Fri, 19 May 2006 18:27:38 +0900 >>>>> "Ben" == Ben Tilly <btilly@gmail.com> writes: Ben> Metcalfe's Law (BTW, note the spelling) is going to apply Ben> best in networks where people actively wish to connect with Ben> fairly random strangers. Very few networks look like this. That's because historically commodity markets were the only practical place where that would work. It is not an accident that the huge complex of networks we call the stock market are concentrated in maybe a dozen institutions worldwide, and they coalesced into an Intermarket so long ago that we no longer think of them when we talk about networks. What search technology *may* make possible is scaling up search in a heterogeneous network to a billion participants or so. Ben> I suggest that you should look up Benford's Law and explain Ben> why it doesn't apply to this type of content when it does to Ben> other kinds. What in the world do you think Benford's Law has to do with anything here? Benford's Law is about decimal representations of a monotonically increasing sequence, not about values. Eg, if I were going to bet on leading "digits" in web content, I'd go for 60 base 256, wouldn't you? Ben> This argument, like Metcalfe's original argument, has the Ben> implicit assumption that everyone is equally interested in Ben> all potential connections. I'm not. Nor is anyone else. Oh, come on, Ben, give us a little! You're capable of presenting the math explicitly, and it's worth doing so here. The question is "given an increasing size of network, how does the value distribution change?" If we assume that to some agent (note the abstraction!) the probability distribution over (gross) value of connection to a random other agent is constant, then the expected (gross) value of the network *is* proportional to network size for each individual. So what you need is marginal cost of connecting bounded below that average value, and voílà! net value per agent is linear in network size, and you got Metcalfe's Law. At one extreme of marginal cost, we have the marriage market. For me it was zero for q=1 and infinite for q=2. So then the value of the network to me was an order statistic (max Vi over i = 1, ..., N). Anything that increases network size increases personal value, but definitely not linearly (bounded above by "10", of course!) So effectively we've increased the value of a rival good, but not changed it into a network good á là Metcalfe. Can we get to the other extreme, constant marginal cost below average value? If we're talking about communication networks among humans, no. Social contact of positive value requires time bounded below on the order of a second or so (however long it takes to perceive your baby's smile), and there are only a finite number of seconds available to each of us. However, anonymous networks (eg, markets) or machine-mediated communication (sorry, no examples---that's Tom's job! ;-) might come pretty close, at least up to population levels feasible for humans on this planet. I think the jury's out on just how far Metcalfe's Law can extend its domain---but the financial markets are a pretty compelling example for "hey, this really does work sometimes!" Note that a market is a pretty trivial network topology (star). But then, the basic property of the Internet that these laws depend on is that it turns Life, the Universe, and Everything into a star topology. Ben> You've just counted connections and assumed that they all Ben> have equal value. But that assumption flatly contradicts Ben> observations of how people really network in social Ben> situations. First, it's not an assumption of equal value, really. It's an assumption of equal expected value per connection, independent of the number of connections. Apply Law of Large Numbers, live Metcalvian ever after. Second, consider Mr. Lynch's problem. Have you ever watched a headhunter network a LUG meeting? ;-) If such people (excuse the exaggeration) are a constant fraction of the population, whammer jammer, Metcalfe's Law! (In the long run, nobody else will matter if they obey Metcalfe's Law---their Metcalfe value is a lower bound on social value.) Third, remember that many participants in networks are not people, they are firms. To the extent that a firm can have a bounded labor force with the additional costs of networking borne by capital, a network of firms (or capital-enhanced humans, for that matter) could exhibit rather different networking behavior. (Then again, maybe they wouldn't. That was Jamie Zawinski's point about groupware.) Ben> And Google's power derives from the fact that good searching Ben> allows people to come closer to perfectly extracting the Ben> value of being on the network. Uh, what are you trying to say here? I think it's arguable that Google increases the size of the network in some sense; I'd like to see a careful argument why that isn't so. Ben> As for your defence of Reed's Law, well what you're saying is Ben> that you think that it is reasonable that adding one person Ben> to the internet could double everyone's perceived quality of Ben> life. If you think that that's reasonable, then I don't Ben> really know what to tell you. "The King!" ;-) >> This is not to say that at some point, every new member has >> that effect. The constant factor on these approximations >> matter and we are lucky/happy when a few great minds per >> generation or 10 have such effect. Ben> Oh no. Reed's Law *explicitly* says that every new member Ben> has that effect. Which is why it doesn't pass the "smell" Ben> test. Strawman! Strawman! Out to the cornfields with you! Reed's Law would work if the fraction of great minds per generation were constant regardless of size of generation, and everybody else was worth precisely zero in this sense. I think that's unlikely but arguable. The problem with Reed's law is elsewhere. It is that it assumes that all needed groups actually form. If there's one great mind, then she has to participate in all (N-1)(N-2) possible groups for her to get Reed's law to work. Again, you could adjust this for some sort of average or probabilistic formation, but I don't think that makes much more sense than assuming all groups form---you still need the number of groups forming to be proportional to the number of possible groups, and eventually Ms. Great Mind will get tired of all that networking. And it needs to be the case that having formed the group-of-one-billion, *you* would get an equal increment of value from also forming the group-of-one-billion-less-*me*. Hardly seems likely! :-) Please note, I agree with you, not Tom. My intuition is that in the long run Metcalfe's law will fail, and network values will be sublinear. But I strongly believe that the reason why they fail matters, both to the degree to which Metcalfe's Law approximates value for "small" networks, and to long-run industrial organization of networks that start out small enough for that approximation to be valid. -- Graduate School of Systems and Information Engineering University of Tsukuba http://turnbull.sk.tsukuba.ac.jp/ Tennodai 1-1-1 Tsukuba 305-8573 JAPAN Economics of Information Communication and Computation Systems Experimental Economics, Microeconomic Theory, Game Theory