Metcalfe’s Law in Referral Networks

2key Network is a blockchain-based referral network intended to reward referrers through smart contracts. Referrals progress through the network thanks to people sharing them through their regular browsers. Therefore, 2key generates off-chain cryptographically signed links, which propagate among users without reaching the blockchain. Upon conversion, a user submits the signed link to the smart contract. The smart contract rewards the chain of referrers as represented in the signed link. The smart contract is deployed per referral campaign by the campaign initiator, depositing a referral reward, as an amount of cryptocurrency, from which the smart contract will later pay the referrers.

The Influence Graph

A campaign conducted over the 2key network starts with a source seeding, and proceed through referrals. The graph of referrals from the source seeding is called the influence graph. Each node in the graph is called an influencer. The node which makes an actual conversion, such as purchasing a product, is called a converter. A referrer leading to conversion is to be rewarded from the campaign budget. Not only is the referral leading to the conversion is to be rewarded but also referrals that had some impact on anyone leading to the conversion.

Now for a bit of terminology. The user creating a campaign is called a contractor, the users doing referrals are called influencers, and a user actually doing a conversion is called a converter.

2key is currently working on developing the algorithms for computing rewards which may not be uniform but depend on a reputation model.

2key Smart Contracts

Each campaign deploys a new contract. Rewards are paid through the campaign contract. While each campaign using the 2key service deploys its own contract, a shared collection of admin contracts govern the 2key token economics.

Naturally, the reputation of contractors, influencers, and converters will be managed by a shared smart contracts facility. So users can accumulate reputation.

Metcalfe’s Law

This section is excerpted for Wikipedia.

Metcalfe’s law states the effect of a telecommunications network is proportional to the square of the number of connected users of the system, n2.

Network effects

Metcalfe’s law characterizes many of the network effects of communication technologies and networks such as the Internet, social networking and the World Wide Web. 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.

The law has often been illustrated using the example of fax machines: a single fax machine is useless, but the value of every fax machine increases with the total number of fax machines in the network because of the total number of people with whom each user may send and receive documents increases. Likewise, in social networks, the greater the number of users with the service, the more valuable the service becomes to the community.


In addition to the difficulty of quantifying the value of a network, the mathematical justification for Metcalfe’s law measures only the potential number of contacts, i.e., the technological side of a network. However, the social utility of a network depends upon the number of nodes in contact. If there are language barriers or other reasons why large parts of a network are not in contact with other parts then the effect may be smaller.

Metcalfe’s law assumes that the value of each node n is of equal benefit. If this is not the case, for example, because the one fax machines serve 50 workers in a company, the second fax machine serves half of that, the third one third, and so on, then the relative value of an additional connection decreases. Likewise, in social networks, if users that join later use the network less than early adopters, then the benefit of each additional user may lessen, making the overall network less efficient if costs per users are fixed.

Modified models

Within the context of social networks, many, including Metcalfe himself, have proposed modified models in which the value of the network grows as n log n rather than n2.

Validation with Actual Data

Despite many arguments about Metcalfe’ law, no real data-based evidence for or against was available for more than 30 years. Only in July 2013, Dutch researchers managed to analyze European Internet usage patterns over a long enough time and found n2 proportionality for small values of n and n log n proportionality for large values of n. A few months later, Metcalfe himself provided further proof, as he used Facebook’s data over the past 10 years to show a good fit for Metcalfe’s law (the model is n2 ).

In 2015, Zhang, Liu, and Xu extend Metcalfe’s results utilizing data from Tencent, China’s largest social network company, and Facebook. Their work showed that Metcalfe’s law held for both, despite the difference in audience between the two sites; Facebook serving a worldwide audience and Tencent serving only Chinese users. The Metcalfe’s functions of the two sites given in the paper were for Tencent:

7.39 * 10^-9 * n^2

and the value of Facebook is:

5.70 * 10^-9 * n^2


The Value of a Referral Network

A referral network value is measured by its ability to reach a significant amount of conversions as referrals propagate through the network. Campaign effectiveness is measured by being able to reach conversion fast, starting from the source seeding. We would like to achieve a high ratio of conversions to referral network size. It is definitely the case that we intend to avoid casting a wide referral net by someone just to reach a few conversions. Not only is it not effective, but it will also cost the campaign in terms of rewards. Moreover, wide useless referral fanout might cause the campaign to be labeled as spam and damage the reputation of the contractor and the 2key network.

These considerations raise the question of whether Metcalfe’s law is applicable to referral networks.

Do Referral Networks Depend on Network Effects?

If you consider a population of n potential users, and you plan a campaign that the above describe measures of progress imply that its effectiveness in reaching a significant conversion depends on the reach of its source seeding. So it would seem we need a reach network effect in order to cover the potential converters as soon as possible. So the desired effect is really a small degree of separation between the source seeding and the group of converters.

But what is the network effect that we need? It is not the network of referrals but the network of connections of the user population. This network of connections is the substrate on which our referral network runs.

Hence, Metcalfe’s law is definitely applicable to referral networks.

The Reputation Model Represents the Network of Connections

The 2key global reputation model which is accumulated over many campaigns is our representation of the network of connections of anyone ever involved in the 2key network.

The reputation model, albeit being partial as it represents only referrals that actually occurred over campaigns, is our approximation of the network of connections. We do not know the full network of connections that exists in the real world, but we can use our approximation to bring network effects into a bear when conducting a referral campaign.

Moreover, we intend to incorporate the reputation model into computing rewards for referrals. The reason is that as you bring through a referral new users into the fold of our reputation model you increase the network effect. This is due to each new user being a potential influencer or converter into a subsequent campaign.


We have distinguished two network effects that are in operation in a referral network:

  1. The network of connections of users through which they propagate referrals.
  2. The reputation model of 2key that approximates this network of connections.

It is clear Metcalfe’s’ law is applicable to these two networks.

However, the network effects we seek for the influence graph of a particular campaign is not a Metcalfe’s law effect but should be measured with more goal-directed measures. We intend to describe these in the future in detail.