A source of personal frustration since 1995. Back then I was giving a background talk on some simple basics of the Internet to some business executives at the company I was with. Someone asked about Metcalfe's Law - something I was unaware of.. I had to ask what it was and was told that it stated the value of a network increased as the square of the number of people in the network. Standing at the whiteboard for 30 seconds I was able to come up with how they got it and also a big flaw in the statement...

Since then there has been a huge amount of blind belief in Metcalfe's Law and Reed's Law leading to some rather spectacular valuations on building out fiber and and new network connections in the late 90s along with companies that were just looking for users - forget the little issue of being profitable and, more recently, in social networks.

More recently people are wondering about the network effect and the value of investments with the non-performance of some social media companies...

A bit of background is probably in order..

There are three "laws" involved (they really aren't laws, but the people who invoke them treat them as if they are solid truth - usually without understanding what they are based on and what they imply)

• **Sarnoff's Law** - after David Sarnoff of RCA fame in the 20s and 30s. It implies the number of possible network connections increases as the number of users. It usually speaks of broadcast networks where a message is sent via radio or TV and anyone with a receiver can "tune in" ... The implication is the more viewers, the greater the value. Note that nothing is said about the strength or value of the individual connection. Note also that it could be newspaper or magazine subscriptions...

• **Metcalfe's Law** - after Bob Metcalfe, inventor of ethernet. It is the number of pairs that can be made in a group. For n members, it is possible to assemble n*(n-1)/2 distinct pairs. If n is a large number, this is very close to n^{2}/2, which is the normal statement of the "law" - people also say it increases as n^{2} ... Since we are used to thinking about communication connections between a pair of people, this is a natural way of thinking about the number of connections on something like a telephone network, but it ignores group conversations. Note that it says nothing about the strength of the connections - just that they are possible.

• **Reed's Law** - after David Reed, an MIT prof who is also something of an Internet boffin. David talks about the number of distinct subgroups that can be formed in a larger set of members. For a group of n members it is possible to have 2^{n} distinct subgroups, but n of them have just one member (themselves) and it includes the group with no members (null set). So the number of subgroups that have at least 2 members is 2^{n} - (n + 1). As n gets large, the first term dominates and the number of subgroups gets close to 2^{n}. It says nothing about the value of these subgroups to their members, but is just the mathematical limit of the number of possible subgroups.

They aren't wrong - as long as they are carefully restricted to their specific domains, but somehow people interpret these as the value of networks and likelihoods rather than limits. Additionally Reed's and Metcalfe's "Laws" are frequently confused.

When talking about these it is probably important to mention Dunbar's Number. The assertion is the maximum number of stable and persistent social relationships you can manage is determined by some cognitive limit and for humans this is about 150. This determines effective organization size, the scale of your meaningful social network and so on. It is highly debatable, but it is clear the number of relationships is finite - people don't have 1,000 close personal friends and certainly not 10,000,000 (a few Facebook phenoms have millions of "friends"). More careful analyses suggest we have layers of friends and other social connections and the average limit is on the order of 100 to 300 people. We usually only have a handful who constitute close personal relationships and maybe a few dozen we might trust.

Back in 1995, and how these "laws" are used today, the question was "what is the value of the network?" Here one can immediately dismiss Reed's Law - if the value of any subgroup is the same, the value of going from n to n+1 members doubles the total value of the network. So if 1,000 users was worth - say $10 - 1,001 would be worth $20. Another 9 members increases the value to a bit over $10,000 and adding another 30 members increases the value by well over a billion times. It would make real sense to find those additional 20 members. It wouldn't take too many more to increase the network's value by more than the total economy of the Earth. Most of these subgroups will never exist and you need to consider their value on a case by case basis.

Metcafe's Law doesn't see such a dramatic increase, but it does increase much faster than what is observed as n becomes large. Back in 1995 I thought a bit about the problem when I was standing at the white board and suggested, half jokingly, that it probably scales as n*log(n) as so much of nature seems to follow a logarithmic increase and we clearly see more "value" than a linearly increasing broadcast network (Sarnoff's Law).

Since then Andrew Odlyzko and others have suggested a limit of value probably increases as - surprise - n*log(n) and show some fairly convincing arguments, but not a proof.

To figure out value, social effects must be included - linkages vary greatly in value. I'm inclined to believe that, but also belief there is a plateau effect going on - a manifestation of something like Dunbar's Number along with the strength of social connections. There also seems to be a burnout effect - particularly among people who begin to get real lives (although there are a large number of people who just waste their idle time or who look for ways to spend bits of time when everything else seems boring).

In any case it is complex. Sociology and anthropology are very complex subjects and there is enormous room for discovery. I suspect that the nature of electronically mediated social connections as well as hybrid connections with machines and real human interactions, will keep evolving. I think that services like Facebook and Twitter happened to offer an "improvement" over what currently existed for many - but the deep social connections just aren't there and the long term prospects (say 10 or more years) are iffy at best.

Perhaps the huge number of new services being tried are just an inefficient monte carlo-ish mechanism that, over time, will help us sort out some rather thorny sociology and anthropology.

For some people and organizations these can be useful and effective tools. For others they can be not terribly useful to wastes of time and money. There will be an evolution, but if you find something useful by all means do so and work on learning how to increase its value. But also appreciate there are limits. If you are in a situation where Sarnoff's broadcast networks are the end of the road there may be value here, but nothing is trivial.