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Metcalfe's Law Refutation Explained 79

sdpinpdx writes "According to this article in the July 2006 IEEE Spectrum Metcalfe's Law (that the value of a network is n^2) is wrong (it's probably only n log(n)). The authors speculate this had something to do with the .com bubble, and that their more conservative model might help alleviate the next one. The article includes an interesting quote from Metcalfe: 'The original point of my law (a 35mm slide circa 1980, way before George Gilder named it...) was to establish the existence of a cost-value crossover point--critical mass--before which networks don't pay. The trick is to get past that point, to establish critical mass.'" This would seem to be an update to a story we ran more than a year ago.
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Metcalfe's Law Refutation Explained

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  • by KingEomer ( 795285 ) on Friday July 14, 2006 @01:24PM (#15720169) Homepage
    Hrm. The editors can't recognize that an article is a dupe of one submitted two days ago, but they can recognize that an article is related to one posted a year ago? Weird. :P
  • by JPribe ( 946570 ) <[jpribe] [at] []> on Friday July 14, 2006 @01:32PM (#15720225) Homepage
    So what if I spend 10 minutes devising some silly and arbitrary, yet very simple formula to place value on something as subjective as the value of a network. But, I'll do one for patent lawsuits...though this formula will give the relative ignorance of the original patent, measured in PES-Bs (patent examiner smoke-breaks.) I think I will square it by LEMIs (Large Entity Monetary Incentives) and divide that by the total number of patents submitted in that CY (Calendar Year.)

    So we get: (PES-B ^ LEMI) / CY Patent total.

    Can I get that formula named after me??
  • Look, just tell me how many pipes I need and when to buy a truck. Anything else is beyond our comprehension. Plus, does this even compensate for the poker chips and horses? I don't think so.
  • Refutation? (Score:3, Funny)

    by Otter ( 3800 ) on Friday July 14, 2006 @01:34PM (#15720248) Journal
    I don't think this is as much a "refutation" as it is three people's assertion countering some other guy's assertion. Since the one guy is richer than the other three put together, I'd say the burden of proof is on them.
    • Re:Refutation? (Score:3, Informative)

      by treeves ( 963993 )
      You're probably right, but I thought I'd mention that one of the author's names I recognized as a top-notch mathematician: Andrew M. Odlyzko []. I read about him in a book [] about the race to prove the Riemann Hypothesis [].

      I'd say he's a pretty smart guy - I don't about practical or "street" smarts - but some smart people don't value money so highly.
  • by Doc Ruby ( 173196 ) on Friday July 14, 2006 @01:37PM (#15720273) Homepage Journal
    That law treats a network as if its only value is its interconnectedness. Especially while some nodes send more info than they receive, some nodes are more valuable, and some connections are more valuable. Unless the actual information transmitted has no value to the network.

    Which is what I've gathered from Metcalfe's InfoWorld columns since then.
  • by Kaenneth ( 82978 ) on Friday July 14, 2006 @01:40PM (#15720295) Homepage Journal
    Depending upon the uniqueness of each node.

    Having two different Legal dictionaries offers less definitions than having both a Legal and a Medical dictionary.

    Two bricklayers or two Carpenters may build a house slower than one carpenter and one bricklayer.

    And a car wouldn't get very far if all it's wheels spun clockwise.

    Back when computers were more specific purpose (This one is for Payroll, this one for Budget, this one for Customer tracking, this one for the actual Job...) linking them together had amazing potential, but now when an entire operation could be run off one machine (Quickbooks, Photoshop, Coreldraw, Web Browser, Fax server were all together one one machine I know of, and all critical for the business) there's not that much data that needs to move over a network to run the business.

    Wikipedia, for example, would still be very useful even if it had zero links to external sites, because in itself it encompasses so much. Amazon does not need to offer links to other retailers, because they sell near everything.
    • Wikipedia, for example, would still be very useful even if it had zero links to external sites...

