Metcalfe's Law and those left behind
Not that I’ve searched (feel free to leave comments about what I’ve missed), but I haven’t seen any discussion about the number of non-users of the network.
Back in the 1980s when people started to put answering machines on their home phone lines, I hated leaving messages on them. So I didn’t buy a machine myself — until some of my friends started complaining that if I wasn’t home when they called they’d have to keep calling back. I was now subjecting them to an annoyance they were no longer used to (it didn’t matter that, not long earlier, having to phone people repeatedly had been a normal part of life). I had moved from the norm to the exception, even though I had done nothing. So I bought an answering machine.
My father didn’t buy a computer until he was 70 or so. My impression was that what pushed him over the edge was the computer-savviness of a friend of his who was 86: computers now seemed de rigueur for everyone, not just the younger folks.
Do you see what I’m driving at? When “everybody’s doing it”, people feel pressure (external or internal) not to be left out, and as a result often “do it” too. The popular term for the external stuff is “peer pressure”.
For social networks, this force is very relevant. People join MySpace not just because of the “positive” network effects but also because of the “negative” effects of being left out of where their friends are. So while Metcalfe’s Law applies, and Reed’s Law, so does, hmm, dare I call it Jayasekera’s Law?
Oh, I guess that would require some math! (I have a math degree, but I don’t make use of it very often. The last time was to make shoulders fit better in made-to-measure suits tailored by The Fitter system.) OK, so let’s say that the number of participants in whatever is N, out of a total possible number of participants U (for universe). If N is a small percentage of U, there is little pressure to join in. But if that percentage grows, and grows so much that N dwarfs U-N (which is the number of those left behind), the pressure can become great.
How great? The closer the percentage gets to 100%, the greater the pressure. So we’re interested in how close N/U is to 1 (or “unity” in fancy math terminology). As in Metcalfe’s Law, I would expect the relationship to be more than linear, e.g. square as in Bob Metcalfe’s own formulation. In which case pressure P ~ (N/U)^2. Or maybe it’s more aggressive, e.g. P ~ (N/U)^3. Or even, if the pressure is quite low at first but ramps up spectacularly when the percentage gets really close to 100%, an inverse relationship based on how close 1-N/U is to zero: P ~ 1/(1-N/U) . While I don’t know what an accurate formula would be, I suspect the curve would fall somewhere in the range from square to inverse. In the time-honoured phrase of mathematicians: the actual formula is left as an exercise for the reader.
While applicable to similar situations, this is a somewhat different kettle of fish from Metcalfe’s Law and Reed’s Law, which have no limit on their independent variable (the size of the network): the dependent variable (the value of the network) can just grow forever. Jayasekera’s Law (hahaha — having your own blog means you can say whatever you want) “ends” when everyone’s a participant (which shouldn’t actually happen because there are always rebels). But if the potential membership is growing, as is MySpace’s for example as its appeal continues to broaden geographically, demographically and broadband-penetrationically, all these forces can apply simultaneously: the invisible hands of Metcalfe, Reed and Jayasekera (hahaha — Adam Smith may be long deceased but our hands are all quite visible) can all be active at once. (Rant: people are sometimes tempted to reduce a complex situation to a single force, e.g. Reaganomics was based on the idea that supply-side economics controlled the U.S. economy, and the Kyoto Protocol is based on the idea that the greenhouse gases generated by humans control global temperature. Such oversimplifications are appealing because they suggest that a difficult problem can be solved by a single course of action, but appealing doesn’t mean effective.)
I do believe that this effect has wider applicability than just to social situations. For example, in the original Metcalfe’s Law context of Ethernet, a device that is not connected loses perceived value relative to those that are, more so as it becomes more of an unconnected oddball.