October 17, 2005

Bubble 2.0 – Reed’s Law is Wrong (I think)

Dr. David P. Reed is a very smart and thoughtful person.  Reed’s Law (see wikipedia and Reed’s paper on Group Forming Networks) is often cited as a principal source of value for Web 2.0 companies and their successful Web 1.0 predecessors.  Reed is almost certainly right that networks increase in value faster than would be explained by Metcalfe’s Law alone and that group forming ability is the source of this extra value.  But I think Reed’s Law is wrong (note: any objective observer would say that there is a better chance that I am the one that is wrong).

BR (Before Reed) many of us said that the value of a network was calculated by the equation: V= a*n+b*n^2 where V is the total value of the network, n is the number of endpoints (users) attached to the network, and a and b are arbitrary constants with a>>b (which just means that a is a lot bigger than b).

In English, the “a term” says that a network has value directly proportional to the number of users – pretty easy to understand in terms of a content delivery network like one-way cable, for example: each subscriber is worth so much.

The “b term”, the Metcalfe Value, says that a two way network in which each endpoint can connect with each other endpoint has added value in proportion to the number of possible connections between pairs of possible endpoints: n*(n-1)/2 to be precise.  When there are just a few users, the b term value is very small.  Imagine a closed phone network with only a few users: it has little value to you because it is unlikely that you will be able to use it to reach anyone you are interested in talking to.  Now take Skype with millions on millions of users: whether eBay overpaid or not, huge value has been created by assembling this large number of endpoints.  Just what we’d expect from Metcalfe’s law.

David Reed says that a network also has value in the number of groups that can be formed on it.  The Internet itself is a prime example of a GFN (Group Forming Network).  Businesses like eBay, Google, and Skype are groups formed as subsets of Internet endpoints and each of these groups has value to its members and adds to the total value of the Internet.  Even more interesting, some of these large groups are GFNs themselves.  For example, eBay can be configured into special purpose auctions for subsets of users.  Applications are being deployed by third-parties for groups of Skype users.

So Reed’s law adds a third term to the equation above: V = a*n+b*n^2+c*2^n. The “c term” is the potential for forming groups and a>>b>>c.  In other words, the “c term”, let’s call it the Reed value, starts out very small; but, because the value of this term doubles with every new endpoint added to the network, it eventually becomes huge and dwarfs the other terms.

Reed derives this third term by calculating the number of “nontrivial” subsets that can be created from n members: 2^n-n-1.  Of course, if you remember the course on combinatorials you took in college (or high school if you were really smart), you already knew this.

In making the Excel spreadsheet to create the chart above to illustrate Reed’s law, I was doing a gedanken experiment without, at first, realizing it.  c here is set to a very tiny number – ten to the -200 power (one over 10 with 199 zeros after it).  Nevertheless, instead of being a nice curve, the Reed value comes rocketing up vertically even in a very small network.  Even smaller values for c just postpone the problem: once it appears the Reed value is virtually horizontal vertical.

“Duh,” I said to myself after a while.  No matter what the value of c, the Reed value is going to double for every endpoint added to the network.  In real life, this means that the total value of a large network like the Internet or eBay doubles every time a new person joins!  Frankly, I don’t think so (but I don’t know how to prove this isn’t true).

Reed is right that GFNs do gain value even faster than Metcalfe’s law would predict but his law overstates this rate of gain.  Certainly he correctly calculates the number of subgroups that can be formed and it IS true that, each time you add a new member,  the theoretical number of subgroups doubles.

So the Group Forming Value must NOT be proportional to the number of theoretical subgroups.  The likely reason is that all subgroups do not have the same value nor do they have an average value proportional to the total number of subgroups.  For example, groups with only a small number of participants have no significant Reed or Metcalfe value, yet small groups are a huge part of the total number of theoretical groups.  If two groups have nearly 100% overlap, it may be that they don’t each add as much unique value as if they had almost no overlap.

The Group Forming Value may be proportional to the number of groups above a certain size and with a certain degree of nonoverlap.  It could well be an even more complex term.  This is well beyond my math but that doesn’t stop me from guessing.  I postulate the Group Forming Value is approximated by (1+c)^n instead of Reed’s c^n where c<<1.  I’d certainly welcome more mathematically qualified comments on this.

Fred Wilson blogged a much easier to understand explanation of Reed’s Law here.