      Yeah, but what if there were no incoming links -- HTTP_REFERRER and HUMAN_EYE links? All of those links give Wikipedia its value. If there were no readers, there would be no value.
    • a car wouldn't get very far if all it's wheels spun clockwise.

      Uh, they all DO spin in the same direction (unlike the other examples where things are actually different) - what changes is your point of reference, looking from a different side ;)

    • Two bricklayers or two Carpenters may build a house slower than one carpenter and one bricklayer.

      However, 27 carpenters and 46 bricklayers may build a house slower than one "special class" shop student who isn't allowed to handle anything "pointy."

      Unless they "cooperate" by assigning 70 people to go for coffee and donuts, at which point the house may stall at the meeting stage when all the time available gets used up argueing over who gets to drive.

    • I agree, the value of nodes (and even a single node) must be variable. The carpenters/bricklayers example is good but I think the specific/general purpose computers one is less so. The nodes have moved from the host to the applications themselves but the number of nodes hasn't reduced to one in the case of special purpose computers being replaced by a general purpose computer. That aside, there is lots of evidence that I find supportive of variable value nodes. The contents of a box that can't be opened (

    • And a car wouldn't get very far if all it's wheels spun clockwise.

      It's called NASCAR, and you're right: the race tends to tend exactly where it began.
    • In other words, dupes [] reduce the value of the network.
  • by fermion ( 181285 ) * on Friday July 14, 2006 @01:52PM (#15720375) Homepage Journal
    One thing that people tend to do is take first approximations and limited domain fits and try to expand them to be a rule of life, the universe, and everything. Then when you try to explain that the chicken is not in fact spherical, they get really mad and call you a liar.

    The thing we seem to know from things like process control, is that it takes a finite amount overhead to manage any group, and a very finite amount of resources to bring an outsider into a group. This is Brookes; Law, that says bringing more people onto a late project will only make it later. We see this action around us right now.

    What I find most fascinating is how easily people will allow themselves to be deluded by a model, even though the reality is all around them. If we look at something like graph theory we see certain features. For instance, no one has an extremely large number of close friends. Most of us have what can be considered concentric circles of people we know, each group out is usually bigger, but more loosely connected. Communicating with the outer circles are very inefficient. Business are arranged the same way.I think what confused people is that the internet, like the telephone, made geographic distances less important, so it is easier to keep up communications with someone across the world, but that does not mean that the person's ability to relate has been increased.

    Additionally, not everyone, or everything, can competently complete all tasks, and not all processes can be factored to take advantage of all resources. At some point one is paying for overhead that does not deliver any added efficiency. I think this is what we are seeing in many international corporations. The corporation supports non-productive real estate, managers, IT, which forces the productive parts of the company to work harder and be less responsive to market forces.

    I would say that that a network initially has a n^2 benefit, but quickly transitions to nlog(n). This is not so. If anything cause the dot com crash, it was not understanding that at some point the overhead begins to be the dominant factor, and efficiency is lost.

  • by Opportunist ( 166417 ) on Friday July 14, 2006 @01:56PM (#15720399)
    Either I don't get it or this is virtual dick-waving. For what I'd say, it's not size that matters, it's how you use it.

    It's not the number of connected hosts that tell you about the value or quality of a network, or how much can be accomplished with it. You can network the biggest LAN in the world and have it play Quake all day, I'd put my money on the 5 computers calculating some more primes back in the basement.

    The value of a network lies in what it connects. Not in its size.
    • I hate to burst your bubble, and I don't disagree with your statements.

      However the network that he is talking about is not analogous to a LAN. Think of it more as a social network, like the six degrees of separation thing. Who you know matters very little until your network of friends gets to a certain critical size.

      What he was trying to say with algorithm is that you can measure how valuable a social network, not computer network, is.

      • The analogy works in social networks too. If your social network consists only of people of the same group (like, say, you're in some sect that only allows contact with others from the sect), your group may be huge but it still offers little insight or input, or "value". If your network consists of people with a very narrow field of expertise, the value is quite limited too.

