# Hacking Metcalfe’s Law

May 15, 2016 · 9 min read

How to use piggybacking to boost a lot the value of your social network.

Metcalfe’s law is a theoretical informatics statement that says that “the value of a telecommunications network is proportional to the square of the number of connected users”. It was first formulated in 1993 by George Gilder and is often used to describe network effects. Its name comes from Robert Metcalfe who created Ethernet. It basically means that the strength of a social network grows a lot when its number of users increases a little.

But why?

What you should care about when creating any kind of social network is not its number of users but the amount of connections between them. That is where the real value is. It is not obvious at first, but it is actually kind of logical after all. Facebook is not good because of its features, Facebook is great because all your friends are already on it. Having a phone is cool because you can actually call people with it and that is probably why the French Minitel (which was maybe better technologically speaking) has never been a success. Finally, if everybody had an email address but nobody sent any email ever, the value of “emailing” would be zero. The only growth rate you need to take into account to understand the engagement of your customers is not the increase in the number of users but the growth of the number of interaction between them.

In this article, we will call social network any kind of product that creates one-to-one relationships between users. So we will not only consider apps such as Facebook, Twitter or Instagram but also more classical communication media such as the cell phone network, emails, online payment… We could make a far more precise and detailed study, considering all the cases and variables possible but this article is just a simplification of the reasoning.

Now, let’s do some math! We can consider two kinds of social networks. First, the ones like Facebook, and Linkedin. We can call them friendly-social networks. In this case, the relation between two users is bilateral. If user A is friend with user B, then and only then user B is friend with user A. For this kind of network, the total number of potential relations among a set of n users is n(n-1)/2 (We assume that a user can’t be friend with himself).

The other sort of social network could be called following-social networks. It involves products such as Twitter, mail services, and Paypal. If user A follows user B, it does not necessarily means that user B will follow user A. Similarly, if you mail a friend, it sadly does not mean that your friend will mail you back. Even more sadly, if you send some money to someone with Paypal, you will not get the same amount of money back. Here, the number of potential connections is n(n-1) because each set of two users can create two connections (one in each way).

In both cases, the total number of potential connections between users is proportional to n^2. When the user base of a social network grows by 2x, its value increases by 4x.

There are two direct implications to this:

• For a big network, the total number of connections (so the value of the network) will be huge. It is the part of the Metcalfe’s law we always talk about. That is why Facebook is so big and powerful, as an example.
• For very small networks (the ones we always forget to talk about), the total number of connections will be extremely low and the growth of its value will be very slow. Our goal in this article will be to understand why it is so and how to boost this number of potential connections and, consequently, the value of the network you are creating.

First of all, let’s consider your segment market is… well, a segment. The green bar represents your market share and the red part the part of the market you still need to conquer.

Now, let’s take two of these bars and make a graph out of them.

The green area then depicts all the potential connections your users can actually make inside the network: that is people from your the network they can reach directly: it can be two people becoming “friends”, the sending of a message, a transaction… The red part shows all the potential connections that could eventually be made in your total available market.

We can quickly deduce that the value of the network is proportional to this green area.

Now, let’s see why the assumptions we made earlier are way more important than they appear to be by considering the following graphs.

On the left graph, the market share is about 3%. On the right graph, it is about 95%. So if the number of users of the right graph is 30 times higher, the total value of the network is almost 1000 times bigger. Most of the time, when you create a social network of any kind, you are in the first situation at the beginning. Your users will only be able to communicate with people who are also using the network.

If you take the case of Facebook or emailing, which are examples of the second situation, almost all the potential connections of the total market are possible inside the network. You can virtually send an email to anybody who has an internet connection if only you know their email address and you can start a conversation in Facebook Messenger with almost anybody providing you know their name.

On this graph, you can see that the size of the network grows very slowly at the beginning and then increases faster and faster as the number of users grows until it reaches the size of the market (1,000 users, in this case). But this is not what we want. What we want is quick growth right from the beginning.

## Piggybacking

For a social network, piggybacking is the fact of using a far more mature network as a vehicle for growth. Here are few examples:

• Farmville is piggybacking on Facebook
• Facebook piggybacked on the real life “social network” of Harvard students
• Instagram piggybacked on Twitter at first then on Facebook when they got acquired
• Paypal used to piggyback both on Ebay and email addresses
• Whatsapp piggybacks on the users phone contacts (using clever onboarding)
• Airbnb piggybacked on Craigslist

When developing a whole new social network, entrepreneurs often focus on two different areas: one the one hand, the relations between two members (the actual product) and, on the other, the acquisition of new users. Piggybacking is about merging these two parts of the job by letting all active users interact with potential customers who are not members of the network yet.

Let’s take get into detail about Paypal. If they had launched the service just like their competitors did, they would only have allowed transactions between their members and then would have bought ads. If they acquired, say, 1,000 users, the value of the network would have been something like 1,000^2 = 1,000,000. The growth would have been slow and difficult. What they actually did is that they allowed members to send money to non-members only using their email address. Let’s say there were 100,000,000 internet users at this time having an email address. The potential number of connections would then be 100,000,000 x 1,000 = 100,000,000,000 which is 100,000 times higher. They then just had to give positive incentives for non-members to sign up when a member wanted to interact with them. That is what they did offering \$10 to each freshly invited customer.

Empirically, piggybacking appears to be the best way to start growing a social network. Let’s get our graph back to understand why.

On this graph, the orange part depicts the connections between one user of the network and one user of the piggybacked network.

Actually, one of the orange bars represents connections from a customer to a non-customer and the other one from a non-customer to a customer. Depending on the situation and on the kind of social network, we will need to consider only one or both of them. For a matter of simplicity, we will always consider both of them for the rest of the article.

The beautiful thing about these bars is that the smaller your market, the bigger they proportionally are! Mathematically speaking, their area represents 2*x*(n-x) potential connections (where n is the size of the market and x the number of customers). Here it is on a graph — You might have noticed that I like graphs :)

If you keep acquiring users at a unchanging rate, the value of the network will not grow faster and faster but slower and slower. By using piggybacking, we just made the growth rate function concave instead of convex. The two curves meet again once you have conquered your whole total available market.

## But what’s the point?

Well, just looking at the graph, the utility of this article does not seem obvious because the result will be the same at the end. But the end is sadly never where we think it will be. Even if Peter Thiel tells us to aim at monopoly, most of us will never ever reach it. Most of the time when your total available market is about several dozens/hundred million people, having half of it as customers will be huge. Actually, even Apple can’t sell an iPhone to everybody on earth, and Facebook “only” has 1,5 billion active users out of 3.5 billion internet users in the world.

So let’s crop the graph.

The impact of piggybacking becomes more obvious there.

It is important to understand that you will not need growth ten years from now but as soon as possible. That is what piggybacking allows you to do.

Do not be sectarian. Let users talk to non-users. Let them interact together because if the interaction goes well, the other member will come in and sign up.

Feature development and customer acquisition should work together.

## But…

We only speak here about potential connections. Growth hacking is nothing if you have nothing to growth-hack about. Managing a network with many potential connections is nice but it worth zero if you do not make these connections real. Piggybacking is just a way to increase this potential number, it’s your job to convert them into reality.