Part 1: Network Effect & Value Per the Number of Connections between Nodes
Let’s begin with an example provided by Sebastian Purcell, who illustrates:
“Imagine for a moment that the telephone is just being invented. You are setting it up in the “Wild West” with one person across town. How valuable is that “network” of two people?
Well, let’s say value, V = 1 because the two of you just have one connection.
What if you added 1 more person, how much more valuable would that be?
Well, V = 3 because even though you can call 2 different people, they can call each other, so there are three connections.”
To help visualize how this calculation works, think of each node as a dot or point, with each connection as a line between them. In Purcell’s example, we’re thinking about 3 user nodes and the resulting unique connections amongst them:
Formally speaking — that is, in terms of a mathematical formula — this image can be expressed as:
- n (n-1) / 2
So with respect to 3 nodes or users:
- 3 (3–1) / 2 = 3
The network value is based on the number of unique connections.
For a network of 3, as we confirmed above, the network value is 3.
Now, let’s try a larger network. How about a network of 9 nodes? Think of 9 nodes as dots or points. Can you visualize how many unique connections there are between those nodes/dots?
Pretty difficult, even if you sit down and draw them out:
Care to count those connected lines? Instead, just plug 9 into Metcalfe’s equation:
- 9 (9–1)/2 = 36
What are the implications of this first part of Metcalfe’s Law?
On the face of it, we can derive a numerical value to act as a comparative measure of networks. But there’s more. Let’s turn to Part 2 to get a sense of the potential of a network effect when there is no upper limit to new users.
Part 2: To Infinity and Beyond
If you browse around the internet in attempting to understand Metcalfe’s Law, you’ll often not find much with regard to Part 1. Instead, you’ll find the statement that Metcalfe’s Law is basically the observation that the value of a network grows non-linearly. In other words, it does not proceed from 1 to 2, from 2 to 3, etc.
We saw above that with 9 nodes, the network value is 36 based on the number of unique connections between the nodes. A simplified version of Part 1 captures the basic non-linear growth in terms of a square of the number of nodes. This simplified version encapsulates Part 2, or “n squared”:
So with our example of 9 nodes, the non-linear growth can be understood as 81, compared to Part 1’s value of 36.
Why this difference between the two parts of the Law?
Mathematically speaking, there is no upper limit to the number of nodes that can join a network. The number can increase to infinity. So to account for this absence of an upper limit, whatever value we derive from Part 1 of Metcalfe’s Law can be represented proportionally in relation to infinity.
Here’s how the Wikipedia entry puts it:
“Metcalfe’s Law is related to the fact that the number of unique possible connections in a network of n nodes can be expressed mathematically as the triangular number n (n-1) / 2, which is asymptotically proportional to n2.”
You may have noticed a peculiar term in that account — i.e. “asymptomatically”. We need not, for our purposes, go into the technical mathematical details of the term. For one, I am not a mathematician! We can simply rely on a basic definition from Dictionary.com. “Asymptotic” means:
“(of a function, series, formula, etc) approaching a given value or condition, as a variable or an expression containing a variable approaches a limit, usually infinity.” (my bold emphasis)
In other words, if we want to think of the value of a network in relation to (but not reaching) an infinite number of nodes, we can simply take the number of known nodes and square that number. Hence, 9 nodes have a network value of 81 in view of a growth and adoption rate of new users approaching but not reaching infinity.
Graphically rendered — that is, when plotted on the X and Y axes— we get the following:
What you will often find, therefore, are quite a few articles stating that Metcalfe’s Law is simply “n squared” or “n2” (the “2” would be in superscript).
And this reduction can be misleading since it tends to overstate limitations of the law with respect to
- uniformity of connections in a network,
- uniformity of value for each node, and
- the assumption that networks actually do not experience a declining growth rate due to “the most valuable links” being “formed first”.
For more on limitations, see Wikipedia and Peterson.
How This Can Be Applied
Never underestimate the power of networks. For example, one can say that the value of a social media site is worth much more than the revenue gained from its advertisements, or even its monetization of personal data (barring some unforeseen dystopian capitalist exploitation!).
This is because value has many different senses when it comes to networks, in terms of the various ways in which utility is provided through the often immeasurable and surprising largesse of human, social intercourse.
So, when thinking about creating or being a part of a network, a strong physical connection to others and to utility is what the technological platform needs to recreate. Friction, visibility, and moderation are but three factors that have become most prominent in recent times. Reduce user friction, facilitate visibility, provide a reasonable form of moderation.
Stayed tuned for the second part of this article, where we look at Metcalfe’s Law in relation to cryptocurrencies.
This article is a part of the Crypto Industry Essentials educational program presented by The Art of the Bubble.
Though this article is credited to me, it contains some written material by Sebastian Purcell, PhD from his The Art of the Bubble education series on cryptocurrencies.
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