        If your network consists of few people who are a source of tremenduous insight, every single one of them, your network is small but its
    • "Either I don't get it or..."

      You don't get it.

      The point is not that 10 granny smith apples tied together with string form a more or less valuable network than 5 crays. The comparison is between different networks of the same thing but of different sizes. And it is about the value of the network itself, not what the networked things accomplish. There is an underlying assumption that the metric under consideration values the networking; it is irrelevant that one can always think of a task for which netw

    • by MountainLogic ( 92466 ) on Friday July 14, 2006 @03:40PM (#15721117) Homepage
      This all came about because Metcalf was trying to make a case for networking (e.g., ethernet). Back then the ethernet cards he was selling were expensive. The decision maker would go, "gee, if it cost $x to network two people why can't Bob just walk down the hall to Jan's office?" If X is greater then the cost of Bob "walking down the hall" (or snail mailing or flying...) then there is no busines case for installing a network. More to the point:

      If the node cost, x, is $100 and there are 100 users, n, then the cost for the network is $10,000.

      If the single user business value, v, of the network is $10 for one user then the ROI for different valuation methods is:

      Linear: vn = $1,000 -- no business case, don't even think about it

      Metcalf's Law: (n(n-1)=2)v = 49,500 -- winner

      Metcalf's Law as misused by dot-bombers: N^2 * V = 100,000 -- "Proves" selling frozen mud on the net is a winner

      As restated by the authors: n long (n) * v = 2000 -- no business case, but better than a flat linear

      There really are two problems here. The scaling formula and setting the business value. If you set the business value for a single connection greater than the cost of the network then it is a no brainer, but back when Metcalf as pushing networking that was a hard case to make and given how many people use /. at work that may still be the case.

      • there is no difference between n(n-1)/2 and n^2 you fool. the only difference is a constant since n(n-1) ~ n^2. since metcalfe didn't include a constant of proportionality in his theory this makes no difference, the point is about asymptotics.

        there is a difference in your numerical example only because it is misconceived.
      • Metcalfe's Law: (n(n-1)=2)v = 49,500 -- winner

        Metcalfe's Law as misused by dot-bombers: N^2 * V = 100,000 -- "Proves" selling frozen mud on the net is a winner

        As restated by the authors: n log (n) * v = 2000 -- no business case, but better than a flat linear

        Um. That's not how Big-O notation [] works. O(n(n+1)/2) is the same as O(n^2). Constant terms don't matter. So your n log n might as well be 10n log n. Or n ln n. Or whatever. You can't plug in your n into the function and expect a useful number out of it.

  • by honkycat ( 249849 ) on Friday July 14, 2006 @01:59PM (#15720425) Homepage Journal
    According to the article (and common sense, because Metcalfe is not a short-sighted fool), Metcalfe acknowledges that his original reason for stating his "law" was simply to illustrate that even though small networks might not be interesting, once a certain size was reached, they would become compelling. For this, the distinction between n^2 and n.log(n) is pretty irrelevant -- the significant feature is that both are superlinear (as the article notes). Metcalfe was absolutely correct.

    This is not to say he was unique in recognizing this, or that it'd be surprising for someone invested in selling networks to claim they'll become important. The point is he was not attempting to carefully quantify the scaling effects of networking. Rather, he had an instinct that said networks will be big when they get big. The quickest back-of-the-envelope estimate of the scaling law says n.(n-1)~n^2, so he used that for his talk.

    When networks started to catch on, someone (the name is in the article but I'm too lazy to go back and look it up) grabbed ahold of this tidbit and named it Metcalfe's Law. Doing anything quantitative with this is ridiculous. It's obvious to everyone involved, Metcalfe included, that his "law" was just the simplest superlinear curve, not some carefully constructed value function. Even the new estimate -- n.log(n) -- is on pretty crude footing. I'm sure you can find a good analysis that gives this result, but there is so much ambiguity in what the value function should actually measure that it's hard to know you're doing the right thing.

