## Beckstrom’s Law: FAIL

### 2009/04/08

I stumbled upon Beckstrom’s Law due to a message at SOCNET. As the paper states in its very first sentence, Beckstrom’s Law tries to answer the question «What is the value of a network?». The claim is that it does a better job at that, than Metcalfe’s Law and Reed’s Law. The paper begins with a really nice idea:

Beckstrom’s Law solves the valuation problem by looking at how valuable the network is to each user.

Beckstrom uses the transactions that a user performs when using the network to valuate it and reaches to a formula that reads «The net present value (V) of any network (j) to any individual (i) is equal to the sum of the net present value of the benefit of all transactions less the net present value of the costs of all transactions on the network over any given period of time (t)»:

$V_{i,j} = \sum_{k=1}^n \frac{B_{i,k}}{(1+r_k)^{t_k}} - \sum_{l=1}^n \frac{C_{i,l}}{(1+r_l)^{t_l}}$

Note that in the paper the first expression contains a minor typo since $r_k$ is simply referenced as $r$.

He then proceeds and defines a simplified version:

$V_{i,j} = \sum B_{i,k} - \sum C_{i,l}$

and declares the value of the entire network as the sum of the network values as seen by each individual user.

For the above expressions we read on this slashdot comment:

There are indices simply missing. The letter l (ell) is clearly not a good index. He uses n for number of transactions, users and networks. He even uses n for networks and users in the same formula, which must mean that number of users and networks are identical. In the summation of the users he leaves the denominators simply away.

And I want to add a question: Since every transaction that a user performs comes with a benefit (B) and a cost (C) why not define the (user) network value as:

$V_{i,j} = \sum_{k=1}^n (B_{i,k} - C_{i,k})$

where k represents the user’s transactions on the network?

Before proceeding to the second part of the paper, let us see what Bob Metcalfe himself wrote about his law at a guest blog post over at VCMike in 2006:

While they’re at it, my law’s critics should look at whether the value of a network actually starts going down after some size. Who hasn’t received way too much email or way too many hits from a Google search? There may be diseconomies of network scale that eventually drive values down with increasing size. So, if V=A*N^2, it could be that A (for “affinity,” value per connection) is also a function of N and heads down after some network size, overwhelming N^2. Somebody should look at that and take another crack at my poor old law.

And again, as we can see from this slashdot comment, Beckstrom in fact restated Metcalfe’s Law, only in an unusable way.

When using Metcalfe’s Law (and especially the $n^2$ expression) to evaluate a network you do not get a result in dollars. What you get is a number that you can use to compare networks. That way it is easily explained why your home network is of smaller value than that of your laboratory and why their value increases dramatically when they connect to the Internet while on the other hand the Internet couldn’t care less.

When you try to use Beckstrom’s Law to reach to a certain result you have to either use trivial transactions where you can calculate the benefits and costs, or make assumptions for non-trivial cases. In that case, as Metcalfe writes, I prefer to stick with $n^2$.

Beckstrom then proceeds to offer an extention of his formula to include security investments: «The net benefit value of a network is equal to the summation of all transaction benefits, less all transaction costs, less security costs, and less security related losses to a user»:

$V_{i,j} = B_{i,k} - C_{i,l} - SI_{i,o} - L_{i,p}$

He then states that a goal should be to minimize $SI_{i,o} + L_{i,p}$ and writes:

This leads to an important insight. One dollar of security investments is only a benefit when it reduces expected losses by more than a dollar.

Please excuse me, but isn’t this is the very definition of investment anyway? He then continues by rediscovering the Paretto principle as applied to security investments, namely that 80% percent of the problems can be dealt with fairly easily, while dealing with the rest 20% becomes increasingly expensive with every step. Please point me to at least one system administrator or security professional that is unaware of this (admittedly empirical) fact, regardless of whether they know of Paretto or not.

While summarizing, Beckstrom argues that his law answers the network value question. This is not true. Beckstrom’s Law introduces the really nice concept that the same network has different value for different users. This fact is established by bringing the transactions that the users perform into the picture. However, as the last statement says «how can we best value the benefit of transactions?».

### 6 Responses to “Beckstrom’s Law: FAIL”

1. thanos Says:

So basically the dude has discovered that Value = Profit – Loss, and then he tries to minimize Loss for a given Profit, so as to maximize Value.

Good thing nobody else in 10,000+ years of human history never thought of that!

