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The Opposite of Metcalfe's Law
(This was my original commentary on a draft version of a July 2006 IEEE Spectrum Article refuting Metcalfe's law. Some of my criticisms have been fixed in the final article.)
As the authors suggest, Metcalfe's law is indeed wildly overoptimistic, and Reed's even more so. However, as they point out, both Metcalfe and Reed were aware of this. The "Laws" are more metaphors than rules, intended by their creators as such. The authors suggest the real damage has come from others taking the "laws" more seriously, and using them as a formula for business.
That may have happened but it was quite frequently not so. During the dot-com boom, companies were valued based on linear relationships to quantities like "eyeballs," "pageviews," and of course multiples of revenue rather than earnings. This applied even to community sites which purported to have a network, rather than a one to many broadcasting system. The authors do not consider if a site with 10,000 members truly was valued in the market as 4 times a site with 5,000 members, or any other such quadratic relationships. Nor do they show examples of companies being valued by Reed's law, or through their nLog(n) suggested rule.
The most telling networks for network effects are communities. This means E-mail networks (for which islands have entirely vanished) and Instant Messaging networks, which mostly remained islands. The biggest IM island (AOL) persists, though Yahoo and Microsoft have agreed to federate, and Google launched using an open standard, pre-federated with Jabber.
In fact, the authors suggest that the markets were consistent with a disbelief in Metcalfe's law, in that they did not always amalgamate. The authors need to present an example of a network that somebody really attached value to under Metcalfe's Law. If nobody actually believed the law when it came to putting their money where their mouth is, a refutation of the law is uninteresting.
There is also a far more dramatic refutation of Metcalfe's law. Many networks, particularly online communities such as newsgroups and mailing lists, suffer from negative factors as the number of members and connections increases. These increases cause two factors. First, the level of noise (entirely undesired messages) increases. This also includes deliberate noise such as spam, "trolling" and flamage, as well as accidental noise.
Furthermore, as such networks grow, the amount of signal also grows. It does not grow with O(n^2) as Metcalfe's law suggests, and it would be a subject of future research to discover any rules regarding the growth of signal. However, as signal grows, it begins to become too much. Soon both the message volume and noise volume grow too much, beyond the capacity of a reader to handle. One cannot handle 500 wonderful well reasoned opinions on a topic, no matter how good they are. Even in the unlikely event that they are not mostly redundant, one can't handle them.
In this case, the value of the network actually decreases in proportion to the number of people on it.
This phenomenon is well known, if not well understood. There are many attempts to find solutions to reduce the signal. These include editors and moderators who limit signal and reduce noise, and even community moderation systems where various members of the community recommend messages and they percolate up in value. (eg. Slashdot.)
However, it is clear that the inherent value of the network is not increasing with size, and that only through the use of these tools is it able to grow usefully at all. Nonetheless many prefer smaller networks of selected individuals of "higher quality" to larger networks subject to editing or moderation, and it is an undecided question whether the value truly increases with size.
It is generally accepted that the problem of "flamewars" is an artifact of larger networks, particularly because the participants are less likely to know one another personally, but also because of the viral nature of flamewars, where one insult or controversial statement is likely to touch off others. While the name "flame" was chosen to express the anger of the comic character Johnny Storm, the ability of fire to spread when given enough fuel is an apt metaphor. A small group of people can discuss abortion or war or politics in a meaningful way. A larger group simply cannot, not as a fully connected network. One might suggest the probability of a flamewar is proportional to the square of the number of members of the network -- the number of possible channels of anger, and thus the value of the network inversely proportional to the square of the number of people on it.
A common early use of online communities was to ask questions. The larger the community, the more likely it could answer your question. In theory, if the probability increase were linear with the number of members, and all those members had questions, the value would be O(n^2). However, in reality expertise is redundant, and for many matters, the probability of an answer may increase with the log of the network size, or even less. However, for many questions, a near 100% probability of answer will be reached quickly. Having more members provides little to no benefit. The authors admirably refute the O(n^2) claim but in fact even their proposed O(n log n) fails after some number.
It does, however, have a downside. The more members there are asking questions, the more noise (uninteresting questions) there is, and the more likely those who can answer questions will not read the messages, or leave the community altogether. After a certain size, the community becomes less valuable as the size of the network grows. Only advanced tools again can try to restore value. In many cases they must do this by sending questions only to a subset of those who can answer them. At best, such a network grows in value linearly with size, simply because more people are able to come for problem solutions.
In truth there is no one "law" for the value of the network. It is highly dependent on the purpose of the network, the type of nodes or people that join it, and the technologies that exist to mitigate the effects of too much signal and too much noise in a large network.
Few networks will grow in value with the square of the size. Some will be n log(n). Some will grow in value when the size is small but then will decrease, value becoming inversely proportional to the size of the network or some factor of it. Determining the form of this "hump" curve of value is a matter for further research.