    Basically, Metcalfe was right. Networks grow in value faster than they grow in node size. His "Law" may be wrong, but it was just a heuristic to begin with. Anyone basing a business model on the details of that law deserved to have their bubble burst.
    • Actually, the precise formula is


  • by E++99 ( 880734 )
    (Aside from the fact that 2^n and n^2 are both absurd in any kind of network I can think of), n log(n) has the advantage over all the other models mentioned in that it correctly gives a zero value for a network of one, which is not a network at all and obviously adds no value. Or if you want the combined value of the network and the networked, maybe it would be n + n log(n).
    • I think you are reading too much into the accuracy of the formula. It is only trying to show a general "shape" for the value, not an exact number. The value for n=1 is really irrelevant. If it were, we would simply add a constant to make it work. (n-1)^2 still gives us the same shape and the zero value.
      • True, but generally I think that simple formulas that reflect laws of nature (or economics or whatever) tend to still make sense when you plug in a one or a zero, if they are valid to begin with.
        • Not if we're talking about asymptotic limits, as we are here. We're looking at the large n limit, when n is far from 0. Constant factors, scaling, etc, are thrown out. That's why we call Metcalfe's Law n^2 instead of n*(n-1)/2 (which, incidentally, also gives 0 at n=1, but that part is ignored).
    • Why is n^2 such an unbelievable function? Add n users to a group. How many relationships do you have? n(n-1)/2. As n becomes large, this reduces to n^2/2...
      • Why is n^2 such an unbelievable function? Add n users to a group. How many relationships do you have? n(n-1)/2. As n becomes large, this reduces to n^2/2...

        Maybe because as n becomes large, a smaller percentage of those n(n-1)/2 relationships are used in any real-life network. If you were going to start with n^2 there would have to be some inverse component added to the function to account for that.

        With n^2, the nth user adds 2n-1 value. That would mean that the next Internet user (the approx 1e9th)

  • Value? (Score:3, Funny)

    by RinzeWind ( 413873 ) <chema@rinzew[ ].org ['ind' in gap]> on Friday July 14, 2006 @02:01PM (#15720449) Homepage
    So... what are the units of the result? Dollars? Web 2.0 beta credits?
  • It is possible to refute this hypothesis, given that "value" is so vaguely defined? Stated as the original "law" states it, I'm not sure that value is something you can state in dollars and cents. If it isn't, I'm not sure what it means.

    Different aspects of the network which have "value" scale different ways.

    For example while the Internet has probably grown considerably since 1999, I don't really use more web sites regularly, or buy from more vendors than I used to. Yes there is Froogle and so on, but I'd
    • This really depends on the network, and what you are trying to accomplish with it.

      Let's consider two major online sales venues (The models will be idealized a bit, but...)
      Amazon: Adding customers to Amazon does not really increase the value much to each customer, maybe Amazon is capable of getting better rates of bulk purchases from publishers, and is financially able to stock more books, but the growth of the value of Amazon is fairly linear with respect to number of users.
      eBay: Adding users to eBay h
  • The article is a moderately interesting and loooooong account of how to tweak a growth law based on observations that the previous law does not explain history well enough. For those among us who have fortunes depending on this (let's see a raise of hands, please), it may be very important. But for the rest of us it really doesn't matter much. My concern is that /. may take Nlog(N) seconds longer to load (where N=number of posts), not whether CmdrTaco will make a bazillion instead of a gazillion next yea
  • ...which everyone seems to be forgetting, was for Metcalfe to raise the value of 3COM stock. If networks weren't seen as "valuable," then people wouldn't buy networking equipment...
  • never even realizes that the boat existed.

    That is: the authors' analysis is fundamentally flawed in a couple of different respects.

    (1) They don't even attempt to establish an actual metric for the value of a network. Without that, any counterarguments to any previous assertions regarding network value that one might make are basically so much handwaving. (One can of course make the same objection to Metcalfe's Law, but saying "My hand-wavy claim is better than yours!" isn't much of an argument.)

