Friday, July 17, 2009

A Corollary to Metcalfe's Law

This is an update of a posting from a while back. I'd love to hear thoughts, comments, pointers to research on the topic, ...


Moore's "Law" is one of the most famous ideas in the computer business. Moore's law drove great benefits and value in this industry for quite some time.

Slightly less well known, but driving a great deal of today's business and Internet innovation and value, is Metcalfe's "Law." Again, this is not a law, but an idea -- that the value of a network increases with the square of the number of participants in a system (nodes, users, etc.). This is probably not true as the number of participants gets larger beyond some saturation point -- for example, temperature sensors are more valuable when they are in every zip code, but are they 100 times more valuable when there are 10 in any zip code? Is it (ten thousand times) more pleasant to live in a city of 100 million people than one of 1 million? Probably not.

Nonetheless, this "network effect" is pretty easy to understand -- if there are two phones, only two calls can be made (A to B, B to A); if there are three, then six calls can be made (A to B, A to C, B to A, B to C, C to A, C to B); and so on. This creates the potential for value.

A Corollary:

However, the growth of a network also tends to create a surety of cost, and here is the basic idea behind it. Simply put, this corollary states that the cost of participating in a network increases with the number of participants. There are reasons and consequences of this corollary to Metcalfe's Law. The most important consequences are that the cost/benefit ratio of a community tends to increase (becoming less desirable) as the community gets larger, and that a community cannot survive unless the cost/benefit ratio is low enough. Of course, when sentience is involved, exogenous factors can intervene.

Reasons for This Corollary:

The reasons for this corollary are related to the varying strength of relationships, inhomogeneity of the community and its interests, network overlaps, the cost of broadcast and "stay awake" messages, the cost of rules for a network (which generally vary from network to network and are often not explicit), and -- not to forget -- the costs on network (or ecosystem) participants placed by abusive participants (predators, parasites, criminals, scammers, practical jokers, enemies, etc.).

Think of networks like Facebook, cable/satellite television, the Internet, e-mail, a private forum like "The OracAlumni Network," or an open forum like Twitter. When the first user joins the network, there is no value. As friends or colleagues or those with similar interests join, the network can grow in value (proportionately to the square of the number of participants). Initially, the network may be not really valuable, or it might be quite valuable, per unit of time invested to participate. Over time, as there are more participants and more content, the cost to participate increases -- more redundant messages, more messages that are important to just a subset of the community, more "flame wars," more people on the To and CC lines, more CYA messages, more "people who don't bother to search," more sp-m, more attempts by scammers and other undesirables to penetrate the network, and more abuse. The cost to participate rises.

When most human networks or communities form, they form around some common experience (communities of interest) or purpose (communities of practice). The initial participants are often very homogeneous, with strong relationships between them. This applies to frontier towns in early American history, Usenet, mySpace, or LinkedIn. However, as the community grows, there is stratification in the participants (native San Franciscans vs. immigrants, descendants of Plymouth Rock vs. newer immigrants, newbies vs. experts), and the stresses that come from the new participants repeating the history of the old -- or interfering with the status quo ante. Messages and rules are created to manage change, and violators (even unwitting ones) are often made to pay a high prices (those who carried guns into Dodge City, a newbie who doesn't read the FAQ). Worse yet, parasites, criminals, and enemies realize that there is value being created in the community, and they come to feed off that value without creating any of their own (Nigerian bank scammers, other bank robbers, flamers, sp-mmers, and frauds).

The Math:

Let's define the Value of a network to be V, the cost of a network as C, the cost of participation in a network (for any individual i) as c(i), the benefit of participation in a network (for any individual i) as v(i), and the number of participants in the network as n. The number of unique connections in a network of a number of nodes (n) can be expressed mathematically as n*(n-1)/2.