And then of course there is the problem of discovering Profit & Loss for individual cases which is impossible to do in a systematic & generic way so as you say this «law» is worthless.

2. Dear Yiorgos,

I enjoyed your piece and wanted to respond to several points.

1) The SlashDot criticism re. the overuse of the «n» notation is valid and I will correct the model appropriately.
2) Your attempt to allign the Benefit B and Cost C on each transaction does not work. There are some cost transactions (paying cash for monthly internet access), the yield thousands of benefit transactions (page downloads). Thus the benefit and cost transactions are not paired.
3) the model is in no way based on Metcalfe’s Law which is a nice concept but which has never been used to value any network
4) the notion of using transaction accounting for network valuation is what is new.
5) Yes, the model can be used to value any network, one users at a time, or statistically, by sampling some set of users and scaling up. Yes it is surprisingly simple. What amazes me is that the solution is so simple and yet to my knowledge was not proposed or published.
6) if anyone has or proposes a better model that has been used to value a network, please send me the link and the data for the network that it actually was used to value.

Thanks again for your thoughtful piece that pushes all of our thinking forward.

Rod

Hello Rod and thanks for the comments. Here is my commentary on the issues that you focus:

– I think my attempt of pairing the benefits and costs for each transaction stands. If I understand it correctly R.H. Coase was awarded the Nobel prize for bringing this into economics: Every transaction has a cost. Since every transaction has a benefit (which can even be zero) the pairing stands.

What I think you are trying to model by not pairing them is the fact that for certain transactions to be even considered, there exist transactions that have to be executed first. To use your example, if you do not have internet access, you cannot download. But there is nothing that guarantees that I will do anything internet related just because I am paying for access. It may even be a backup connection that is never used, or I am abroad and I am not using it for the given period.

The very fact that I have a connection does not mean that I will use it, although I agree that it is what most people will do.

– I never meant to say that your model is based on Metcalfe’s Law. However if you do the math you end up with the generalized version of it. And by the way, in the dot boom era people were incorrectly applying Metcalfe’s Law to explain network value. Metcalfe himself points this out.

– I do not think that your model is simple. If it is as simple as you say, then in the revised version of the paper please provide an example of a real network with thousands of users, and how calculating its value was possible. I really want to see how to apply your model in the network that I manage, without oversimplifying on the transaction costs for any given user.

– I understand that you are trying to answer a very important managerial question: How much does the network cost? But this is a question that cannot be answered meaningfully unless implied questions that lead to it are also asked: Why are you asking this question? Do you want to shut it down? Do you want to buy it? Do you want to build a second network from scratch? And more importantly: Why do you need a monetized metric that changes by the day instead of an inalterable value?

If I were asked this question by my manager, I would reply back: «I will give you a value only if you tell me what you are going to use it for».

4. Thanks for your thoughts on this.

1) Your last sentence directly above is the proves the model «If I were asked by my manager…. “I will give you a value only if you tell me what you use it for»

In other words, network value is driven by transactions, in this instance, those transactions valued by your manager. It is not driven by the number of other parties that happen to be on the network, most of whom are irrelevant to your manager as he would never do transactions with them.

2) Coase’s Thereom generally refers to «transactions costs» such as the frictional cost of executing transactions. Here is the Wikipedia entry: http://en.wikipedia.org/wiki/Coase_Theorem I have not found reference to his pairing of benefits and costs you refer to it, but trust it exists. Do you have a link?

3) Watch my Black Hat presentation for demonstration of some examples of valuing networks both empirically as well as conceptually with the model.

Good luck.

Rod

Rod, again thank you for your reply. Here is some more commentary:

1) Agreed. Please put what you wrote above in the paper.

2) Coase was the one who introduced the idea that every transaction has a price. Therefore, every transaction has a benefit and a cost. But forget Coase. Just ask any DBA around you and they will tell you the same thing: Every transaction has a price.

But lets forget the above paragraph. You say that «There are some cost transactions (paying cash for monthly internet access), the yield thousands of benefit transactions (page downloads).» I say that the pairing works! It is as simple as putting a cost that equals to zero for such transactions. Whenever a partner in the pair is missing, just put zero. That will simplify the notation a lot:

Every transaction has a benefit (even when it is zero) and every transaction has a cost (again even when it is zero).

3) Looking forward to allocate time to watch it!

6. Fred Krueger Says:

Intriguing discussion.