    On a
    • Either you missed the boat or you didn't read the article.

      (1) They don't even attempt to establish an actual metric for the value of a network. Without that, any counterarguments to any previous assertions regarding network value that one might make are basically so much handwaving.

      Both models discussed are GROWTH models. You do not need a metric to model growth. If a child increases his height by 10% over the course of a year, he increases his height 10% in feet, meters, parsecs, lightyears, cubits. In

      • I read the article. I stand by my positions.


        (1) You do indeed need a metric to model growth, because unless you can agree on what you're measuring, you don't know how to tell whether (or how much) it's growing. That is: we know how fast the _network_ is growing; the question is how fast the _value_ is growing, and that's not meaningful unless you define "value". Your objection is not on point because you appear to implicitly assume that it's obvious what's being measured (which is true of a
  • this applies to communities as well as networks (a community being a kind of network).

    For example, does the community that forms around an open source project have n log(n) value where n is the number of members in that community?

  • Here's what Metcalfe had to say about this same paper from these same academics when they first circulated it in '05: []

  • It kind of annoys me: that author throws together some "rules of thumb" sentences and "it seems plausible to assume", and thinks it makes something worthwhile to publish? I don't know if Metcalfs Law is correct or wrong, and I don't care as long as my internet bandwidth doubles every year, but I am a little bit shocked by this display of extremely low standards. Perhaps the author wanted to prove his own point, though: clearly not all additions to the internet are very valuable...
  • I got a chance to review this article before publication and in my commentary on the draft version [] I point out not only that Metcalfe's law is wrong, but that often any positive law is wrong, because in many cases, particularly mailing lists, the value of the network eventually starts dropping as the size increases, due to noise and excess signal.

    That's why may people prefer smaller mailing lists to larger communities, and in fact some topics simply can't be handled properly in large groups, even with moder
  • At least you should differentiate between personal communication networks and information networks. If I want to talk to David D. Johnson on IM, the metrics are simple - either he's on the network or he isn't. On the other hand, if I wanted sports results it wouldn't matter if 90% of all news sites got knocked off the Internet - the remainding 10% would do. Networks are also self-specializing, you get on a network because your friends/coworkers/contacts are there. You have also issues like multi-networking
  • You can connect each node to one less than the total, but don't double count your paths:

    The number of lines to connect 3 dots = 3*(3-1)/2 = 3
    The number of lines to connect 4 dots = 4*(4-1)/2 = 6
    The number of lines to connect 5 dots = 5*(5-1)/2 = 10
    The number of lines to connect n dots = n*(n-1)/2 = Hey, let's just use an approximation this time.

  • I sort of see the key insight of Briscoe, Odlyzko, and Tilly that, if you are going to pull a function out of your ass, it makes more sense if the differential of the function flattens out rather than slopes linearly upwards forever, because there is ultimately a decreasing value of each connection as the number of connections increases.

    So they were correct to pull a log function of of their ass, but they could have just as easily pulled out n*ln(n) or some other base. They made no attempt to "calibrate" th
  • I wish I could remember the name so as to give proper credit, but someone pointed out that Metcalfe may have severely understated matters.

    The option value of a network depends on how many groups can form using it. Every time a thousand specialized message boards like "people who audit for security in CUPS on Solaris" form, the network becomes more valuable. The number of possible groups is easy to calculate. A group can be represented as a bitmap with as many bits as there are endpoints, each bit representi
  • Back in my day, it was really hard to come up with a "Law." You had "hypotheses" (otherwise known as wild-ass guesses), "theories" ("well we tested it and it *looks* right") and after those theories have been tested over and over again without failing ONCE... you called it a "Law." It is a testament of today's liberalization that laws are passed without any supporting evidence for the hypothesis. This apparently now applies to science as well as politics.

Never buy from a rich salesman. -- Goldenstern