The total value of the network would be the sum over i of v(i), = a * n^2 (according to Metcalfe), where a is the unique benefit of that community or network (as in helping find a job, defend against enemies, solve technical problems, or provide enjoyment). This would imply that the average benefit or value of the network to each particupant would be v(i) = total value divided by number of participants = a * n -- Metcalfe may not have been right here, as it stands to reason that a network does not always increase in value as additional nodes are added, and a community does not increase in value forever as its size increases. Of course, some participants could benefit far above v(i) and some could have a lesser benefit - or even zero benefit, or a negative benefit (those who got killed in Dodge City, those who get scammed on the Internet, those few who get stuck with the entire income tax bill of the United States).Similarly, C = sum over i of c(i), which I claim increases with the number of participants and related not to the specific community, but instead to the media of participation, and also to the effectiveness of the rules system in minimizing costs (moderation of a forum, ability to block scammers, lynchings in Dodge City). Media that have high costs will increase the cost of participating -- in-person meetings or cross-country migrations on foot would be more costly than videoconferences, which would be more costly than phone calls or e-mails (which are asynchronous, non-interrupting, and require little time for parsing), which are more costly than SMS text messages.

Let's define m as the cost of the medium for the network or community. A community moderator or good community policing would tend to reduce the cost of participation as well -- let's define r as the effectiveness of rules enforcement in the community or network. The cost to participate in the community goes up with the cost of the medium for participation, and down with the effective implementation of a common rules system. Thus C = the sum over i of c(i), where the average of c(i) = m / r * n. This implies that C = m / r * n^2, which would mean that the cost / benefit ratio of a network or community is C / V = (m / r * n^2) / (a * n^2) = m / r / a.


If the community offers enough benefits to the participants at low enough cost, then the community can survive. If the cost / benefit ratio is too high, the community dies off; if the cost / benefit ratio is low enough, the community will grow. This is why there were boom towns on the American frontier, as well as ghost towns -- also why many cities survive in the same locations after centuries and even millenia. It is also why fads like mySpace, AOL, and communes, decline over time -- and why they grew in the first place. "Coolness" can be a benefit, but that benefit can wane over time.

I contend that Metcalfe's "Law" is wrong -- the value of a network increases in proportion to the square of its nodes (or participants) only until it reaches some critical size, at which point the marginal contribution of additional users becomes smaller. If this is true, and if this corollary is also true, then the cost / benefit ratio in a network decreases as participation increases, to a point where it may become flat and then start to rise without limit. The cost / benefit ratio must be less than unity (1) for participation in a network to continue for any rational participant.

Summary and Next Work:

I need to do a lot more thinking here, to refine this to a workable model. Individuals will not value the network's benefits equally, and individuals will continue to join the network if they perceive the benefits of participation to be greater than the costs of participation (including the opportunity cost). Individuals will stay in the community until the costs of participation exceed the benefits of participation, at which point they will exit the community. There is a stochastic analysis I need to do to refine this concept. But I do believe there is an interesting and valuable kernel of an idea here ...

Please add comments if you can point me to work that has been done already on this idea, or if you have any contributions to make. Thanks!


  1. Hi there.

    I came across this post whilst checking out Metcalfe's Law. I am not a mathematician, but I am interested in trying to work out in real numbers what your corollary means. According to Wiki, Metcalfe claims 5 connections (friends) means 10 pssible other connections, 10 equates to 45, 12 to 66 etc etc.

    If this is wrong, what figures would you put in place? Could your Corollary be explained in simple terms for us non mathematical types?

    What you say, however inasmuch as it relates to organisations relying on Metcalfe to justify massive investments etc (Dot Com boom and bubble) is, I belive very important, and should be widely shared.

    Thanks for your post.

  2. Mobius Loops -

    The number of connections in a network of "n" nodes is n*(n-1)/2. If you have 5 nodes, then the number of connections is 5 * (5 - 1) / 2, or 10; if you have 10 nodes, then the number of connections is 10 * 9 / 2 or 45. My point is that each node has potential value, but also potential cost. In a computer network, there is some overhead to keep the network alive - checking for connections, two connections sending simultaneously etc. In a human network, there are the costs I mention above. If Metcalfe's Law were correct, networks like MySpace and AOL would never stop expanding, because they would create value forever; my point of view is that, while they may create value overall for the network while the network grows, for each individual there is a cost to participate in the network. This cost can be controlled by limiting anti-network behaviors as described in the blog entry. Thanks for your questions - let me know if you have any other questions, if you take this thinking further, or if you find any issues with the analysis. Thanks!